Representations of Algebras: 224

representations of algebras

PURE

AND APPLIED

MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, NewJersey

,

EDITORIAL M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Zuhair Nashed University of Delaware Newark, Delaware

BOARD Anil Nerode Cornell University DonaM Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitiit Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES 1N PURE AND APPLIED

MATHEMATICS

1. N. Jacobsen, ExceptionalLie Algebras 2. L.-/~. LindahlandF. Poulsen,Thin Setsin Harmonic Analysis 3. I. Satake,ClassificationTheoryof Semi-Simple AlgebraicGroups 4. F. Hit-zebruch et al., DifferentiableManifolds andQuadratic Forms 5. I. Chavel,Riemannian SymmetricSpacesof RankOne 6. R. B. Burckel,Characterization of C(X)Among Its Subalgebras 7. B. R. McDonald et aL, RingTheory 8. Y.-T. Siu, Techniques of Extension on Analytic Objects 9. S. R. Caradus et aL, CalkinAlgebrasandAlgebrasof Operatorson Banach Spaces 10. E. O. Roxineta/., Differential Games andControlTheory 11. M. OtzechandC. Small, TheBrauerGroupof Commutative Rings 12. S. Thornier,Topology andIts Applications 13. J. M. LopezandK. A. Ross,SidonSets 14. W.W. ComfortandS. Negrepontis,ContinuousPseudometdcs 15. K. McKennon andJ. M. Robertson,Locally Convex Spaces 16. M. CarmeliandS. Malin, Representations of the RotationandLorentzGroups 17. G. B. Seligman,RationalMethods in Lie Algebras 18. D. G. deFigueiredo,FunctionalAnalysis 19. L. Cesadetal.,NonlinearFunctionalAnalysis andDifferential Equations 20. J.J. SchlJffer, Geometry of Spheres in Normed Spaces 21. K. YanoandM. Ken,Anti-lnvadantSubmanifolds 22. W.V. Vasconcelos,TheRingsof Dimension Two 23. R. E. Chandler,HausdorffCompactifications 24. S. P. FranklinandB. V. S. Thomas, Topology 25. S. K. Jain, RingTheory 26. B. R. McDonald andR. A. Mords,RingTheoryII 27. R. B. MuraandA.Rhemtulla,OrderableGroups 28. J. R. Graef,Stability of Dynamical Systems 29. H.-C. Wang,Homogeneous BranchAlgebras 30. E. O. Roxinet al., Differential Games andControlTheoryII 31. R. D. Porter,Introductionto FibreBundles 32. M. Altman,ContractorsandContractorDirectionsTheoryandApplications 33. J. S. Golan,Decomposition andDimension in ModuleCategories 34. G. Fairweather,Finite Element GalerkinMethods for Differential Equations 35. J. D. Sally, Numbers of Generators of Idealsin LocalRings 36. S. S. Miller, Complex Analysis 37. R. Gordon,Representation Theoryof Algebras 38. M. GoreandF. D. Grosshans, Semisimple Lie Algebras 39. A. L Arrudaet al., Mathematical Logic 40. F. VanOystaeyen, Ring Theory 41. F. VanOystaeyen andA. Verschoren, ReflectorsandLocalization 42. M. Satyanarayana, Positively OrderedSemigroups 43. D. L Russell, Mathematics of Finite-Dimensional ControlSystems 44. P.oT.Liu andE. Roxin,Differential Games andControlTheoryIII 45. A. GeramitaandJ. Seberry,OrthogonalDesigns 46. J. Cigler, V. Losert, andP. Michor, BanachModulesandFunctorson Categoriesof Banach Spaces 47. P.-T. Liu andJ. G. Sutinen,ControlTheoryin Mathematical Economics 48. C. Bymes, Partial Differential Equations andGeometry 49. G. Klambauer, Problems andPropositionsIn Analysis 50. J. Knopfmacher, Analytic Arithmeticof AlgebraicFunctionFields 51. F. VanOystaeyen, Ring Theory 52. B. Kadem,Binary TimeSedes andR. A. Artino, HypoellipticBoundary-Value Problems 53. J. Barros-Neto 54. R. L. Stemberg et aL, Nonlinear’PartialDifferential Equations in Engineering andAppliedScience RingTheoryarid AlgebraIII 55. B. R. McDonald, Overa Noncommutative Ring 56. J. S. Golan,Structure Sheaves 57. T. V. Narayana et aL, Combinatortcs, Representation TheoryandStatistical Methods in Groups andDifferential Equations in Biology 58. T.A. Burton,Modeling 59. K. H. KimandF. W.Roush,Introduction to Mathematical Consensus Theory

60. J. Banasand K. Goebel,Measures of Noncompactness in BanachSpaces 61. O.A.Nielson,Direct Integral Theory 62. J. E. Smithet al., Ordered Groups 63. J. Cmnin,Mathematics of Cell Electrophysiology 64. J. W. Brewer,PowerSeries OverCommutative Rings 65. P. K. Kamthanand M. Gupta, Sequence Spacesand Sedes 66. T. G. McLaughlin, Regressive Setsandthe Theoryof Isols 67. T. L. Herdman et aL, Integral andFunctionalDifferential Equations 68. R. Draper,Commutative Algebra 69. W.G. McKay andJ. Patera, Tablesof Dimensions,Indices, andBranchingRulesfor Representationsof SimpleLie Algebras 70. R. L. Devaney andZ. H. Nitecki, Classical Mechanics andDynamical Systems 71. J. VanGeel, PlacesandValuationsin Noncommutative Ring Theory 72. C. Faith, Injective Modules andInjective QuotientRings Programming with DataPerturbationsI 73. A. Fiacco,Mathematical 74. P. Schultzet aL, AlgebraicStructuresandApplications andPreradicals 75. L Bicanet al., Rings,Modules, 76. D. C. KayandM. Breen,ConvexityandRelatedCombinatorialGeometry 77. P. FletcherandW.F. Lindgren,Quasi-Uniform Spaces 78. C.-C. Yang,FactodzationTheoryof Meromorphic Functions 79. O. Taussky,TernaryQuadraticFormsand Norms 80. S. P. SinghandJ. H. Burry,NonlinearAnalysisandApplications 81. K. B. Hannsgen et aL, VolterraandFunctionalDifferential Equations 82. N. L. Johnson et aL, Finite Geometries 83. G. L Zapata,FunctionalAnalysis,Holomorphy, andApproximation Theory 84. S. GrecoandG. Valla, Commutative Algebra 85. A. V. Fiacco,Mathematical Programming with DataPerturbationsII et al., Optimization 86. J.-B. Hidart-Urruty andM. A. Picardello, Harmonic Analysison FreeGroups 87. A. Figa Talamanca 88. M.Hatada,FactorCategories with Applicationsto Direct Decomposition of Modules andComplex StdctConvexity 89. V. L Istrz~tescu,Strict Convexity 90. V. Lakshmikantham, Trendsin TheoryandPracticeof NonlinearDifferential Equations 91. H. L. Manocha andJ. B. Srivastava,AlgebraandIts Applications 92. D. V. Chudnovsky and G. V. Chudnovsky,Classical and QuantumModelsand Arithmetic Problems 93. J. W.Longley,Least Squares Computations UsingOrthogonalizationMethods 94. L. P. de Alcantara,Mathematical Logic andFormalSystems 95. C. E. Aull, Ringsof Continuous Functions andProbability 96. R. Chuaqui,Analysis,Geometry, 97. L. FuchsandL. Salce, ModulesOverValuationDomains 98. P. FischerandW.R. Smith, Chaos,Fractals, andDynamics AlgebraicStructures 99. W.B. PoweflandC. Tsinakis, Ordered 100. G. M. RassiasandT. M. Rassias,Differential Geometry,Calculusof Variations, andTheir Applications 101. R.-E. Hoffmann andK. H. Hofmann, Continuous Lattices andTheir Applications 102. J. H. Ughtboume III andS. M. RankinIII, PhysicalMathematics andNonlinearPartial Differential Equations 103. C. A. BakerandL. M.Batten,Finite Geometrics 104. J. W.BreweretaL, Linear SystemsOverCommutative Rings 105. C. McCroryandT. Shifdn, Geometry andTopology Logic andTheoretical Computer Science 106. D. W.Kuekeet aL, Mathematical 107. B.-L. Lin andS. Simons,NonlinearandConvex Analysis 108. S.J. Lee, OperatorMethods for OptimalControlProblems 109. V. Lakshmikantham, NonlinearAnalysisandApplications 110. S. F. McCormick, MultigddMethods 111. M. C. Tangora,Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, SearchTheory and R. D. Jenks, ComputerAIgebra 113. D. V. Chudnovsky 114. M. C. Tangora,Computers.inGe0meby and Topology andIntegral Equations 115. P. Nelsonet al., TransportTheory,InvadantImbedding, 116. P. Cldment et aL, Semlgroup TheoryandApplications 117. J. Vinuesa,Orthogonal Polynomials andTheir Applications 118. C. M.Dafermos et aL, DifferentialEquations 119. E. O. Roxin, Modem OptimalControl 120. J. C. Dlaz, Mathematics for LargeScaleComputing

121. P. S. Milojevi~NonlinearFunctionalAnalysis 122. C. Sadosky, AnalysisandPartial Differential Equations 123. R. M. Shortt, GeneralTopology andApplications 124. R. Wong,AsymptoticandComputational Analysis 125. D. V. Chudnovsky andR. D. Jenks, Computers in Mathematics 126. W.D. Wallis et aL, Combinatorial DesignsandApplications 127.S. Elaydi,Differential Equations 128. G. ChenetaL, Distributed Parameter ControlSystems 129.W.N Evefitt, Inequalities 130. H. G. KaperandM. Garbey,AsymptoticAnalysisandthe NumericalSolution of Partia~Differential Equations 131. O. AdnoetaL, Mathematical PopulationDynamics 132. S. Coen,Geometn] andComplex Variables 133.J.A. Goldsteinet aL, Differential Equations with Applications in Biology,Physics,andEngineering 134. S.J. Andima et aL, GeneralTopologyandApplications 135. P Cldment et al., Semigroup TheoryandEvolutionEquations 136. K. Jarosz, FunctionSpaces 137.J. M.Bayod et aL, p-adic FunctionalAnalysis 138. G.A. Anastassiou,Approximation Theory 139. R. S. Rees,Graphs,Matrices, andDesigns 140. G. Abrams et aL, Methods in ModuleTheory 141. G. L. MullenandP. J.-S. Shiue,Finite Fields, CodingTheory,andAdvances in Communications and Computing 142. M. C. Joshi andA. V. Balakrishnan, Mathematical Theoryof Control 143. G. Komatsu and Y. Sakane,Complex Geometry 144. I.J. Bakelman, Geometric AnalysisandNonlinearPartial Differential Equations 145. T. Mabuchi andS. Mukai,Einstein MetdcsandYang-MillsConnections 146. L. FuchsandR. Gt~bel,AbelianGroups 147. A. D. Po/lington andW.Moran,Number Theorywith an Emphasis on the MarkoffSpectrum 148. G. Doteet al., Differential Equations in Banach Spaces 149. T. West, Continuum Theoryand DynamicalSystems 150.K. D. Bierstedtet al., Functional Analysis 151.K. G. Fischeret al., Computational Algebra 152.K. D. E/worthyet al., Differential Equations,Dynamical Systems, andControlScience 153. P.-J. Cahen,et al., Commutative RingTheory 154. S. C. CooperandW.J. Thmn,ContinuedFractionsandOrthogonalFunctions 155. P. Clement andG. Lumer,EvolutionEquations,Control Theory,andBiomathematics 156. M. GyflenbergandL. Persson,Analysis,Algebra,andComputers in Mathematical Research 157.IN’. O. Brayetal.,FourierAnalysis 158. J. BergenandS. Montgomery, Advances in HopfAlgebras 159. A. R. Magid,Rings, Extensions,andCohomology 160.N. H. Pavel,OptimalControtof Differential Equations t61. M. Ikawa,SpectralandScatteringTheory 162. X. L/u andD. Siegel, Comparison Methods andStability Theory 163.J.-P. Zoldsio, Boundary ControlandVariation 164. M.K’fl2ek et al., Finite Element Methods 165.G. DaPratoandL. Tubaro,Controlof Partial Differential Equations 166.E. Ba//ico, ProjectiveGeometry with Applications 167. M. Costabe/etal., Boundary ValueProblems andIntegral Equationsin Nonsmooth Domains 168. G. Ferreyra,G. R. Goldstein,andF. Neubrander, EvolutionEquations 169. S. Hugge~t, TwistorTheory 170. H. Cooket al,, Continua 171. D. F. Anderson andD. E. Dobbs,Zero-Dimensional Commutative Rings 172. K. Jarosz,FunctionSpaces 173. V. Anconaet aL, Complex Analysis and Geometry 174.E. Casas,Controlof Partial Differential Equations andApplications 175.N. Kaltonet al., InteractionBetween FunctionalAnalysis,Harmonic Analysis,andProbability 176.Z. Deng et al., Differential Equations andControlTheory 177.P. Marcelliniet al. Partial DifferentialEquations andApplications 178. A. Kartsatos,TheoryandApplicationsof NonlinearOperatorsof AccretiveandMonotone Type 179. M. Maruyama, Moduliof VectorBundles 180.A. Ursini andP. AglianO,LogicandAlgebra 181.X. H. Caoet al., Rings,Groups,andAlgebras 182. D. Arnold and R. M. Rangaswamy, Abelian GroupsandModules 183.S. R. Chakrevarthy andA. S. Alfa, Matrix-AnalyticMethods in StochasticModels

184. J. E. Andersen et al., Geometry andPhysics 185. P.-J. Cahen et al., Commutative Ring Theory 186.J.A. Goldsteinet al., StochasticProcesses andFunctionalAnalysis 187. A. Sorbi, Complexity,Logic, andRecursion Theory 188. G. DaPrato andJ.-P. Zol(~sio, Partial Differential EquationMethods in Control andShape Analysis 189. D. D. Anderson, Factodzation in Integral Domains 190. N. L. Johnson, MostlyFinite Geometries 191. D. Hintonand P. W.Schaefer,SpectralTheoryandComputational Methods of Sturm-Liouville Problems 192. W.H.SchikhofetaL, p-adic FunctionalAnalysis 193. S. Sert5z, AlgebraicGeometry 194. G. CadstiandE. Mitidied, ReactionDiffusionSystems 195. A. V. Fiacco, Mathematical Programming with DataPerturbations 196. M. A?l~eket aL, Finite ElementMethods:Superconvergence, Post-Processing,andA Postedod Estimates 197. S. Caenepeel andA. Verschoren,Rings, HopfAlgebras,andBrauerGroups 198. V. Drenskyet aL, Methods in Ring Theory 199. W.B. JonesandA.SdRanga,OrthogonalFunctions, Moment Theory,andContinuedFractions 200. P. E. Newstead, Algebraic Geometry 201. D. DikranjanandL. Salce, AbelianGroups,Module Theory,andTopology 202. Z. Chenet aL, Advances in Computational Mathematics 203. X. CaicedoandC. H. Montenegro, Models,Algebras,andProofs 204. C. Y. YiIdlrtm andS. A. Stepanov, Number TheoryandIts Applications 205. D. E. Dobbset aL, Advances in Commutative Ring Theory 206. F. VanOystaeyen,Commutative AlgebraandAlgebraic Geometry 207.J. Kakolet al., p-adicFunctional Analysis 208. M. Boulagouaz andJ.-P. 77gnol,AlgebraandNumber Theory 209. S. Caenepeel andF. VanOystaeyen,HopfAlgebrasand Quantum Groups 210. F. VanOystaeyenandM. Saodn,Interactions BetweenRing Theoryand Representations;of Algebras 211. R. Costaet aL, Nonassodative AlgebraandIts Applications 212. T.-X. He,WaveletAnalysisandMultiresolutionMethods 213. H. HudzikandL. Skrzypczak, FunctionSpaces:.TheFifth Conference 214.J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis 215.G. Lumer andL. Weis,EvolutionEquationsandTheir Applicationsin PhysicalandLife Sciences 216. J. CagnoletaL,ShapeOptimizationandOptimalDesign 217. J. Het’zogandG. Restuccia,Geometric andCombinatorial Aspectsof Commutative Algebra 218.G. Chertet al., Controlof NonlinearDistributedParameter Systems 219.F. AfiMehmetietal., Partial Differential Equations onMultistructures 220. D. D. Anderson andL J. Papick,Ideal TheoreticMethods in Commutative Algebra 221.,4. Granjaet al., RingTheoryandAlgebraicGeometry 222.A. K. Katsaras et aL, p-adic FunctionalAnalysis 223. R. Salvi, TheNavier-Stokes Equations 224. F. U. CoelhoandH. A. MerMen, Representations of Algebras 225.S. AizJcoviciandN. H. Pavel,Differential Equations andControlTheory 226. G. Lyubeznik,Local Cohomology andIts Applications Additional Volumesin Preparation

representations of algebras prooeedings of the held in S&o Paulo

edited

oonferenoe

by

Fl~vio Ulhoa H~ctor A. Merklen Universityof S~oPaulo S~oPaulo-SP,Brazil

MARCEL

MARCEL DEKKER,INC. DEKKER

NEW YORK. BASEL

ISBN:0-8247-0733-8 Thisbookis printedonacid-freepaper. Headquarters MarcelDekker,Inc. 270 MadisonAvenue, NewYork, NY10016 tel: 212-696-9000; fax: 212-685-4540 EasternHemisphere Distribution Marcel DekkerAG Hutgasse4, Postfach812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 WorldWide Web http://www.dekker.com Thepublisheroffers discountson this bookwhenorderedin bulk quantities. Formoreinformation, write to SpecialSales/Professional Marketing at the headquarters addressabove. Copyright©2002by MarcelDekker,Inc. All RightsReserved. Neitherthis booknor any paxt maybe reproducedor transmittedin any formor by any means,electronic or mechanical,includingphotocopying,microfilming,and recording,or by any information storageandretrieval system,withoutpermission in writingfromthe publisher. Currentprinting(last digit): 10987654321 PRINTEDIN THE UNITEDSTATESOF AMERICA

Preface

The Conference on Representations of Algebras-Sao Paulo (CRASP)was held at the Instituto de Matem~iticae Estatfstica of the Universidadede Sao Paulo. The Scientific Committeeconsisted of: ¯ Michael Butler (Liverpool) ¯ Jose Antoniode la Pefia (Mexico) ¯ Idun Reiten (Trondheim) Claus Ringel (Bielefeld) ¯ Fl~ivio Ulhoa Coelho(S~o Paulo) while the local Organizing Committeememberswere ¯ ¯ ¯

Fl~ivio Ulhoa Coelho Eduardo N. Marcos Maria Izabel R. Martins H6ctor A. Merklen

Seventy-two researchers from 17 different countries attended this conference. There were 14 invited talks and 32 contributed talks. Manyof the contributions are presented in these proceedings. All papers published here were refereed and consist mainly of original research results. Wewouldlike to express our thanks to all the participants of the conference and the contributors to these proceedings. Also, our sincere thanks go to our colleagues whoserved as referees. Our thanks are also extended to Joelma Martins Gomesand Sueli Aparecida Paschoal Dian, whose secretarial work was essential to the conference and for the publication of this volume. The conference was supported by FAPESP, CNPq, CAPES(through its program PROAP), IME-USP, CCInt-USP, CPq-USP, SBM, and CBA. Withouttheir valuable support, this conference wouldnot have been possible. Fldvio Ulhoa Coelho H~ctor A. MerMen ooo

111

Contents Preface Contributors Invited Talks Participants

V

vii ix xi

On the Existence of Left and Right Almost Split Morphisms Lidia Angeleri-Hiigel Actions of Hopf Algebras on QuantumPolynomials Vyacheslav A. Artamonov

11

Strongly Simply Connected Derived Tubular Algebras Ibrahim Assem

21

H~ and Presentations of Finite DimensionalAlgebras Michael J. Bardzell and EduardoN. Marcos

31

TameTilted Algebras with Almost Regular Connecting Components 39 Grzegorz Bobidski

10.

Reflexive Modules Are Not Closed under Submodules Gabriella D’Este

53

Fibre SumFunctors and the Bimodule Ext Peter Dri~xler

65

Smooth Automorphism Group Schemes Daniel R. Farkas, Christof Geiss, and EduardoN. Marcos

71

A Combinatorial Characterization of Hereditary Categories Containing Simple Objects Dieter Happel and Idun Reiten

91

Symmetric Quasi-Schurian Algebras Octavio Mendoza Herndndez

99

¥

vi

Contents On Lattices at the Ends of ConnectedComponentsof the Auslander-Reiten Quiver Alfredo Jones

117

12.

Factorisations of Morphismsfor Wild Hereditary Algebras Otto Kerner

121

13.

A Note on Concealed-Canonical Artin Algebras Dirk Kussin and Zygmunt Pogorzaly

129

14.

Koszul Algebras and the Gorenstein Condition Roberto Martfnez- Villa

135

15.

SomeRemarksabout the "Double Extension" Algebra of a Finite Poset Teresita Noriega

157

16.

Coil Algebras that Are Derived-Tame Jose Antonio de la Pe~a and Bertha Tom~

17.

One-Point Extensions of Quasitilted Algebras by Modules on Stable Tubes Jose Antonio de la Pe~aand Sonia E. Trepode

177

Combinatorial Partial Tilting Complexesfor the Brauer Star Algebras MarySchaps and Evelyne Zakay-Illouz

187

AlmostSplit Sequencesin Categories of Representations of Quivers II ~ Sverre O. Smal¢

:209

18.

19.

20.

Cotilting Objects and Dualities Robert Wisbauer

21.

Coherent Componentsof Auslander-Reiten Quivers whose DTr-Orbits Are Finite Hailou Yao

22.

Twisted Hopf Algebras Pu Zhang and Li-Bin Li

165

215

235

269

Contributors Lidia Angeleri-Hiigel Universitat

Munchen, Munich, Germany

Vyacheslav A. Artamonov MoscowState University,

Moscow, Russia

Ibrahim AssemUniversit6 de Sherbrooke, Sherbrooke, Quebec, Canada Michael J. Bardzell Salisbury State University, Salisbury, Maryland Grzegorz Bobi~iski Nicholas Copernicus University, Toruri, Poland Gabriella d’Este Universith di Milano, Milan, Italy Peter Dr~ixler Universit~it Bielefeld, Bielefeld, Germany Daniel R. Farkas Virginia Polytechnic Institute and State University, Blacksburg, Virginia Christof Geiss Instituto M6xico D.F., Mexico

de Matem~iticas, UNAM, Ciudad Universitaria,

Dieter Happel Technische Universit~it Chemnitz, Chemnitz, Germany Octavio MendozaHern~ndez Universidad Nacional del Sur, Bahfa Blanca, Argentina Alfredo Jones

Centro de Matemfitica, Montevideo, Uruguay

Otto Kerner

Heinrich-Heine-Universit~it, Diasseldorf, Germany

Dirk Kussin

Universit~it Paderborn, Paderborn, Germany

Li-Bin Li University of Science and Technologyof China, Hefei, P. R. China Eduardo N. Marcos Universidade de S~o Paulo, Sao Paulo, SP, Brazil vii

Contribwlors

Viii

Roberto Martinez-Villa D.F., Mexico

Universidad Nacional Aut6nomade M6xico,, M6xico

Teresita Noriega Universidad de la Habana, Havana, Cuba Jose Antonio de la Pefia UNAM, Ciudad Universitaria,

M6xico D.F., Mexico

ZygmuntPogorzaly ¯ Nicholas Copernicus University, Toruri, Poland Idun Reiten Norwegian University of Science and Technology, Trondheim, Norway MarySchaps Bar-Ilan University, Ramat-Gan, Israel Sverre O. Sinai0 NTNU,Trondheim, Norway Bertha Tom6UNAM,Ciudad Universitaria,

M6xico D.F., Mexico

Sonia E. Trepode Universidad de Mar del Plata, Mar del Plata, Argentina Robert Wisbauer University of Dtisseldorf,

Dtisseldorf,

Germany

Hailou Yao Beijing Polytechnic University, Beijing, P. R. China Evelyne Zakay-lilouz Jordan Valley College, Jordan Valley, Israel Pu Zhang University of Science and Technology of China, Hefei, P. R. China

Invited

Talks

M. BAROT,A characterization

of positive unit forms.

R. BAUTISTA,Representations over rational functions. C. CIBILS, Noncommutativetensor products of sets. Y. DROZD, No~e-omrnutativenodes and their derived categories. C. GEISS, Derived clannish algebras. E. GREEN,Some results

on Hochschild cohomology and related topics.

H. KRAUSE,Morphisms determined by objects and Brown representability. H. LENZING,Two-orbits of tubular type. H. MERKLEN, Standardly stratified M. I. PLATZECK, Module of finite algebras.

algebras and the characteristic projective

module.

dimension over standardly stratified

J. SCHR6ER,Module varieties

over some canonical algebras.

A. SKOWROI~SKI, Selfinjective

algebras of quasitilted

type.

S. SMALO, Almost split sequences in categories of finite tions of quivers.

dimensional representa-

P. ZHANG, x-Hopf algebras, Hall algebras and Lusztig-Green-Ringel classes. ix

Participants

ALVAP~ES,ED$ONRIBEIRO Departamento de Matem~tica-IME, Universidade de S£o Paulo, Rua do MatZo, 1010, CEP 05508-900, S~o Paulo, Brazil. e-mail: [email protected] ANGELERI-HOGEL, LIDIA Matematisches Institut 39, D-80333, Miinchen, Germany. e-mail: [email protected]

der Universit~it

Theresienstr

AQUINO,REGINAMARIADE Departamento de Matem£tica-IME, Universidade de S~o Paulo, Rua do MatZo, 1010, CEP 05508-900, S~o Paulo, Brazil. e-mail: [email protected] ASSEM,IBI%AHIMDepartement de Mathematiques et Informatique, de Sherbrooke, Sherbrooke, Quebec, J1K 2R1, Canada. e-mail: [email protected] BAROT,MICHAEL Instituto de Matematicas, ico, D.F., C. P. 04510, Mexico. e-mail: [email protected]

Universit~

UNAM,Ciudad Universitaria,

BAUTISTA, RAYMUNDO Instituto de Matematicas, taria, Mexico, D.F., C. P. 04510, Mexico. e-mail: [email protected]

Mex-

UNAM,Ciudad Universi-

BEKKERT,VIKTORFaculty of Mechanics and Mathematics Kyiv Taras Shevchenko, University Vladimirskaya Str, 64 252033 Kyiv, Ukraine. e-mail: [email protected] BOBII~SKI, GRZEGORZ Faculty of Mathematics and Informatics, nicus University, Ul. Chopina 12/18, 87-100, Torufi, Poland. e-mail: [email protected] xi

Nicholas Coper-

xii

Participants

BOVDI, VICTORDepartamento de Matem~itica-IME, Universidade Rua do Mat£o, 1010, CEP05508-900, S~o Paulo, Brazil. e-mail: [email protected]

de S~o Paulo,

BRAGA,CLI~ZIO APARECIDO Departamento de Matem£tica-IME, Universidade de S~o Paulo, Rua do Matho, 1010, CEP 05508-900, S~o Paulo, Brazil. e-mail: [email protected] BRENNER, SHEILADepartment of Mathematical Sciences, pool, Liverpool, L69 3BX, UK. e-mail: [email protected] BUAN, ASLAKBAKKEInstitutt Trondheim, Norway. e-mail: [email protected]

for Matematiske

of Liver-

Fag, NTNU, Lade, N-7491,

BUSTOS, CRISTIAN PATRICIO NOVOAUniversidade 227-A Ntlmero 72 apto 1204, setor leste universitario, e-mail: [email protected] BUTLER,MICHAEL C. R. Department of Mathematical Liverpool, Liverpo0]L69 3BX, UK. e-mail: [email protected] CIBILS, CLAUDE Departement de Mathematiques, F-34980, Montpellier, Cedex 5, France. e-mail: [email protected]

University

Cat61ica de Goi~s, Rua 74610-096, Goiania, Brazik

Sciences,

Universit6

University

of

de Montpellier

2,

COELHO,FL/~VIO ULHOADepartamento de Matem£tica-IME, Universidade S~o Paulo, Rua do Matho, 1010, CEP05508-900, S~o Paulo, Brazil. e-mail: [email protected] DMYTRENKO, VASYLDepartment of Mathematical Sciences, ware, 19711 Newark, DE, USA. e-mail: [email protected]

University

DROZD,YURYbept. of Mathematics, Kieve Taras Shevchenko University, mirska 64, 252033 Kiev, Ukraine. e-mail: [email protected]

de

of Dela-

Volodi-

ESCUDER,CECILIA TOSARD~partarnento de Matem~tica-IME, Universidade de S~o Paulo, Rua do Matho, 1010, CEP 05508-900, Sho Paulo, Brazil. e-mail: [email protected] FACCHINI,ALBERTO Dipartimento

di Matematica,

Universit~

di Udine 1-33100

ooo Xlll

Participants e Informatica, Via Delle Scienze, 206, Italy. e-mail: f~¢[email protected] FARKAS,DANIELMath Department, Virginia USA. e-mail: [email protected]

Tech, Blacksburg, VA24061 - 0123,

FERNANDEZ, ELSA A. Universidad de la Patagonia, Puerto Madryn, Chubut, Argentina. e-mail: [email protected]

Auda Roca 1890, (9120)

FERRAZ,RAULDepartamento de Matem£tica-IME, Universidade Rua do MatZo, 1010, CEP 05508-900, S$o Paulo, Brazil. e-mail: [email protected]

de S~o Paulo,

FERREIRA, VITOR DE OLIVEIRA Departamento de Matem~tica-IME, Universidade de S~o Paulo, Rua do MatZo, 1010, CEP05508-900, S£o Paulo, Brazil. e-mail: [email protected] FERRERO,MIGUELInstituto Brazil. e-maih [email protected]

de Matemgtica, UFRGS,Porto Alegre,

GASTAMINZA,SUSANADepartamento de MatemAtica, Del Sur, Av Alem 1253 8000, Bahia Blanca, Argentina.

90420-160,

Universidad

Nacional

GEISS, CHRISTOFInstituto de Matematicas, ico, D.F., C. P. 04510, Mexico. e-mail: [email protected]

UNAM,Ciudad Universitaria,

GREEN,EDWARD L. Dept of Math., Virginia Blacksburg, USA. e-mail: [email protected]

Tech.,

24061-0123,

HOUARI, MOHAMMED EL Departamento de MatemAtica-IME, Universidade S~o Paulo, Rua do Matfio, 1010, CEP05508-900, S~o Paulo, Brazil. e-maih [email protected] HUARD,FRAN(~OISB]shop’s University, e-mail: [email protected]

Lennoxville,

Mex-

de

Quebec, Canada.

IKEMOTO,LUCIADepartamento de MatemAtica-IME, Universidade Rua do Matfio, 1010, CEP 05508-900, S£o Paulo, Brazil. e-maih [email protected]

de S~o Paulo,

xiv JENSEN, BERNTTOREInstitutt Trondheim, Norway. e-mail: [email protected]

Participants for Matematiske

Fag, NTNU, Lade, N-7491,

JONES, ALFREDO Centro de Matem~tica, Facultad de Ciencias, tevideo, Uruguay. e-mail: [email protected] KERNER,OTTOMathematisches Institut, Dusseldorf, Germany. e-mail: [email protected] ~

Iguel 4225, Mon-

Heinrich-Heine-Universit~/t,

KRAUSE,HENNINGDepartment of Mathematics, 33501, Bielefeld, Germany. e-mail: [email protected]

University

D40225,

of Bielefeld,

D=

LANZILOTTA, MARCELOAMt~RICO Departamento de Matem~tica-IME, Universidade de S$o Paulo, Rua do MatZo, 1010, CEP 05508-900, S~o Paulo, Brazil. e-mail: [email protected] LENZING, HELMUT Fachbereich Mathematik 33095, Informatik, Paderborn, Germany. e-mail: [email protected] LOCATELI,ANACLAUDIA Departamento de Materraitica, Espirito-Santo, Vit6ria, Brazil. e-mail: [email protected] LOPES, ANATERESATAVARESAv. Caxingui 000, S~o Paulo, SP, Brazil. e-mail: [email protected]

Universit£t

Universidade Federal do

95, ap 62, Butant$,

CEP 055,79-

MADSEN,DAGInstitutt for Matematiske Fag, NTNU,Lade, N-7491, Trondheim, Norway. e-mail: [email protected] MALICKI,PIOTRFaculty of Mathematics and Informatics, University, Ul. Chopina 12/18, 87-100, Toruri, Poland. e-mail:[email protected]

Nicholas Copernicus

MARCOS, EDUARDODO NASCIMENTODepartamento de Matem~tica-IME, Universidade de S£o Paulo, Rua do Mat£o, 1010, CEP05508-900, S£o Paulo, Bra.zil. e-mail: [email protected] MARTINS,MARIAIZABEL R. Departamento

de Matemdtica-IME,

Universidade

Participants

xv

de S~to Paulo, Rua do Matg~o, 1010, CEP05508-900, S~to Paulo, Brazil. e-mail: [email protected] MENDOZA,OCTAVIOHERN/~NDEZHumboldt 2870, 8000, Bahia Blanca, Buenos Aires, Argentina. e-mail: [email protected]

prov.

MERKLEN,HI~CTOR ALFREDODepartamento de Matem~tica-IME, Universidade de Sgo Paulo, Rua do Matg~o, 1010, CEP05508-900, Sg.o Paulo, Brazil. e-mail: [email protected] MICHELENA,SANDRA Departamento de Matemdtica, Sur, Av Alem 1253 8000, Bahia Blanca, Argentina. e-mail: [email protected]

Universidad

NORIEGA,TERESITAFacultad de Matematica y Computacidn, La Habana, San Lazaro y L La, Habana 4, Cuba. e-mail: [email protected]

Nacional Del

Universidad

de

OLIVEIRA, ALEGRIA GLADYSCHALOMDE Departamento de Matem~iticaIME, Universidade de Sgo Paulo, Rua do Matgo, 1010, CEP 05508-900, Sgo Paulo, Brazil. e-mail: [email protected] PEN~, MARIAIN~S Dpto. de Matem~tica, Fac.Cs.Ex., Mar del Plata, Funes 3350, 7600, Argentina. e-mail: [email protected]

Universidad

PLATZECK,MARIAINES Departamento de Matemg.tica, Del Sur, Av Alem 1253 8000, Bahia Blanca, Argentina. e-mail: [email protected] PRATTI, NILDAISABELDpto. de Matem~tica, Fac.Cs.Ex., de Mar del Plata, Funes 3350, 7600, Argentina. e-mail: [email protected] REDONDO,MARIAJULIA Departamento de Matem~tica, I)el Sur, Av Alem 1253 8000, Bahia Blanca, Argentina. e-mail: [email protected]

Nacional de

Universidad

Nacional

Universidad Nacional

Universidad

Nacional

for Matematiske Fag, NTNU,Lade, N-7491, Trondheim, . REITEN, IDUNInstitutt Norway. e-mail: [email protected] RODRIGUES,VIRGINIA SILVA Rua Amambaf, no. 107, Monte Castelo,

Juiz de

xvi

Participants

Fora, MG, CEP 36081-060, Brazil. RODRIGUES,WALTERMARTINSDepartamento de Matem~itica-IME, Universidade de S£o Paulo, Rua do MatZo, 1010, CEP05508-900, S~o Paulo, Brazil. e-mail: [email protected] S/~ENZ, EDITHCORINAFacultad ata, Gto. M(ixico. e-mail: corina@fracta|.cimat.mx

de Guanajuato,

Apdo. Postal

402, Guanaju-

SALAZAR, HERNANALONSO GERALDODepartamento de Matem~tica-IME, Universidade de S~o Paulo, Rua do MatZo, 1010, CEP05508-900, S~o Paulo, Brazil. e-mail: [email protected] SALORIO,MARIAJOS]~ SOUTOFaculdade Inform£tica, Campusde Elvina, 15017, Corufia, Spain. e-mail: [email protected] SAORIN,MANUEL Departamento de Matematicas, 4021, 30100, Espinardo, MU, Spain. e-mail: [email protected]

Universidade

Universidad

La Corufia,

de Murcia, aptdo.

SAVIOLI, ANGELAMARTAPEREIRA DAS DORES Departamento de Matem~itica-IME, Universidade de S~o Paulo, Rua do MatZo, 1010, CEP 05508-900, Paulo, Brazil. e-mail: mar [email protected] SCHR(~ER,JAN Department of Mathematics, University Bielefeld Germany. e-mail: [email protected]

of Bielefeld,

SKOWROI~SKI,ANDRZEJFaculty of Mathematics and Informatics, Copernicus University, Ul. Chopina 12/18, 87-100, Toruri, Poland. e-mail: [email protected] SLUNGAARD, INGER HEIDI 7491, Trondheim, Norway. e-mail: [email protected]

Institutt

D-33501,

Nicholas

for Matematiske Fag, NTNU,Lade, N-

SMALO,SVERREO. Institutt for Matematiske Fag, NTNU,Lade, N-7491, Trondhelm, Norway. e-mail: [email protected] SOLBERG,OYVINDInstitutt helm, Norway.

for Matematiske Fag, NTNU,Lade, N-7491, Trond-

xvii

Participants e-mail: [email protected] TREPODE, SONIA ELISABET Dpto. de Matem£tica, Fac.Cs.Ex., Nacional de Mar del Plata, Funes 3350, 7600, Argentina. e-mail: [email protected] TOMI~, BERTHADepartamento de Matematicas, Mexico. e-mail: [email protected]

Facultad

Universidad

de Ciencias,

UNAM,

VARGAS, ROSANARETSOS SIGNORELLI Departamento de Matem~tica-IME, Universidade de S~o Paulo, Rua do Mat£o, 1010, CEP05508-900, S~o Paulo, Brazil. e-mail: [email protected] ZACHARIA,DANDept. Mathematics, e-mail:[email protected]

Syracuse

ZHANG,PU Department of Mathematics, of China, Hefei 230026, P. R. China. e-mail: [email protected]

NY 13244, USA.

University

of Science and Technology

ZUAZUA,RITA E. Instituto de Matematicas, UNAM,Ciudad Universitaria, ico, D.F., C. P. 04510, Mexico. e-mail: [email protected] ZWARA,GRZEGORZ Faculty of Mathematics and Informatics, nicus University, U1. Chopina 12/18, 87-100, Torufi, Poland. e-mail: [email protected]

Mex-

Nicholas Coper-

On the existence phisms

of left

and right almost split

mor-

LIDIA ANGELERI-HOGEL Mathematisches Institut der Universit~it, Theresienstrafie 39, D-80333 Mfinchen, e-mail: [email protected]

ABSTRACT Wediscuss the existence of left and right almost split morphisms for a skeletally small category Mof modules over an arbitrary ring R. To this end, we associate to A4 a certain R-module Mand investigate finiteness conditions on M viewed as a modul~ over its endomorphismring. INTRODUCTION Left and right almost split morphismsare usually studied for categories of finitely generated modulesover artin algebras. In this small note, we consider left and right almost split morphismsfor a skeletally small subcategory A4 of the category of all (right) modulesover an arbitrary ring. If the modulesMs, i E I, are representatives of the isomorphismclasses of A4, then the existence of left and right almost split morphisms can be interpreted in terms of finiteness conditions for the modules M= Hie1 Ms and N = I-I~el Ms viewed as modules over their endomorphism rings S and T, respectively. An important role in this context is played by the radicals r(M, C)s and Tr(A, N)s where A and C are R-modules. In case that 2P[ is a finite subcategory consisting of modules with local endomorphismring, we can restrict ourselves to the Jacobson radical J(S) of S, and we obtain for instance that 2¢/ has left (respectively, right) almost split morphismsif and only if J(S) is a finitely generated left (respectively, right) S-module. V~re also study generalized right almost split morphisms, a concept introduced by Brune in [3]. For a right artinian ring R, the existence of such maps means that R is right pure-semisimple, see [11]. This is a consequence of Brune’s work on the so-called Kulikov property, relying on a functorial approach. Wenow obtain a direct, module-theoretical proof of this result and also a dual characterization of pure-semisimple rings in terms of generalized left almost split morphisms. The paper is divided into three sections. The first section is devoted to some

2

Angeleri-Hfigel

preliminaries. In Section 2, we relate finiteness conditions for sM and TNto the notion of a finitely (co)generated family of homomorphismswhich was introduced by Auslander in [1], These results are then applied in Section 3 to the study of left and right almost split morphisms. The author acknowledges a HSPIII-grant of the University of Munich. 1

PRELIMINARIES

Let us first introduce some notation. If R is a ring, we write J(R) for the Jacobson radical of R, and denote by ModR the category of all and by modRthe category of all finitely presented right R-modules. Throughout the paper, we fix a skeletally small subcategory Mof ModRwhich is closed under isomorphic images, and let {Mi [ i E I} be a complete irredundant set of representatives of the isomorphism classes of M. Further, we set M= [Iiel Mi with S = EndRM, and N = l-Iiet Mi with T = EndRN. Moreover, we denote by add ~ the class consisting of all modules isomorphic to direct summands of finite direct sums of modules of Recall that for two modules XR, YRthe radical r (X~ Y) denotes the collection of all homomorphismsf : X ~ Y such that there is no isomorphism of the form Z -~ X ~ Y ~ Z where Z is a module with local endomorphism ring. Then r(X, Y) is an EndR Y- EndRX - subbimodule of HomR(X, Y). Let us collect easy properties of this bimodule. LEMMA 1.1. Let Y~ be a module with endomorphism ring E = Endn Y, and let XR be an indecomposable module. (1) J(E) C r( Y, I0, with equality if Y = LIin=~ Yi and all Yi have local endomorphism ring. (2) Homn(X, Y)/r(X, Y) is either zero or a simple left E-module. (3) r(X, Y) is a noetherian left E-moduleif and only if Homl~(X,Y) is a noetherian left E-module. Proof. The first assertion in (1) is well-known. For the second assertion, assume that Y has a decomposition as stated. This means that E is semiperfect, that is, idy = y].~n=~ ei for local idempotents e~,... ,en ~ E. Then every f ~ r(X,Y) has the form f = ~in__~ f ei where Ef ei is properly contained in Eei and therefore lies in J(E) ei, which shows f e J(E). (2) Assume that there is a nonzero element ~ ~ Hom.~(X,Y)/r(X,Y). Since is indecomposable, s t is a split monomorphism and therefore generates the left Emodule HomR(X,Y). This yields the claim.

(3)

Apply

statement

(2)

on the

exact

sequence

of E-modules

0 --+ E r(X, Y) ---4/~ Hom/~(X,Y) ---+F~ Hom/~(X,Y)/r(X, Y) --40.

Left andRight AlmostSplit Morphisms 2

FAMILIES

3

OF HOMOMORPHISMS

Following Auslander [1, §1], we say that a family of homomorphisms (ak : A ~ Xk)keK is finitely cogenerated if there is a finite subset K0 C K such that the product map a : A -~ ]-I~eKo Xk induced by the ak with k E Ko has the property that all ak, k ~ K, factor through a. Westart out with two propositions describing when families of homomorphisms in [.Jiez r(A, Mi) are finitely cogenerated. PROPOSITION 2.1. The following

statements are equivalent

for a module A.

(1) The family of all homomorphisms in [JieI r(A, Mi) is finitely (2) The left T-module r( A, N) is generated by finitely contained in a finite subproduct I-Iielo Mi of N.

cogenerated.

many maps whose images

(3) There is a map a ~ r(A, X) such that X ~ addM and all maps h ~ r(A, Y) where Y ~ addA4 factor through a. If A is finitely generated, the following statement is further equivalent. (4) r(A, M) is a finitely

generated left S-module.

Proof. (1)=~(3): By assumption there are indices il,... ,it E I and maps r(A, Mi~), 1 _< k _< r, such that all maps in (Jielr(A, Mi) factor through the product map a : A -~ X = Hk=~Mi~ ~ addA/[ induced by the ak, and of course a ~ r(A, X). (3)=~(2) : By the universal property of products, every map f ~ r(A,N) factors through a. Moreover, there is a split monomorphism~ : X -~ ~k=l Xk for some X~,... ,X,~ E J~/, and for any 1 <_ k _< n there is i~ ~ I such that X~ -~ Mi~, giving rise to a split monomorphisma~ : X~ -~ N. Then the maps n

aa : A--% X ~ ~ Xk ~--~ X~ -~ N, k----1

1 < k < n, form a generating set of r(A, N) over T with the required property. (2)=~(1): Let a~, < k _
4 PROPOSITION 2.2. The following statements

Angeleri-Hiigel are equivalent

(1) Every family of homomorphismsin Uiel r(A, Mi) is finitely

for a module A. cogenerated.

(2) r(A, N) is a noetherian left T-module. If A is finitely generated, then (1) and (~) are further equivalent (3) r(A, M) is a noetherian left S-module. Proof. Denote by pi : N -r M~and ei : Mi ~ N, i E I, the canonical projections and injections, respectively. (1)=t,(2) : For a submodule TU C r(A, N) we consider the family of homomorphisms {Pif I f e U,i ~ I} in I~lietr(A,M~). By assumption there are indices il,... ,in ~ I and maps fk ~ U, 1 < k < n, such that the product ~nap a : A -~ l-]k=l Mi~ induced by the Pi~ fk has the property that all other maps of the form pi f with i ~ I and f E U factor through a. Then also all f 6 U factor through a, and so the f~, 1 < k < n, form a generating set of TU. (2)=~(1) : Let (ak : A ~ Mi~)kel( be a famil y in Uielr (A, Mi), and c onsi der the T-submodule U = ~eK T fk of r(A, N) given by the maps fk : A ~ Mi~ ~ N in r(A, N). By assumption TU = Y~et¢o T ft~ for some finite subset K0 C K. This implies that all f~, and therefore also all a~, factor through the product map a : A --+ L[k~KoMi, induced by the a~ with k ~ K0. If A is finitely generated, the equivalence of (1) and (3) is proven with similar arguments, taking into account the fact that each map in r(A, M) has its image a finite subcoproduct of M. [] Recall that a module is said to be endonoetherian if it is noetherian as a module over its endomorphism ring. As observed by Huisgen-Zimmermann [7], the module N is endonoetherian if and only if for all finitely generated modules An every family of homomorphismsin I.Jiet Homn(A,Mi) is finitely cogenerated. Over a semilocal ring, we can now restrict ourselves to families of homomorphisms in ~Jiel r(A, Mi) with A indecomposable. PROPOSITION 2.3. Assume that R is semilocal. are equivalent.

Then the following

statements

(1) N is endonoetherian. (2) M is endonoetherian. (3) For all finitely generated (or equivalently, for all finitely presented) indecomposable modules Aa every family of homomorphismsin [.Jiet r(A, Mi) is finitely cogenerated.

5

Left and Right AlmostSplit Morphisms

Proof. (2)¢~, (3): Mis endonoetherian if and only if HomR(A,M) is a noetherian left S-modulefor every finitely generated (or equivalently, for every finitely presented) module Ate. Since R is semilocal, every finitely generated module has finite decompositionin indecomposables[6, 1.14]. So, it suffices to consider finitely generated (or finitely presented) indecomposable modules A, and the statement follows from 2.2 and 1.1. The equivalence (1)¢~ (3) is proven by the same arguments. [] We now consider the dual situation. A family of homomorphisms (bk : Xk ~ C)kel~ is said to be finitely generated [1, §1] if there is a finite subset K0 C K such that the coproduct map b : L[keKo X~ -+ C induced by the b~ with k 6 Ko has the property that all b~, k 6 K, factor through b. Let us describe when families of homomorphisms in I.J~E I r(M~, C) are finitely generated. The arguments are dual to those employed above, and we will therefore omit the proofs. PROPOSITION 2.4. The following statements are equivalent

for a module C.

(1) The family of all homomorphisms in [.JiEl r(Mi, C) is finitely (2} The right S-module r(M, C) is generated by finitely contain a cofinite subcoproduct LIiet\Io Mi of M.

generated.

many maps whose kernels

(3} There is a map b e r(X, C) such that X ~ addA4 and all maps h ~ r(Y, where Y 6 addJ~/[ factor through b. PROPOSITION 2.5. The following statements are equivalent (I) Every family of homomorphismsin (Jiet r(Mi, C) is finitely

for a module C. generated.

(2) r(M, C) is a noetherian right S-module. Wenow give a dual version of Proposition 2.3. Recall that R is said to be a right Morita ring if it is right artinian and the minimal injective cogenerator of ModRis finitely generated. PROPOSITION 2.6. Assume that R is semilocal. Further, let Wa be a minimal injective cogenerator of ModR, and M~= HomR(M, W) s. The following statements are equivalent. (1) M~is noetherian. For all finitely cogenerated indecomposable modules C~ every family of homomorphismsin [.Jiez r(Mi, C) is finitely generated. If R is a right Morita ring, then following statement is further equivalent. (3) For all finitely generated indecomposable modules Cn every family of homomorphisms in ~Jiex r(Mi, C) is finitely generated.

6

Angeleri-Hiigel

Proof. Since R is semilocal, there are only finitely many simple right R-modules $1,-.. ,Sn up to isomorphism. So, W ~ I_L~I ci where C~ = E(Si) is an injective envelope of Si. (1)=~(2): If C is finitely cogenerated, then HomR(M,C)s is finitely M*-cogenerated, and the claim follows from Proposition 2.5. (2)=~(1) : By Proposition 2.5 we knowthat r(M, Ci)s is noetherian for all 1 < i < n, hence by the dual version of statement (3) in Lemma1.1 also HomR(M,C~)s is noetherian for all 1 < i < n, and M~is noetherian. Assumenow that R is a right Morita ring. Since R is right artinian, all finitely generated modules are finitely cogenerated, which yields (2)=~(3). Moreover, since all C~ are finitely generated, we deduce (3)=~(1) as above from 2.5 and the version of statement (3) in Lemma1.1. 3

LEFT

AND RIGHT

ALMOST SPLIT

MORPHISMS

Assume now that C is a module in A~ with local endomorphism ring. Recall that a homomorphismb : X -~ C with X E addA~ is said to be right almost split in add ¢~A if b is not a split epimorphism and any homomorphismh : Y --~ C where Y E addag[ and h is not a split epimorphism factors through b. Wehave seen in Proposition 2.4 that C has a right almost split morphismin addA/l if and only if the family of all homomorphisms in [J~e~ r(Mi, C) is finitely generated. Inspired Brune’s work [3], we will further say that C has a generalized right almost split morphismin add ~ if every family of homomorphisms in [.J~el r(M~, C) is finitely generated. The notions of ~. left almost split morphism and a generalized left almost split morphism in add ~ are defined dually. Finally, if ~ consists of modules with local endomorphism ring, we say that A/I has (generalized) right~ respectively left~ almost split morphisms if every module belonging to 3/[ has a (generalized) right, respectively left, almost split morphismin addJ~/l. The results in Section 2 can now be interpreted as characterizations for the existence of (generalized) left and right almost split morphisms. In particular, have the following two Corollaries. Observe that statement (3) in Corollary 3.1 has independently been obtained by Dung[4, 2.3], [5, 3.11]. COROLLARY 3.1.

Assume that all M~ have local endomorphism ring.

(1) A/I has generalized right almost ,split noetherian right S-module for all i ~ I.

morphismsif and only if r(M, Mi)

(~) Jr4 has generalized left almost split morphisms if and only if r(M~, N) is noetherian left T-modulefor all i ~ I. (3) lf sgl consists of finitely generatedmodules, then A/[ has (generalized) left abnost split morphismsif and only if r(Mi, M) is a finitely generated (noetherian) left modulefor all i ~ 1.

Left and Right AlmostSplit Morphisms COROLLARY 3.2. Assume that A/~ is a finite with local endomorphismring.

7 subcategory consisting

of modules

(1) 3/1 has left (respectively, right) almost split morphismsif and only if J(S) finitely generated left (respectively, right) S-module. (~) ]P[ has generalized left (respectively, right) almost split morphismsif and if J(S) is a noetherian left (respectively, right) S-module. (3) Assume that all Mi have finite length. Then Ad has left (respectively, right) almost split morphismsif and only if S is left (respectively, right) artinian if and only if A4 has generalized left (respectively, right} almost split morphisms. Proo]. Note first that we have M = N and S = T. Then (1) and (2) follow 2.1 and 2.2, or 2.4 and 2.5, respectively, and the fact observed in Lemma1.1 that J(S)= r(U, M)-~ ~]~i=xnr(Ui, M)-~ ~:=1r(U, (3) If all Mi have finite length, then so does M, and is the refore sem iprimary. HenceS is left artinian if and only if sJ(S) is finitely generated if and only if sJ(S) is noetherian, and the symmetric statement holds on the right side. This proves the claim. [] Note that there are categories with left and right almost split morphismswhere the module Mis not endonoetherian. EXAMPLE3.3. Let R be a hereditary artin algebra, and denote by 79 = {Pj I J E J}, respectively Z = {Ik I k E K}, a complete irredundant set of representatives of the isomorphism classes of the indecomposable preprojective right modules, respectively of the indecomposable preinjective left modules. Set Pa = I_[jeg Pj, and aI = L[kel~ Ik. Then add79 is closed under submodules, hence it is covariantly finite in modRby [2, 4.7], and it has right and left almost split morphisms. Dually, add2: is contravariantly finite in Rmodand has right and left almost split morphisms. However, Pa is endonoetherian if and only if R is of tame representation type, and aI is endonoetherian if and only if R is of finite representation type. To prove the latter statements, we argue as in [12, Corollary 12]. Observe first that for the usual artin algebra duality D : modR--+ Rmodwe have D(Pa) ~IIjej D(Pj) ~- l~aeK I~. Moreover, it follows from [12, Proposition 3] that Pa satisfies the ascending (descending) chain condition for finite matrix subgroups and only if the module D(Pa) satisfies the descending (ascending) chain condition for finite matrix subgroups, which is further equivalent to the fact that E1 satisfies the descending (ascending) chain condition for finite matrix subgroups. But know from [12, Observation 8] that for the pure-projective modules PR and aI the ascending chain condition for finite matrix subgroups is equivalent to endonoetherianness. So, Pa, respectively air, is endonoetherian if and only if aI, respectively Pa, is E-pure-injective. Now,it was shown in [9, 4.6] that aI is E-pure-injective if and only if the algebra is tame. Furthermore, if PR is ~-pure-injective, then we knowfrom [12, Proposition 4] that the class of homomorphismsbetween modules in

8

Angeleri-H~iigel

:P is noetherian, and this implies by [1, 5.6] that R is of finite representation type. Conversely, if R is of finite representation type, then every moduleis endofinite [12, Theorem 6] and hence endonoetherian. So, the proof is complete. Weclose this chapter with an application to pure-semisimple rings. Namely, we can nowrediscover a characterization of these rings in terms of generalized right almost split morphisms given by Brune in [3] and later improved by Zimmermann in [11]. Moreover, we give a dual characterization in terms of generalized left: almost split morphisms. Recall that a ring R is said to be right pure-sernisirnple if every right R-moduleis a direct sum of finitely presented modules. THEOREM 3.4. (1) (cp. [3, §3, Corollary 1] and [11, p. 372]) A right artinian R is right pure-semisimple if and only if every finitely presented indecomposable right R-module has a generalized right almost split morphism in modR. (2) A semilocal ring R is left pure-semisimple if and only if every finitely presented indecomposable right R-module has a generalized left almost split morphism in rood R.

Proof. (1) Let {Mi [ i E I} be a complete irredundant set of representatives of the isomorphismclasses of the finitely presented right R-modules, and set M= L[ie~ Mi with S = EndRM.Further, let WRbe a minimal injective cogenerator of ModR, and E = EndoW. As observed by Zimmermann in [11, p. 372], we can assume that R is a right Morita ring. In fact, by [11, Theorem 4], this property follows from the existence of right almost split morphisms in modRfor each simple non-projective module(7~, and is therefore satisfied whenever every finitely :presented indecomposable right R-module has a generalized right almost split morphism in modRor when R is right pure-semisimple. So, the module Wginduces a Morita-duality modR ~ Emod, and we have that M* = ~Hom,~(M,W)s is an E-S-bimodule with E ~ HomR(HomR(M, W) ®s M, W) ----- EndsM* S ~- (EndEM*)°p. Moreover, the E Hom~(Mi, W) form a complete irredundant set of representatives of the isomorphismclasses of the finitely presented left. Emodules. Further, we know from [8, Theorem 7] or [10] that R is right puresemisimple if and only if so is E, which means by [12, Theorem 6] that every pure-projective left E-module is endonoetherian, or in other words, that the left E-module I-[ie~ Hom~(Mi,W) is endonoetherian. But by Proposition 2.3 the latter is equivalent to ILe~ Hom~(Mi,W) ~ ~M*being endonoetherian. So, we see that R is right pure-semisimple if and only if M~is noetherian, and by Proposition 2.6 the proof is complete. (2) Take Mas in (1). By 2.3 we knowthat every finitely presented indecompos;~ble module ARhas a generalized left almost split morphism if and only if Mis endonoetherian. But the latter means that every pure-projective module is endonoe~herian, which by [12, Theorem6] is equivalent to R being left pure-semisimple. []

Left and Right AlmostSplit Morphisms

9

REFERENCES [1] M. AUSLANDER: A functorial approach to representation Notes in Math. 944 (1982), 105-179. S. O. SMALO:Preprojective [2] M. AUSLANDER., Algebra 66 (1980), 61-122.

theory,

Lecture

modules over artin algebras,

J.

[3] H. BI~UNE,On a theorem of Kulikov for artinian rings, Comm.Alg. _10 (1982), 433-448. [4] N. V. DUNG,Preinjective modules and finite rings, to appear in Comm.Algebra. [5] N. V. DUNG,Strong preinjective preprint.

partitions

representation

type of artinian

and almost split

morphisms,

[6] A. FACC~IINI, Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Math. 167, Birkh~user (1998). [7] B. HUISGEN-ZIMMERMANN, unpublished. Stable equivalence and rings whose modules are a direct [8] H. L. HULLINGER: sum of finitely generated modules, J. of Pure and Appl. Algebra 16 (1980) 265-273. Homologicaltransfer from finitely presented to infinite [9] H. LENZING: Lecture Notes in Math. 1006 (1983), 734-761.

modules,

Pure semisimple categories and rings of finite representation type [10] D. SIMSON: (Corrigendum), J. Algebra 67 (1986) 254-256. [11]

W. ZIMMERMANN: Auslander-Reiten sequences over artinian 11.._9.9(1988),366-392.

rings, J. Algebra

On the sparsity of represen[12] B. ZIMMERMANN-HUISGEN, W. ZIMMERMANN: tations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), 695-711.

1Actions of Hopf algebras

on quantum polynomials

VYACHESLAV A. ARTAMONOV Department of Algebra, Faculty of Mechanics and Mathematics, MoscowState University, Moscow, 119899, RUSSIA, email: [email protected] To the memory of my mother

ABSTRACT In this paper we classify actions of quantum groups on quantum affine spaces in terms of their coordinate algebras of affine spaces and quantum groups. In other words we classify coactions of commutative Hopf algebras on quantum polynomials.

Let k be a field with a fixed matrix Q = (qij) E Mat(n, k) whose entries qlj k*, i, j = 1,..., n are called multiparameters. It is assumed that multiparameters satisfy the relations qii = qijq~ = 1 for all i,j. Let r be an integer 0 < r < n. DEFINITION 1. Denote by ~1 A k~:~ IX1 ,...,Xr =

4-1

,Xr+l,...,...

(1)

,

the associative k-algebra with a unit element generated by elements Xl,

x~-l,...

,

Xr,

X~-1,

Xrq-l,...

, Xn

subject to defining relations X~X~-1 = X~-~ = 1,

xixj = qi~x~x~,

i =1, .... r;

1<_i,j
(2)

The algebra (2) is an algebra of quantum polynomials. ties

1Research partially of Russia"-552/

supported

by Grants

RFFI 99-01-00382,

ll

00-15-96128~

Program

"Universi-

12

Artamonov

It is pretty clear that every element of A has a unique representation as a linear combination of monomials X~l

i~ ¯ ..Xi. r i.Xr+~...X~,

ijEZ,

ir+l

....

, ,z~>_0.

DEFINITION 2. The algebra A is a general algebra of quantum polynomials if all multiparameters q~j, 1 < i < j _< n, are independent in the multiplicative group k* of the field k. It meansthat each equality I-Ii_~_
(3)

Ifr
a(xo # o. Suppose that a(X~), a(X~) ~ Observe tha t the smallest (the leadi ng) term of a product of non zero polynomials in A is equal to the product of the smallest (the leading) terms of factors¯ Thus (2) implies aiaj = qijajai.

(4)

Suppose that

= xI’...

taj = 7X~ ...X t. , where fl,7 ~ k*. n

Now(2) (see IMP]) implies ¯ ""-n

=

qrs

""Xn

13

Actions of HopfAlgebrason Quantum Polynomials Hence

(5) Suppose that i > j. If r > s then Definition 2 and (5) imply

(6) Consider the matrix

Let for example l n ~ 0 and p ~ j, i. For every index q ~ p By (6) we have

and therefore tq = tplal; 1. In particular ti = tplil; ~, 1, tj = tpljl; that is litj - l~ti = litnljl~ 1 - l~tnlil~t = O, which contradicts (6). t~ = 0 ifp ¢ i,j. Hence

Thus lp = 0 for all p ~ i,j.

Similarly one can prove that

ai = l~X~iX~~, aj --= TX~’X~~, where litj

- ljti

= 1.

By the assumption there exists a third variable X~ such that a(Xu) ~ O. The preceding ar~ment applied to the p~rs of indices (i,u), (j,u) shows au =~X~*X~’ = AX~X]~, where ~,A e k*. Finely r~ = dj = 0, that is au = ~X~~¯ Similarly ai = ~X{i, aj = 7X~~, l,t~

= 1,

and therefore li = t~ = e = ~1. Wehave now to show that the least and the leading terms of a(Xi) coincide and therefore they are equal to ai. If this is not the c~e, then r = n and the le~t term of a(Xi) has the form

x, Then for and index i = 1,... = 7iXi

i=

e

, n + ~’li

~1~1

""

(7)

14

Artamonov

where -"Yi, ~ -"7i (s) k* and the sum is taken over all multi-indices (rail(S),...

,rain(s))

such that i

(0 .... ,0,--1,0,...

,0) < (miz(s),...

i

,mln(s)) < (0,...

,0,1,0,...

,0).

Thus, miz(s) = ....

mi,i-~(s) =

mii(s) = -1,0, 1; =’1, ifmii(s) if rail(S)----- 1.

O, i < n, {XO, i
ml,i+~(s)

Foranyindexi = 1,.. . , n pickthe least monomial a(Xi)a(Xj) = qqa(Xj)a(Xi),i impli es

~ 7~ +7i--~ ~--i

-~-’’-~ qijTj"’X j 7iAi

’

7j’r~

+qij~j"

qqTj ..j

7~"x~m" ... Xnm~. in (7). Then

7j--~

...X~

+

~’’...x~,~ ’X-~ j ~"X i

"’’..n

mii,... , m

7i 7i’’i

Suppose first

(11)

+ "’" ,

~ ~ k*, such that

+7~X~ +

¯ q,j~j II

¯ , min)

i

,0,-1,0,...

I --1 The equalities "riAi - ~-1_ 7~~-1= qqT)X~t -17~X i ~nd (9)

X~tn

(10)

i, j-1, mij -- 1, m i,j +l, ¯ ¯

(0,...

7~X~ 7j~j "’" qij’lj _ ~x-l_ ¯ j "~iIlxmii i

(9)

+’"=

where +... is the sum of monomials JX~~ ~, ...X~ (/~,... , l~) >min((O,... ,

(8)

,O,m~,... ,m~)).

imply

¯ ~ ~ ~..~ mjj

~mjn Xj

+...

II ......

(12)

that (0,...

,O, mii,...

,mid-l,mij i

(0,... ,0,-1,0,...

- 1,mi,j+l,...

,min)

,0,m~,... ,m~,).

Then mii = -1 and i + 1 ~ j - 1 implies then mi,i+ 1 = 0, which contradicts Hence i ~ j - 2 implies (0,...

,O,

mii,...

,mi,j-l,mij-

1,mij+l,..,

,rain)

i

(0,... ,0,-1,0,... ,0,my1,..., and therefore in (12) we obtain ’’--n

717iAi

’

(8).

15

Actions of HopfAlgebrason Quantum Polynomials Then as in (5) we obtain

Thus,

-1, 0,

r =j, r > j,

a contradiction with (8). Wehave shown that if 1 < i < j - 2 < j _< n then either a(X~), or a(Xj) is of the form (3) with e = -1 provided w = i,j. Let (3) holds for some w = 1,... ,n. The preceding considerations show that the leading term of each a(X~) # 0 is -1, i. e. (3) holds for any index. the form 7~X~ THEOREM 2. Let H be a commutative semisimple Hopf k-algebra and A an Hcomodule algebra with a structure morphism p : A -~ H @~A. Assumethat either condition is satisfied: 1)

3
2) r _< 2 and the algebra H is finite dimensional. Then there exist elements al,... p(X~)=a~®X~,

, an 6 H such that, for any i = 1... , n

-1 whereA(a~)=a~®a~,

e(a~)=l,

S(c~)=ai

Proof. By the assumption p is a k-algebra homomorphismand H is a subdirect sum of fields Kj,j 6 J. Let ~rj : H -~ Kj be a corresponding projection. Then ~rjp : A -~ Kj ®k A is a k-algebra homomorphism inducing an embeddingof the field k into the field Kj. Thus ~rjp induces an endomorphismof the general quantum polynomials algebra K~ ®k A over the field Kj. By the assumption ~r~p(X~), ~rjp(X2), ~rjp(X3) # 0. According to Theorem1 there exists an integer e1 = =hl such that for each index j=l,...,n ~rjp(X~) = ai ® X~ , ai 6 K~. Moreover r < n implies ej = 1. Suppose now that r < 2 and the algebra H has finite dimension. By [M, Theorem2.3.1] there exists a finite group G and a field extension F/k such that F ®k H ~_ (FG)* where (FG)* is the Hopf dual of the group algebra FG. In addition F ®~ A is a FG-module algebra [M, Lemma1.6.4]. It means that the group G acts as an automorphism group of F-algebra F®k: A. Let g ~ G. According to Theorem 1 there exist elements 0~,... ,0~ ~ F such that

g(X~) = O~X,,.. . , g(X.) Then (13) holds for every g ~ FG. By [M, Lemma1.6.4],

(13) for each i = 1,...

p(X~) = a~ ® X~ 6 (F ®~: H A) ~q (H ®~ A) ==~a~ 6 and ~i(g) = 0i. It follows p(Xi) = ai ® Xi that for each index i.

,n, (14)

Artamonov

16 In both cases considered in Theoremwe have proved that p(X,) = a~ ® Xi, i = l...

,n.

Now

a, ®a~ ®x~ = (1 ®p)p(x~)= (~x®1)p(x~)= A(c~) and Xi = e(ai)Xi. It follows that

A(~i)= ~i

~(ai) = 1, S(cq) = 1

THEOREM 3. Let H be a commutative Artinian semisimple Hopf k-algebra and A an H-comodule algebra with a structure morphism p : A ~ H ®~ A. Suppose that n = r _> 3. Then H = A ~9 B is a direct sum of ideals A, B such that B C_ ker e and for any i = 1,..., n there exist invertible elements Ai E ]4, #i E B for which

~(x,) = ~, ®x~ + mx~-1.

(15)

Moreover

A(#~) = ~,®#, +U, ® ~’-~ ~ (A®B) $ (B®A) H®H, E()~i) = S(#i) = 0 , S(Ai) = A -~ , S(#, ) = #,

(16)

Proof. It follows from the assumption that H = Kx ~...$Km,

(17)

where K~,... ,Km are field extensions of k. Observe that the counit e : H -~ k is an algebra homomorphism. So we can always assume that K1 = k, kere = K~ ~ ... ~ Kin. From the definition of p it follows that (6 ® 1)p(gi) = for any i = 1,... ,n. Hence e~ = 1. Suppose now that ej = 1 for j = 1,... ,p and ej = -1 ifj =p+ 1,... ,m. Let p < m. Set A = K~ $ . . . $ K~,, B = K~+I $ " " ~ Km. Then B C_ ker~. Moreover, for any index i = 1,...

,n,

p(Xi) = (Ai ® Xi) ~ (#i ® -~) ~ (A® A) (~ (B ® A where A, ~ A,#i ~ B. Since the elements X1,... ,Xn are invertible in A, the elements A~, #i are invertible in A and B, respectively. Moreover in H ® H ® A we have

(A®1)p(Xi)= (1 ®p)p(Xi) [(~ ®a,) ~ (U,®U?l)]®X~+ [(a~®U,)~ (~ ®~;-x)] and therefore

A(#,) = (hi ® Ui) ¯ (#i ® ~-~) e (A ® B) ~

(18) (19)

Actions of HopfAlgebrason Quantum Polynomials

17

Since B C_ ker~, (18) implies Ai = Ai~.(Ai). Therefore e(Ai) = 1 for each i = 1,... , n, because Ai is invertible in A. Moreover(18) and (19) imply 1 = e(Ai) = S(Ai)A~ + S(#~)#~-1;

o = =

+

But HA~ = A, H~ = B. Thus (20),(21) imply S(A~)A~ = IA, S(~)~7~ = IB, S(A~)~ = S(~)A7~ = 0. Sincetheelements A~,pl ~reinvertible in A andB, respectively, we deducethat S(A~)6 A and S(~) 6 B. Thuswe S(A~)=A 7’ cA, S(p~)=p{, ~(A~)=I, ~(~)=0.

COROLLARY I. LetH, A, p, A, B, A~,p~ be fromTheorem3 and H has finitedimension. Thenthereexistsa positive integer d suchthat

Pwof. Since A~ = 0 it follows form (16) that for any m E A(AT)= [(A,@ A~) + (~ @ p~)]~7 @ AT) + (~7 @ PT¯ (23) Denoteby ~ : H ~ A the naturalprojection whichis a k-algebra homomorphism. Sincethe multiplication map m : H @ H ~ H is alsoan algebrahomomorphism the mapping~ = rmA : A ~ A is againa k-algebrahomomorphism and by (23) m. ¢(A~)= Take aminimal po lynomialf(T) ~k[~) for anelement A~6 A. Applying thealgebrahomomorphism ¢ we seethatf(A2 ) = 0, forany s ~ i. But A isa direct sumof somefields K~ from(17).So projections ofinvertible elements A2", s ~ 0, havefinitely manyimagesin eachKj. Clearly thereexistspositive integers m > m~ suchthatA2~ = A2~’. Put d = 2m m’. - 2 By (23)

=

=

= (A~A~)+(,~@~)=(1A~IB)+(~,~).

But A(1A) is an idempotent of H @ H lying in (A @ A) $ (B @B). Hence shows that an invertible element ~ @~ 6 B is an idempotent. An easy exercise shows that ~d@ ~ = 1B @ 1B and therefore ~ = 1~. THEOREM 4. Let H, A, p, a~,..., an be ~ in Theorem 2. The subalgebr~ of coinvariants AH: Ac°His generated byall monomials X~’ .. ¯ X~m~suchthat ~" 1. In particular,

A is a finitely generated left (right) AH-moduleif and only ~1~...

are roots of 1.

(25)

18

Artamonov

Proof. Recall that AHconsists of all elements g E AHsuch that p(g) = 1 ® g. Let =

....,

’"Xn , ~/m~ .....

mr 6k.

By Theorem 2

p(g)= ~ ~’ ...~Y~ m, ®~..... mox~’ ...x~o e H® A. Thus g ~ An if a~d only if (25) holds. If al,... , an are roots of degreed of 1, then X~,... , X~ ~ Aar and therefore A is a finitely generated]eft (right) All-module. Converselyif h is a finitely generatedleft (right) At/-modu]ethen according to lAW, Theorem3.17] there exists an integer d such that X~ ~ At/ for every i = 1,... ,n. Then c~ = 1. [] COROLLARY 2. Let H,A,p, at,... ,a~ be as in Theorem2, and the algebra H has finite dimension.ThenA is a finitely generatedleft (right) A/~-module. Proo]. Let G be the set of group-like elements in H. Then G is a group and a~,... , a,~ E G. Thegroup algebra kG is a subalgebrain H. HenceG is finite and a~,... , an have finite orders. [] Note that Corollary 2 follows also from [M, Lemma 1.7.2, Theorem4.2.1]. THEOREM 5. Let H,p,A,B,A~,~ be from Theorem 3. Suppose that H has a finite dimension.ThenA is a finitely generated]eft (right) moduleover the subalgebra of coinvariants AH. Proof. By Corollary 1 there exists a positive integer d such that (22) holds. Put .f~ = X~ + X~-d. Since )~t~ = 0 wehave

-~ = ~(~) = ~(x~)~ + a(x~) [~ ® X~ + m ® X - i-~] ~ + [:~7 ~ ® X~-~ + ~-~ ® X~]~ =

(~ + ,;-~) ~ x - ~ + (~7~ + ~) ®x7~ = (~ + ~) ®x~+ (1~+ 1~)-~= 1 ® ~. Thus ]¢E Ate. Denote by F the suba]gebra in A generated by the e]eme~ts f~,... , fn. Then F C_ AHand A is a finitely generated F-module [AWl. [] Consider now the dual space H* = homk(H, k). Then H* is an algebra with respect to convolution multiplication l ¯ l’ where, for any h ~ H (l * l’)(h) -- E l(h(~))l’(h(~)), h

and A(h) = E h(1) ® h(2) e h

Moreover A is a left H*-modulealgebra with respect to the action l o X~ = l(a~)X~, i = l,...

,n.

(26)

Actions of HopfAlgebrason Quantum Polynomials

19

If H has finite dimension over k then (H ® H)* = H* ® H* and H* is a Hopf algebra with respect to comultiplication, counit and antipode defined as follows (A/)(hl ® h2) = l(hlh2), COROLLARY 3. Let H, A, p, a~,... algebra H* is commutative.

(~/)(h)

=/(1),

(Sl)(h)

= l(Sh).

, an be as in Theorem 2. Then the convolution

Proof. For any monomial f = X~nl ...

X~m~ we have

p(f) = a"~’ ...an

®

(27)

Therefore if A E H*, then

Suppose now that A, # E H*. Then (~*~)(~...~)

= (~

~)(~(~.-.~))

=

). It means that

(A*~)of=(~.A)of.

Wehave already mentioned above that if H is ~ finite dimensional commutative semisimple Hopf algebra over ~ algebraicMly closed field k then there exists a finite group G such that H ~ (kG)*, [M, Theorem 2.3.1]. Moreover by [M, Lemma 1.7.2] the subalgebra of invariants AH" coincides with subalgebra of coinvariants An. Recall also that a subalgebra in A generated by XI,... , Xa is an H-comodule subalgebra of A. Let be given an arbitrary Z~-graded algebra A A=*A~,

l~Z".

(28)

Then AtAv ~ At+v for any l, l ~ ~ Zn. Take a free multiplicative Abelian group G with free generators g~,... ,g~. By [M, Example4.1.7, p. 41] a grading (28) in is equivalent to an existence on A of ~ structure of a kG-comodule~gebr~ with a structure map r : A ~ kG ~ A, where ~(a) = g~’ ...g~ @a, if a e A(h ..... THEO~M6. Let H be ~ commutative biMgebra with group-like

~nd A from (28). A linear

map p : A ~ H @A such that

....~. @a, whereaEA(h .....~.), definesan H-comodule structure on A.

elements

20

Artamonov

Proof. Let A : kG -~ H be an algebra homomorphismsuch that A(gi) = ai for i = 1,... ,n. Then

(~ @~)5(g~)= ~(g~~ g~)= a~®a~= A(a~)= ±(x(g~)), S(~(gi))1 = ~(S(ai), ~(~(gi)) = ~(~i) = 1 = Thus A is a Hopf algebra homomorphism.For every element a E A(h ..... (A ® 1)T(a):

(A ® 1)(g~’ ..

~,) we have

.g~’®a) ~ n ®a : p(a). = ...c~

It follows that (A ® 1)~- = p, i. e. p introduces a structure of a H-comodulealgebra on A. [] In noncommutative geometry [D] an quantum polynomial algebra A with r = 0 is considered together with quantum Grassman algebra F. The algebra F is generated by elements ~1,... , ~n subject to defining relations (29)

~2 = 0, ~j = p~j~j~

where p~ E k*, and p~ = p~p~ = 1 for any i, j = 1 .... , n. The algebra of functions on matrices of size n is introduced in the book [D] as a universal bialgebra Mp, Q(n) coacting on the pair of algebras A, F. In Theorem 6 we have actually considered universal commutative Hopf algebras coacting on A, F. REFERENCES [A1] Artamonov V. A., QuantumSerre’s conjecture, N4., p. 3-76.

Uspehi mat. nauk. 53 (1998),

[A2] Artamonov V. A., General quantum polynomials: irreducible modules and Morita-equivalence, Izv. RAN,ser. math. 63 (1999), N 5. P. 3-36. [A3] Artamonov V. A., Division ring of quantum rational functions, nauk. 54 (1999), N 5, p. 154-155. lAW]Artamonov V. A., Wisbauer P~., Homological properties mials, Algebras and representation theory (to appear). [D] Demidov E. E., Quantum group, Moscow:Factorial,

Uspehi mat.

of quantum polyno-

1998, 127P.

[MP] McConnellJ.C., Pettit J.J., Crossed products and multiplicative Weyl algebras, J. London Math. Soc. 38 (1) (1988), 47-55.

analogues of

[M] MontgomeryS., Hopf algebras and their actions on rings, P~egional Conference Series in Mathematics, v. 82 - American Math. Soc.: Providence R. I., 1993.

Strongly simply connected derived tubular algebras IBRAHIM ASSEM Math~matiques et Informatique, Universit~ de Sherbrooke, Sherbrooke, Quebec, J1K 2R1, Canada, e-mail: [email protected]

ABSTRACT In this note, we show that a derived tubular algebra is strongly simply connected if and only if it contains no full convex subcategory which is hereditary of type Am,and give several other characterisations of the strong simple connectedness of (derived) tubular algebras. INTRODUCTION The objective of this note is to give a simple (and fairly visual) characterisation the strong simple connectedness of a derived tubular algebra. A finite dimensional algebra A over an algebraically closed field is called derived tubular if there exists a tubular algebra B and an equivalence of triangulated categories between the derived categories Db (modA)--- Db(modB)of bounded complexes of finitely generated right A- and B-modules,respecti’~ely. It is shownin [8] that a derived tubular algebra is simply connected. Wehere give criteria for such an algebra to be strongly simply connected (in the sense of [17]). Wesay that an algebra is strongly k-free if it contains no full convexsubcategory which is hereditary of type A~n, for any m _> 1. Skowrofiski has asked in [17], Problem 2, whether it is true that a simply connected algebra is strongly simply connected if and only if it is strongly ,~-free. The answer to this question is known to be positive if the algebra is iterated tilted of euclidean type [2], or tame tilted [5]. Also, it was shown that there exists a close connection between the strong simple connectedness of an algebra, and the shape of the orbit graphs of the directed components of its Auslander-Reiten quiver [11,4,13]. The main result of this note states that a derived tubular algebra A is strongly simply connected if and only if it is strongly/~-free. Further, we give other criteria (using, amongothers, orbit graphs of directed components), showing that it suffices to consider two particular full subcategories of A. In the case where A is tubular, we (predictably) obtain a much stronger characterisation, using different techniques. The results of this note are applied in [1] to yield a complete characterisation of the (strongly) simply 21

22

Assem

connected tame quasi-tilted 1

STRONGLY

SIMPLY

and semiregular iterated CONNECTED

tubular algebras.

TUBULAR

ALGEBRAS.

1.1. Throughout this paper, k denotes an algebraically closed field. By algebra is meant a basic and connected finite dimensional k-algebra and by module a finitely generated right module. Such an algebra A can be written as a bound quiver algebra A -~ kQA/I, where the pair (QA, I) is called a presentation of A, and may equivalently be considered as a locally bounded k-category, whose object set is denoted by Ao, see [11]. An algebra A is called triangular if QAhas no oriented cycle. A full subcategory C of A is called convex if, for any path ao -+ ¯ .. -~ as, with ao, as E Co, we have ai E Co for all i. Weuse freely properties of the module category modA, the Auslander-Reiten quiver F(modA), and the AuslanderReiten translations T = DTr and T-1 = TrD, as can be found, for instance, in [16]. A component F of F(modA)is called directed if, for all indecomposable modules M in F, there exists no sequence M = M0 -~ M1 -~ ... ~ Ms = M of nonzero non-isomorphisms between indecomposable A-modules. Given a component F of F(modA),its orbit graph O(F) is defined as follows: the points of (9(F) the v-orbits Mr of the modules Min F, there exists an edge Mr --+ Nr if there exist m,n ~ Z and an irreducible morphism TraM and the number of such edges equals dimk Irr(~-mM, vaN) or dima Irr(vnN, TnM), respectively (here, Irr(X, Y) denotes the space of irreducible morphismsfrom X

Y). For tubular algebras, we refer the reader to [16]. Weneed the following facts. Let A be a tubular algebra, then A contains exactly two tame concealed full convex subcategories, denoted by C(°) and C(~), and every object of A belongs to C(°) or to C(°°). Also, A is an extension of C(°), and a coextension of C(~), by truncated branches (in the terminology of [9]), and the tubular type of A is one of (2, 2, 2, (2, 3, 6), (2, 4,4), (3, 3, 3). Further, F(modA)has a postprojective and a preinjective components, which coincide respectively with the postprojective component of F(modC(°)) and the preinjective component of F(modC(°°)). Finally, for simply connected and strongly simply connected algebras, we refer the reader to [17, 6, 3]. 1.2. Wedenote by B[M] the one-point extension of an algebra B by a B-module M. LEMMA. Assume that B is a representation-finite triangular A = B[M] is simply connected. Then B is simply connected.

algebra,

and that

Proo]. Assumethat this is not the case, and let (QB, ~) be apresentation of B such that the fundamental group ~rl (Q~, ~) i s n ot t rivial. T here e xists a presentation (QA, I) of A such that I V~ kQ~ = I ~. Let G be an arbitrary abelian group. By [6] (2.4), there exists an exact sequence of abelian groups Hom(~r~(QA, I), G) Hom(~r~ (Q~, I’ ), G) --+ m (where mis defined as in [6] (2.4)). Since A is simply connected, the first term of sequence vanishes. Since B is a triangular representation-finite algebra, it is stan-

Strongly SimplyConnectedDerivedTubularAlgebras

23

dard, hence the group rl (QB,I’) is free [15](3.9)(4.3), therefore Hom(~rl (QB,I’), 0. Weinfer that m ~ 0, and hence B contains a full subcategory of the form

where each of the shownpaths is non-zero in B. The full subcategory of B generated by the points a~,... , as, b~,... , bs is clearly hereditary of type ,~, a contradiction to the representation-finiteness of B. [] 1.3. gory (a) (b) type

LEMMA. Let A be a tubular algebra, and B be a proper full convex subcateof A. Then: If B is representation-infinite, then B is tilted of euclidean type. If B is representation-finite, then B is tilted of Dynkintype or of euclidean ~

Proof. (a) If B is representation-infinite, then it contains a tame concealed full convex subcategory. NowA contains exactly two tame concealed full convex subcategories C(°) and C(°°). Therefore, B contains one of them (but not both, because, by hypothesis, B ~ A). Wemay assume, by duality, that B contains (°). Now B is a truncated branch extension of C(°) and A is obtained from B by a sequence of one-point (tubular) extensions. In particular, B is not a tubular algebra. Therefore the tubular type of B is domestic. By [16] (4.9), B is tilted of euclidean type. (b) Assumethat B is representation-finite. Since A is tubular, then it is quasitilted, and hence so is B, by [14] (1.15). Since B is representation-finite, it actually tilted [14] (3.6). On the other hand, the Euler quadratic form qs qAl~ of B is positive semi-definite, because qA is. Therefore, B is tilted of Dynkinor of euclidean type. By [10], there remains to show that B is simply connected. Since B is convex in A, there exists a sequence A = Ao ~ A~ ~ ... ~ At = B of full convex subcategories with each Ai either a one-point extension or a one-point coextension of Ai+~. Let j be the largest index such that Aj is representation-infinite (thus, Aj+I is representation-finite). Note that, since A is tubular, then t _> 2 and j _> 1. Assumethat Aj+I is not simply connected. By (1.2) and its dual, A~. is not simply connected. NowAj is a representation-infinite full convex subcategory of A, and it is proper (because j _> 1). By (a) above, A1 is tilted of euclidean type. Since Aj is not simply connected, it is of type ~, by [8]. It is thus a truncated branch extension, or coextension, of a hereditary algebra of type ~, by [7]. Since Aj+~ is representation-finite, the unique point of Aj which is not in A~.+~cannot lie in the branches, hence it lies on the unique cycle in the bound quiver of A1. Since this point is either a source or a sink in Aj~ then Ai+~ is tilted of type A. But then Aj+~ is simply connected, a contradiction. This shows that Aj+I is simply connected. Since B is a full convex subcategory of Aj+~, then B is simply connected [12] (2.8). 1.4. COROLLARY. Let A be a tubular algebra which is not strongly simply connected. Then C(°) or C(~) is hereditary of type ~.

24

Assem

Proof. Let C be a full convex subcategory of A which is not simply connected. Since A itself is simply connected, C ~ A. By (1.3), C is tilted of Dynki~a euclidean type. Since it is not simply connected, then it is of type ~, by [8]. Applying (1.3) again, C is representation-infinite. Therefore C contains e~ (unique tame concealed) full convex subcategory ~ ahereditary al gebra of typ e ~. ]Now A has only two tame concealed full convex subcategories, namely C(°) (°°) and C Therefore C’ = C(°) or C’ = C(°°). [] 1.5. LEMMA. Let A be a truncated branch extension of a tame concealed algebra C. If A satisfies the separation condition, then C is not hereditary of type ~. Proof. This follows from the fact that, for each x E Co, the indecomposable projective modules e~C and e~A (where ex denotes the primitive idempotent corresponding to x) coincide when considered as A-modules. [] 1.6. Weare nowable to state and prove the main result of this section. THEOREM. Let A be a tubular algebra. The following conditions are equivalent: A is strongly simply connected. (a) (b) The orbit graph of each of the directed components of F(modA)is a tree. (c) A is strongly ]~-free. (d) C(°) and C(~) are not hereditary of type ~,. (e) A and A°p satisfy the separation condition. Proof. (a) implies (b). The directed components of F(modA)are the postprojective and preinjective components which are, respectively, the postprojective and the preinjective componentsof a tilted algebra. The result then follows from [2] (1.3) or [13] (4.2). (b) implies (c). The postprojective component of F(modA)is the same as (°)) and its preinjective componentis that of F(modC(°°)). Thus, neither of F(modC (°) C nor C(~°) is hereditary of type ~. However, if A contains a full subcategory C which is hereditary of type .~, then C must coincide with either C(°) or C(~), a contradiction. (c) implies (d). This is trivial. (d) implies (a). This follows from (1.4). (a) implies (e). This is trivial. (e) implies (d). This follows from (1.5) and its dual. 1.7. EXAMPLES. (a) Let A be given by the quiver

bound by a~ = "r~, e7 = A#, v#~ = 0. Then A is tubular of type (3, 3, 3) and strongly simply connected. The orbit graph of the postprojective component of F(modA)

Strongly SimplyConnectedDerivedTubularAlgebras

25

and that of its preinjective componentis

(b) There exist tubular algebras satisfying the separation condition, which are not strongly simply connected (thus, one cannot improve condition (e) of the theorem). Let A be given by the quiver

bound by aa = 0, 7P = 0 and a/~A = 76A. Thus A is tubular of type (3,3,3), °p satisfies the separation condition, but is not strongly simply connected. Here, A does not satisfy the separation condition: C(~°) is hereditary of type (c) The statement of the theorem does not hold for derived tubular algebras. Let A be given by the quiver

bound by As = It’)’, A~ = ItS, a~, = c./~h ~’ = c.Srl (for somec E k \ {0, 1}) Aau = 0. Then A is derived tubular: indeed, reflecting at the unique sink yields a tubular algebra of type (2, 2, 2, 2). Let A_(or A+) denote the full convex subcategory of A generated by all points except the unique source (or sink, respectively). Then the postprojective (or preinjective) component of F(modA) coincides that of F(modA_)(or F(modA+),respectively). Moreover, A_ (or A+) is tilted type l~ and has a complete slice in the postprojective (or preinjective, respectively) component of its Auslander-Reiten quiver. Thus the orbit graph of each of the postprojective and the preinjective component of F(modA)

On the other hand, both A and A°p satisfy the separation condition. But A is not strongly simply connected, because it is not strongly k-free. Finally, notice that A is a quasi-tilted algebra, so the statement of the theorem does not apply either to quasi-tilted algebras (see, however,[1]).

26 2

Assem STRONGLY GEBRAS.

SIMPLY

CONNECTED

DERIVED

TUBULAR

AL-

2.1. By [8], a derived tubular algebra is always simply connected. Hence, if it is representation-finite, it is strongly simply connected [12] (2.8). Weare thus only interested in the representation-infinite case. If an algebra A is representation-infinite, then, by [9] (2.5), it is derived tubular if and only if it is isomorphic to a branch enlargement of a tame concealed algebra (in the sense of [9] (2.2)) and its tubular type equals that of the corresponding tubular algebra. Consequently, there exist a source a+ and a sink a_ (lying in the branches of A) such that each of the full convex subcategories A+ and A_, generated respectively by the objects of A0\ {a_} and A0\ {a+}, is iterated tilted of euclidean type. Notice that the points a+ and a_ are usually not unique. A pair (a+, a_) as above will be called tu bular pair ofA. 2.2. LEMMA. Let A be a derived tubular algebra which is not strongly simply connected, and (a+,a_) be a tubular pair of A. Then one of the algebras A+and A_ is not strongly simply connected. Proof. Let B denote the tame concealed full convex subcategory of A. Since A is not strongly simply connected, then, by [3] (1.3), its bound quiver contains irreducible cycle which is not a contour, or an irreducible contour which is not naturally contractible. Let C denote this cycle. If C lies entirely inside A+or A_, we are done. If not, then both a+ and a_ lie on C. Now, the cycle C cannot lie entirely inside any individual branch and each walk between two branches passes through B. Since a+ lies on C, we deduce that a+ is the root of an extension branch, thus is an extension point of B. Moreover, a+, being a source of A, is also a source of C. Similarly, a_ is the root of a coextension branch and thus is a coextension point of B. Moreover, a_ is a sink of C. Now,a+, being the root of a branch, is a separating point, hence, by [6] (2.2), if (~ : a+ -~ a and ~ : a+ -~ b are the two arrows on C starting at a+, there exists minimal relation ~lC~V+ A2f~w+ EAjuj where hi E k (for all i) and v,w, uj (for j_>a all j) are paths from a+ to c E Bo (say). Let a’, b’ denote the last points of v, respectively, which lie on C, and d be the first commonpoint of v and w. Observe that, since a~, b~,c ~ are predecessors of c ~ B0, and proper successors of a+, then a~, b~, c~ ~ B0. Denoteby v~, w~ the subpaths of v, w, respectively, from a’ to c~ and from b~ to c ~, and by u~,u" the subwalks of C from a ’ to a_ and from b* to a_, respectively. Then ~-1 : a~ ~ .~.’. -- a_ -- .~.". -- b~ ---¢ .w.’. __~c’ +--- .v.’. ~-- a’ C’ =u’u"-lw~v is a closed walk entirely contained inside A_. Weclaim that C~ is an irreducible cycle which is not a contour. Indeed, we notice first that c ~ ¢ a_, because c~ ~ B0 C_ (A+)o, while a_ ¢ (A+)o. There is no path from a_ to c ~, ~ because a_ is a sink of A, and no path from c to a_, because the existence of such a path would contradict the irreducibility of C. On the other hand, since u~u"-~ is a subwalk of C, it has no self-intersections. By definition, v~w~-1 has no self-intersections. Finally, there is no commonpoint

Strongly SimplyConnectedDerivedTubularAlgebras

27

between v’w’-1 and u’u’’-~, because the existence of such a path would contradict the irreducibility of C. Next, C~ is clearly irreducible, because C is, and, finally, C’ is not a contour, because it has at least two different sinks, namely c’ and a_. Wehave established the existence of an irreducible cycle C’ which is not a contour, and lies entirely inside A_. Hence, again by [3] (1.3), A_ is not strongly simply connected. [] 2.3. The above lemma reduces the study of the strong simple connectedness of a derived tubular algebra A to that of the two iterated tilted algebras of euclidean type A+ and A_, which was characterised in [2] (3.3). Weare able to state and prove the main result of this section. THEOREM. Let A be a representation-infinite derived tubular algebra, and (a+, a_) be a tubular pair of A. The following conditions are equivalent: (a) A is strongly simply connected. (b) is str ongly .~-free. (c) A+and A_ are strongly ,~-free. (d) A+ and A_ are strongly simply connected. (e) The orbit graph of each directed component of F(modA+)and F(modA_) a tree. Proof. Clearly, (a) implies (b), which implies (c). It follows from (2.2) implies (a). Thus, the first three conditions are equivalent. Since A+and A_ are iterated tilted algebras of euclidean type, the equivalence of (c), (d) and (e) follows directly from [2] (3.3). 2.4. The above theorem yields further criteria for the strong simple connectedness (°° of a tubular algebra. Let indeed A be a tubular algebra, then clearly a+ E Co and a_ E Co(°), so that A+and A_ are tilted algebras of euclidean type. Wethen have the following corollary. COROLLARY. Let A be a tubular algebra, and (a+,a_) be a tubular pair of The following conditions are equivalent: (a) A is strongly simply connected. (b) A+and A_ are strongly ,~-free. (c) A+and A_ are strongly simply connected. (d) The orbit graph of each directed component of F(modA+)and F(modA_) a tree. (e) The orbit graph of the preinjective component of F(modA+)and the orbit graph of the postprojective component of F(modA_)are trees. Proof. The equivalence of (a), (b), (c) and (d) follows from (2.3). It is clear implies (e). In order to showthat (e) implies (c), we note that, since a+ ~ Co(°°), then A_ is a domestic truncated branch extension of C(°) , hence there is a complete slice in the preinjective componentof F(modA_).Similarly, there is a complete slice in the postprojective component of F(modA+).The result now follows at once from [2] (2.3) and its dual.

Assem

28 2.5. EXAMPLES. (a) Let A be given by the quiver

.

bound by a/~ = ~,~, pa = 0, a/~ = 0, ~a = O and 7#v = 0. Then A is derived tubular of type (2, 3, 6), but is not tubular. Clearly, A is strongly simply connected. have here two possible choices for a+ (and only one for a_). (b) It does not suffice to have A+(or A_) strongly simply connected for be strongly simply connected, as is shown by the example (1.8)(b) above. ACKNOWLED

GEMENTS.

The author gratefully acknowledges partial support from the NSERCof Canada. He is also grateful to Fltivio Coelho and Sonia Trepode for useful discussions. REFERENCES [1] I. Assem, F. U. Coelho and S. Trepode: Simply connected tame quasi-tilted algebras, to appear. [2] I. Assem and S. Liu: Strongly simply connected tilted Math. Quebec 21 (1997), No. 1, 13-22.

algebras,

[3] I. Assem and S. Liu: Strongly simply connected algebras, (1998), 449-477.

Ann. Sci.

J. Algebra 207

[4] I. Assem, S. Liu and J. A. de la Pefia: The strong simple connectedness of a tame tilted algebra, to appear in Comm.Algebra. [5] I. Assem, E. N. Marcos and J. A. de la Pefia: The simple connectedness of a tame tilted algebra, to appear. [6] I. Assemand J. A. de la Pefia: The fundamental groups of a triangular algebra, Comm.Algebra 24(1) 187-208 (1996). [7] I. Assemand A. Skowrofiski: Iterated tilted algebras of type .~,~, Math. Z. 195 (1987) 269-290. [8] I. Assemand A. Skowrofiski: On some classes of simply connected algebras, Proc. London Math. Soc. (3)56 (1988) 417-450. [9] I. Assemand A. Skowrofiski: Algebras with cycle-finite Math. Ann. 280 (1988) 441-463.

derived categories,

[10] I. Assemand A. Skowroxiski: Quadratic forms and iterated tilted Algebra 128 (1990) 55-85.

algebras, J.

Strongly SimplyConnectedDerivedTubularAlgebras

29

[11] K. Bongartz and P. Gabriel: Covering spaces in representation theory, Invent. Math. 65 (1981) 331-378.

[12]O. Bretscher

and P. Gabriel: The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983) 21-40.

[13]S.

Gastaminza, J. A. de la Pefia, M. I. Platzeck, M. J. Redondoand S. Trepode: Finite dimensional algebras with vanishing Hochschild cohomology, J. Algebra 212 (1999) 1-16.

[14]D. Happel, I. tilted

Reiten and S. O. Smalo: Tilting in abelian categories and quasialgebras, MemoirsAmer. Math. Soc., No.575, Vol.120 (1996).

[15]R.

Martinez-Villa and J. A. de la Pefia: The universal cover of a quiver with relations, J. Pure Applied Algebra 30 (1983) 277-292.

[16]C.

M. Pdngel: Tamealgebras and integral quadratic forms, Lecture Notes in Math., 1099 (1984), Springer-Verlag, Berlin-Heidelberg-New York.

[17]A.

Skowrofiski: Simply connected algebras and Hochschild cohomologies, Can. Math. Soc. Conf. Proc. Vol.14 (1993) 431-447.

H1 and Presentations bras

of Finite

MICHAEL J. BARDZELL Salisbury email: [email protected]

State University,

Dimensional

Salisbury,

Alge-

Maryland

EDUARDO N. MARCOS x Universidade De S~o Paulo, S~o Paulo, Brazil en~i: [email protected]

ABSTRACT In this paper we study the first Hochschild cohomology group H1 (A) of certain finite dimensional algebras and how it relates to presentations of A. In particular, we consider this relationship for monomial,directed, and a generalization of Schurian algebras. The relationship between presentations and the fundamental group ~rl (A) is also studied. 1

INTRODUCTION

The purpose of this paper is to study Hi(A) for a finite dimensional algebra A kF/I, where F is a finite quiver, k is a field, and I is an admissible ideal. We will focus primarily on the relationship between H~ (A) and presentation properties of certain classes of algebras. In [AP~S] Open Problem 5 asks for an invariant characterization of monomialalgebras. Such a characterization is provided in [BG]. The approach of that paper is based on H1 (A) and group gradings ( coverings ). general characterization, however, is not algorithmic. It depends on the existence of a certain grading. Analgorithmic solution is provided for a certain class of algebras. An algebra A is said to be constricted if dimk o(a)At(a) = 1 and dimk ray = 1 for each a E F~ and v E F0. The result on constricted algebras is that A is a monomial algebra if and only if dim~ H~(A) --- 1 - IF01 + IFll = x(F) , the reduced Euler characteristic of F. For this class of algebras it is shownthat if A has a monomial presentation, then every presentation is monomial. Constricted algebras include schurian, narrow, and incidence algebras ( see [Ha] ). In the first section of this paper we show that the assumption that dimk vAv = 1 for each v E F0 can be dropped. Thus we obtain an algorithmic solution for Problem 5 for a more general 1Thesecondauthor gratefully acknowledges financial supportin the formof a researchscholarship from CNPq- Brasil 31

32

Bardzell and Marcos

class of algebras than the constricted algebras in [BG]. Wealso give a necessary and algorithmic condition for any algebra to be monomial. A different approach to the monomial characterization problem can be found in [GS]. The algebras satisfying dim~ o(a)At(a) = 1 for each a E F1 also have nice properties regarding the fundamental group 7rl. In section 3 we will see that the fundamental group of an algebra in this class does not depend on the presentation. The final presentation problem we consider is the former conjecture that HI (A) = implies F has no oriented cycles. A counterexamplefor the general case has recently been given in [BL]. Weconsider some algebras where the conjecture does hold even for undirected cycles. Our approach is based on the combinatorics of the quiver and relations of an algebra A. Throughout this paper Fo will denote the vertex set, F1 will denote the arrow set, and R will denote a generating set for I. Weuse [u, v] to denote the set of all paths starting at u and ending at v.We can compute H1 (A) via the complex 0 --+ P~ ~ Pl* ~’~ P~, i.e.

dim~Hl(A)

= dimk ker¢~ -dimkim¢~.

Here

P~=veroII vAv, P~= aeroHo(a)At(a), and P~= r~el~o(r)At(r). Theseterms and maps can be found by applying Horn^, ( , A) to the first three terms of the projective resolution P~ ---+ P1 --~ Po ---+ A ~ 0 as discussed in IBM]. Throughout this paper we also use the notation ~ E P~ to denote the vertex v in the v th component and ~ ~ P~* denote the arrow a in the ath component. 2

DERIVATIONS,

Hi(A),

AND MONOMIAL ALGEBRAS

In this section we provide a necessary homological condition for an algebra to be a monomialalgebra ( Theorem2.1). This is an algorithmic solution for one direction of Open Problem 5 from JARS]. Wealso provide necessary and sufficient conditions for algebras satisfying dim~ o(a)At(a) = 1 for each a 6 F~ to be monomial(Theorem 2.4 ). THEOREM 2.1.

Let A = kF/I be a monomial algebra.

Then dimk Hl(A) > x(F).

Proof. Let {Pl, ...Pn} be a minimal set of generating paths for I. As we stated in the introduction, the first cohomology group can be computed from the complex 0 ---~ II vAv ~-~ H o(a)At(a) ~--~ ~I o(pi)At(pi) v~Fo

IBM]. To simplify

notation,

using the maps described in

i=l

a~F~

write H ray = A~gB. Here A = H kv, B = H rAy, v~Fo

v~Fo

v~Fo

and ray is the k span of all paths starting and ending at v,exclud~ v. Similarly, write H o(a)At(a) = C $ D where C II ka andD -- H o(a )At(a). Note aeF~ a~F~ aer~ the first boundary map can be decomposed as ¢~ = f ~ g. Also, f(A) C_ and f(B) C_ si nce ¢~is multiplication by arr ows, i.e . ¢~ rai ses degrees of Po*elements by 1 or else sends them to 0. Similarly, write ¢~. = h @l. Then the complexbecomes 0 ---~A $ B I_~ C ¯ D ~ ~I o(pi)At(pi).

Now, dim~ imf = IFol - 1 (see the

constricted case in [BG] ). Also, C = II ka C_ ker ¢~. So dim~ ker h - dim~ im f = aer~ 1 - IF0[ + [F~I = X(F). Using the fact that img _C ker/, the result follows.

Ht andPresentationsof Finite Dimensional Algebras

33

The following result was proved in [BM] using weight functions. follows immediately from Theorem 2.1. COI~OLLARY 2.2. Let A = kF/I be a monomial algebra. only if F is a tree.

However, it

Then H1 (A) = 0 if and

An alternate proof of Theorem2.1 can be constructed using the fact that there is a monomorphismHom(~rl(r), -~ H~(A) for any pres entation ( se e lAP] [FGM],[PS] ) and that for monomialalgebras ~r~ is the free group on X(F) letters. Throughout the rest of this section we will assume that dim~ o(a)At(a) for each arrow a. Note that this implies the quiver has no loops and no parallel arrows. In addition, if a path p ¢ a in F is parallel to an arrow a then p E I. Let Der^o (A) denote the set of all derivations on A that fix Ao.That is, Der^o (A) = {5 eDer(A) : 6(v) = v for all v 6 Ao}. Before our next result we need following definition. DEFINITION2.1. (lAd], [FGM]) Let (F,I) be a bounded quiver. A relation ~ Aj’rj e I is called minimal if, for every proper subset L of J, we have ~ )~/t l~L

I. PP~OPOSITION 2.3. Let A be an algebra satisfying the aforementioned hypotheses. Then dirn~ DerAo (A) _< [F~ [. In addition, A is a monomialalgebra if and only if dimk Der^o (A) Proof. Since dimk o(a)At(a) = 1 for each arrow a, given d ~Der^0(A) and c~ ~ F1, d(c~) = Aaa for some Aa E k. From this it follows that Der^o(A) C_ krland dim~ DerAo(A)< IF1 I. It is also easy to see that if I is generated by monomials then all the elements of kr~ are derivations. Nowassume that A is not monomialand dim~ Der^o (A) = IF~[, i.e. Der^o (A) r~. k Let f~ = ~’~A~#i ~ [u,v], where n > 2,be a minimal non-monomialrelation. Since dim~ o(a)At(a) = 1 for each arrow a, we know there are no double arrows in the quiver. Let tt~ = aa~w~and #~ = ao~2w2with al ~ c~2. Let d be the derivation corresponding to the map ~fa~ .That is, d(a) = 6~a, (a~) for any arrow a ( here is the Kroneker delta ). Then (f~ (c~2) = 0. Moreover, (ia~ (f~) = ~ A~#~ Otl ~SuppD~

relation,

contradicting

the fact that/~ is minimal.

[]

The following is a generalization of Theorem4.1 from [BG]. THEOREM 2.4. Let A = kr/I be a connected algebra such that dimk o(a)At(a) for each arrow a. The following are equivalent: i)dimk Der^o (A) = IF~I ii) dim~ Hi(A) = X(F) iii) I is monomial. Proof. From the previous Proposition we have i ¢=~iii. To establish ii ¢==~iii, first assume that I =< p~ .... Pn > is monomial. Then we can compute H~(A) from the complex 0 --~A ~ B ~ C -~ ~ o(p~)ht(p~).

As before dim~ imf = Irol - 1.

34

Bardzell and Marcos

Note that g is the zero map and C = H ka = ker ¢~.. So dimk ker ¢~--dimk im ~ = 1 -IFol + Irll = x(r). To show the other direction,

assume that I is not a monomial algebra.

r -=- ~ A~p~be a non-monomial generator.

Let

Following the argument in the proof

of Theorem 4.1 from [BG], construct an arrow a that divides Pl but not all the other paths P2, ..., Pn. Then ~ ¢ ker ¢~ and it follows that dimk ker ¢~ < IF1 I- Since dim~¢ im¢~ = dimk~mf ----- IFol - 1, we have dim~ Hi(A) < x(F). Note that algebras satisfying the hypotheses of Theorem2.4 can be broken into two classes based on H~(A). All the monomial algebras satisfy dim~ and all the non-monomial algebras satisfy dimk H~(A) < X(F). From the proof Theorem2.4 we see that if there exists a non-monomialpresentation of A, then A is a non-monomialalgebra. This gives us the following result on presentations of this type of algebra. COROLLARY 2.5. IrA is a monomial algebra satisfying 2.~, then every presentation of A is monomial. 3

7~

the hypotheses of Theorem

AND PRESENTATIONS

In this section we will examine the first homotopygroup ~r~ and presentations of certain algebras. Let us first recall the definition of ~r~ and somerelated terminology. Assumethe quiver F is connected. DEFINITION3.1. For an arrow a : u ---+ v, denote by a-~ the formal inverse. A walk in F from u to v is a formal composition a~l...a~* where a~ ~ F~ and ei ~ {+1, -1}. Denote by eu the stationary path at u. DEFINITION 3.2. Define a homotopy relation ~0 on (F, I) to be the smallest equivalence relation on the set of all walks in F satisfying the following: i) For each arrow a : u ~ v in F we have aa-~ ,,~ eu and a-~a ii) For each minimalrelation ~ Ay),j ~ I we have 3’i ~" ")’j for all i, j ~ iii)

If p, q, w, and w’ are walks and p ~ q, then wpw’ ,,~ wqw’ whenever these products are defined.

DEFINITION 3.3. Fix a base vertex u ~ F. Then the group ~r~ (F, I) of all homotopy classes of closed walks which start and end at u is called the first homotopy group of (F, I). See fad] for ,nore details on ~r~. In [FGM]it is proved that one can take any set of minimal relations generating I to define the homotopy group. Note that ~r~ need not be an invariant of the algebra. That is, it is possible for two different presentations of the same algebra to produce two different homotopy groups. A triangular algebra is called simply connected if all presentations of the algebra give the trivial homotopygroup. It can be difficult to determine if a given algebra is simply connected since one has to check the vanishing of the homotopy groups on every presentation. So it. is

H~ andPresentationsof Finite Dimensional Algebras

35

important to describe some classes of algebras where the homotopy groups do not depend on a given presentation. Wewill show that this is the case for algebras that satisfy dimk o(a)At(a) = 1 for each arrow a E F1. This includes schurian algebras and therefore triangular algebras of finite representation type. Before we get to this result we first need some technical lemmas. Throughoutthis section let ~ denote c~ ÷ I for any c~ E kF. LEMMA 3.1. Let A = kF/I and choose any complete set of primitive idempotents {-~i, ...~n}. Then there is an invertible element # such that Ei = p-l~¢(i)p for some

¢(i) sn.

Proof. After reordering we can assume by Krull Schmidt that there is an isomorphism A~ ¢-~ A~i which takes ~i to Ei.So we get a left module automorphism A = HA~i ¢~-~’ IIAEi such that ¢(~i) = Ei. If we let # = ¢(1) then /z is vertible since A = A#. Since ¢(,~) = A/~ for all A ~ A and ¢ is an epimorphism, ~ = ~#. So A~i = A~#. Also, 1 = ~#-1~i# = ~ and P-l~i# ~ A~i. Thus, ~ =/z-l~#. [] LEMMA 3.2. Let h = kF/I have vertex set {~1, ...,~,,}. Let {#-1~#} be any other complete set of primitive idempotents. Then there is an isomorphism ¢ : kF ----r A such that ¢(vi) = #-1~i# and ker¢ = I. Proof. Let 7r : kF ---+ kF/I be the natural projection. Define ¢ : kF ---r A by ¢(7) = #-17#. Then ¢(vi) = #-1~i# and it is clear that 7 ~ kerr if and only [] 7 E ker ¢. The former Lemmahas a nice interpretation. Let A~ = kF/I and {e~, ..., en) be a complete set of primitive orthogonal idempotents. Wealways can assume that {e~, ...en) is the vertex set without changing the ideal I. COROLLARY 3.3. Let h = kF/I and assume dimk o(a)ht(a) <_ 1 for all a ~ F1. 5 : kF ~ A is any epimorphism, then there is another epimorphism ¢ : kF --~ A with ker¢ = kerJ and, for each a e F~, ¢(a) = c~a~ for some c~a ~ k. Proof. We can change J by ¢ and assume ¢(v) = ~ for each v ~ Fo. Since dim~ o(a)At(a) _< 1 for all a E F~, ¢(a) = a~. COROLLARY 3.4. Let A = kF/I and assume dimk o(a)ht(a) <_ 1 for all a ~ Let ¢ : kF ~ A be any epimorphism with ker ¢ = I’. Furthermore, let # = ~ ~i~, c~i ~ k\{O}, be a minimal relation. Then there is a minimal relation ~ ~ ker¢ with i=1

Proof. By Corollary 3.3 there is an isomorphism kF/I ~- kF/ker¢ which takes the class of any nonzero arrow a to a nonzero multiple of g. So 0 = ~ o~i~i goes to 0 = ~/~ where ¢(~) = ~i. Then ~/~i~Ti i=1

E ker¢ and, if it is not minimal, i=1

36

Bardzell and Marcos

then we can assume there

1 <_ j < n such that ~/~ = 0. This means ~ a~ = 0 i=1

and # is not minimal.

i::1

[]

THEOREM 3.5. Let (F, I) and (F, I’) be two presentations of the same. finite dimensional algebra A such that dimk o(a)At(a) <_ 1 for all a E F1. Then (F, I) = ~rl (F, I’). Note that this corollary tells us that ~rl of a schurian algebra A does not depend on the presentation of A. In fact we can even commenton binomial algebras. Recall that an ideal I is called binomial if it has a generating set consisting solely of monomials and k- linear combinations of two monomials. Wehave the following. Let A ~- kF/I;

COROLLARY 3.6. algebra. Then

~ kF/I2 be two presentations

of a schurian

i) I1 is binomial

ii) rl(r, I1) Proof, Let u, v E Fo and ~l, ~2 two classes of nonzero paths in kF/I; from u to v. Then B1 = A~ for some A ~ k since A is schurian. Hence I1 is binomial. Part 2 follows from the previous corollary since schurian algebras satisfy dimk o(a)At(a) 1 for all a ~ F. [] 4 H~(A)

AND CYCLES

In this section we consider the vanishing of H~ (A) and its relationship to whether or not the underlying quiver of an algebra has cycles. It was conjectured for awhile that H1 (A) = 0 implies that F has no oriented cycles. In [Ha] it is shown that the conjecture is true when I is homogeneous and char k = 0 and also when A is 3-nilpotent. In [BM], it is also shown true for monomial algebras ( see also Corollary 2.2 of this paper ). A counterexample to the general case has recently been given in [BL]. In this section we will construct some other cases where the conjecture still holds. In fact, we will be able to say something about quivers with undirected cycles also. Our approach is based on using the boundary maps from IBM] and considering the combinatorics of the quiver. Throughout this section assume that the characteristic of k is zero. As in section 2 we write A = IIkv, B = HvAv, C = IIka, D = IIo(a)At(a), and ¢~ = f $ g. Nowsuppose ro

Fo

r~

that A = kF/I where I is generated by the reduced grSbner basis ( see [FFG] R = {pl,...,pm,ri ..... rs}. Here pi,...,Pm are paths and rl ..... rs are non-monomial generators. Since we know the conjecture is true for monomial algebras, we will ¯ assume that A is not a monomialalgebra and that R contains at least one generator that is not a path. Of course, it is possible for there to be no paths in T/. Then H )At(p~) P~ = H o(r)At(r) = E ~ where E = =IIio ~=l° and F= ~= )At(r~) , . ~en ¢* So we may compute H~ (A) from the complex 0 ~ A (9 B ~ C ~ D ~ E ~[~ Note that ¢~ can send some elements of C ( D ) into E and other elements into

Ht andPresentationsof Finite Dimensional Algebras

37

PROPOSITION 4.1. Suppose that A = kF/I where F contains the cycle ( not necessarily directed ) v: ~ v2 ~ ""ve ~ v:. Let a = 6~i ~ f~ E P~ where i

5~ ~ {0,1} for i = 1,...e

and ~ ~ D. If ~ = 1 for at least one i,then

¢~. Proof. First note that ¢~ = f $ g maps elements of B to zero or linear combinations of radic~ squared elements. So we only need to consider ¢~(A) f( A). Let {a~+~, ..., ar~ } denote F~{a~, ...a~}. Then we have the following:

¢~(v~) = (-i)~,+~ $ (-i)~ A~ : ~hereA:,...,A~ ~ C arecontained in ~he~e+:,...,~r~ components and {0,I}.Thei~ aredetermined by ~heorientation of~hearrows. If~hearrow ~ ends a~ ~ ~hen: ~illhavepositive coefficien~ in ~(:~). If~hearrowa begins a~ v~ ~hen : willhavenegative coefficien~ in ~(:~).Nowle~~ = k:::~ ...@ k~:e.Suppose ~ (a)= a. Thenwe obtain~hefollowing systemof equations: = ~

-k~ + ka :

-k~_:+ k~ = ~_: - k~ = ~

k:

Adding we obtain 0 = ~Si.Since char k = 0 and we have assumed that 51.= 1 for at least one i,we have contradiction. Note that this proposition gives us a recipe for showing that certain undirected algebr~ have nonzero first Hochschild cohomology group. We simply look for elements of the form ~ that are in ker ¢~. The following generalization of Corollary 2.2 is one such application. COROLLARY 4.2. Let A = kF/I and suppose that F contains the cycle ( not necessarily directed ) v~~ v2 ~ ""re ~v~. If there exists some ai, i = 1,...,e, such that i) ai does not divide any support path in R or ii) ai divides only monomialrelations in R or iii)

for each relation

r = ~ a~pj ~ R , there exists a nonnegative integer mr j=l

such that ail pj exactly m~times for j = 1, ...,m~ then H~ (A) ~ Proof. If ai does not divide any support path in R,then ¢~ (~i) = 0. If a~ divides only monomial relations in R, then ¢~ sends ~i into the ai position of any path p~, j = 1, ...,m, that ai divides. So we just obtain scalar multiples of the paths

38

Bardzeil and Marcos

that are zero relations. Finally, if for each relation r = ~ ~jpj E R there exi..sts

a

j----1

nonnegative integer mr such that a~I pj exactly mr times for j = 1, ..., mr, then in the r-component of ¢2(~) E P~ we have ~_,mr~jp~ = m~r = 0. In each of these j=l

cases ~ E ker ¢~ and ~ ¢im¢~.

[]

REMARK: If one examines the proof of proposition 4.1 and corollary 4.2, one can see that the items i) and ii) of proposition 4.2 are valid in any characteristic. I~EFEI~ENCES lAd] Assem, I, de la Pefia, J.A, The Fundamental Group of a Triangular Algebra, Communications in Algebra 24 (1996) 187-208. JARS] Auslander, M, Reiten I, Smalo, S; Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. [BG] Bardzell, M.J., Green, E.L.; An Invariant Characterization of MonomialAlgebras; Communicationsin Algebra, 27(5), 2331-2344, 1999. IBM] Bardzell, M.J., E.N. Marcos, Induced Boundary Maps for the Cohomologyof Monomial and Auslander Algebras; Canadian Math Society Conference Proceedings, Volume24, 47-54, 1998. [BL] Buchweitz, tt.O, Liu, S; Artin Algebras with Loops but no Outer Derivations, Preprint 232, University of Sherbrooke (1999). [FFG] Farkas, D., Feustel, C., Green, E.L.; Synergy in the Theories of GrSbner Bases and Path Algebras, Canadian J. Math 45, 727-739 (1993). [FGM]Farkas, D, Green, E.L., Marcos, E.; Diagonalizabl e Derivations of Finite Dimensional Algebras II, Trabalhos do Departmento de Matemfitica, preprint I~T-MAT99-04 1999. [GS] Guil-Asensio, F, S~orin, M, The Automorphism Group and the Picard Group of a Monomial Algebra, Communications in Algebra 27 (1999) 857-887. [Ha] Happel D., Hochschild Cohomologyof Finite Dimensional Algebras, Springer Lecture Notes, 1404, 108-126, 1984. [PSI de la Pe~a, ~.A, Saorin, preprint 1999.

M, The First

Cohomology Group of an Algebra,

Tame tilted algebras ing components

with almost regular

GRZEGORZ BOBII~ISKI1 Faculty of Mathematics and Informatics, nicus University, ul. Chopina 12/18, 87-100, Torufi, Poland. email: [email protected]

connect-

Nicholas Coper-

ABSTRACT In the paper we classify the tame tilted algebras with almost regular connecting components in the terms of quivers and relations. Wealso use this classifications to describe the selfinjective algebras of Euclidean type which admit almost regular nonperiodic components in the Auslander-Reiten quiver. Throughoutthe paper K denotes a fixed algebraically closed field. By an algebra we will mean a finite dimensional basic associative K-algebra. Finally, a moduleis always a finite dimensional right module. A componentof the Auslander-Reiten quiver FAof an algebra A is called almost regular if it has only one nonstable ~-A-orbit which consists of exactly one vertex. It follows from the description of the Auslander-Reiten quiver of a tame tilted algebra presented in [3] that its connecting component cannot be stable. Our aim in this paper is to classify all representation-infinite tame tilted algebras with almost regular connecting components. Note that the field K is the unique representationfinite connected tilted algebra with almost regular connecting component. The paper is organized as follows. In Section 1 we formulate our main result, in Section 2 we present the proof of it and in Section 3 we use the main result to classify selfinjective algebras of Euclidean type whose Auslander-Reiten quivers admit almost regular nonperiodic components. 1

THE MAIN

RESULT

Before formulating the main result we introduce some families of algebras. For nonnegative integers u, v, w, t we denote by A(u, v, w, t) the following family 1Supportedby the Polish Scientific Grant KBN No. 2PO3A 012 14.

40

Bobidski

quivers

(,)

where each vertex denoted by a square can be replaced by a connected finite subquiver of the following infinite quiver

containing the vertex x, and the original vertex is then identified with the vertex x. Similarly, each vertex denoted by a circle can be replaced by a finite connected subquiver of the following quiver

containing the vertex y, and the original vertex is then identified with the vertex y. The integer u is the sum of p and the number of arrows in the quivers denoted by circles with a vertical bar in the picture (*) and the integer v is the sum of and the numberof arrows in the quivers denoted by circles with a horizontal bar in the same picture. Analogously, using r (respectively, s) and the quivers denoted squares with a vertical (respectively, horizontal) bar we define w (respectively, Similarly for positive integers u, v, w and t we denote by A’(u, v, w, t) the following family of quivers

E~ .....

a~a

e~ .....

e~e

TameTilted Algebraswith AlmostRegularConnectingComponents

41

where the meaning of circles and squares is the same as above. Wealso define the numbers u, v, w, t in the same way as before. Finally, using the same conventions we define the family A"(u,v,w,t), u,v >_ 2, w,t _> 1, as follows

Let u, v, w and t be positive integers. We define the family of algebras A(u, v, w, t) as the family of bound quiver algebras KA’/I’, where the quiver belongs to the family A’(u,v,w,t) and the ideal I’ is generated by ~pPl, ~1 "’" apal ... as - [~ ... ~qPl "’" Pr and all elements of the form c~e, ¢¢. Similarly, for positive integers u, v, w, t, B(u, v, w, t) will denote the family of bound quiver algebras KA’/J’, where A’ belongs to the family A’(u, v, w, t) and the generators of J’ are ~qa~, oo ... o~ppl - ~ ... )~qPl , o~pp~... Pr - o~pa~... as and all elements of the form (~i~, f~j~, 5p~, 5al, ¢¢. For u,v >_ 2 and w,t _> 1 the family of algebras of the form KA"/I", where A" belongs to A"(u,v, w, t) and I" is generated by (~1 ""~p + f~l’" "f~q +71")’2, o~ppl, ~qO’l, 72Pl "" "Pr- 720"1"" "as and all elements of the form aie, ~i e, 5pk, 5at, ¢¢, will be denoted by C(u, v, w, t ). Finally, for nonnegative integers u, w, v, t we have the family D(u, v, w, t) consisting of the algebras of the form KA/I, with A belonging to A(u, v, w, t) and the generators I being O/1 "’" O/p~l¢lP 1 "’" Pr -- ]~1 "’" ~q/]4~4al "’" as, ?~1~1 -- ?~2~2, ~3~3 -- ~4~’4 and all elements of the form c~ie, B~e, 5pk, 5at, The main result of the paper is the following.

THEOREM 1. Let A be a representation-infinite connected tame tilted algebra with almost regular connecting component. Then A or A°p belongs to one of the families A(u,v,w,t), u,v,w,t >_ 1, B(u,v,w,t), u,w,v,t >_ 1, (u,w,v+t-1) (2 ,2,n-2), n _> 4, (2,3,3), (2,3,4), (2,4,3), (2,3,5), (2,5,3), C(u,v,w,t), u,v >_ 2, w,t >_ (u+w-l,v+t-1) (2,n-2), n >_4, (3, 3), (3, 4), (3, D(u,v,w, t), u,v,w,t >_ O, or to one of the families 1-14 defined below (each algebra from the families 1-1~ is described as a boundquiver algebra with relations listed to the right of a quiver).

Family 1.

42

Bobifiski

Family 2.

Family 3.

Family 4. 07 - ~6a

~a - 0c3

Family 5.

~c - A0¢

Family 6.

A~ - #0

Tame Tilted AlgebraswithAlmostRegularConnecting Components

Family7.

Family8.

#0 - we

#0 - w Family9.

"~ "~-’"X"

43

44 Family 10.

Family 11.

Bobidski

TameTilted Algebraswith AlmostR~gularConnectingComponents

45

F~mily 12.

¯ ~- ¯ -g- ¯

~r] - A0~

Family 13.

Family 14.

~7 - #06

~3- r~ac~

#7 - vO

46

Bobi~ski

REMARK. It follows from the proof of the above theorem presented in the next section that the algebras from the families 14 and A(u,v,w,t), u,v,w,t >_ B(u,v,w,t), u,v,w,t >_ 1, (u,w,v + 1) = ( 2, 2, n- 2), n _>4, (2, 3,3), (2,3,4 (2,4,3), (2,3,5), (2,5,3), C(u,v,w,t), u,v >_ 2, w,t >_ 1, (u + w- 1,v + 1) (2,n - 2), n _> 4, (3,3), (3,4), (3,5), and their opposite algebras are all algebras of extended Euclidean types with almost regular connecting components. 2

PROOF

OF THE MAIN RESULT

Let A be a connected representation-infinite tame tilted algebra with almost regular connecting component. Denote by X the unique projective-injective A-module. It is well-known that rad X and X/soc X are indecomposable A-modules and we have

TameTilted Algebraswith AlmostRegularConnectingComponents

47

the Auslander-Reiten sequence of the form 0 --~ rad X ---+ X ~ rad X~ soc X ~ X/soc X --+ O. Let a and b be vertices of the ordinary quiver QA of A such that X = PA(a) IA (b). If we denote by B1 the full subcategory of A formed by all objects except a, and by B2 the full subcategory of A formed by all objects except b, then A = B1 [rad X] = IX~ soc X]B2. Moreover, if B denotes the full subcategory of A formed by all objects except a and b, then B1 = [rad X/soc X]B and B2 = B[rad X/soc X]. It follows also that B1 and B2 are representation-infinite tilted algebras of Euclidean type, and B is a product of tilted algebras of Dynkintype. Wewill call B1 the left end algebra of A. Similarly, B~ will be called the right end algebra of A. Thus our objective is to study the following situation. Let B be a product of tilted algebras of Dynkin type and R a B-module. Weare asking when B[R] and [RIB are representation-infinite tilted algebras of Euclidean type. Wewill also denote by A the algebra

B D ) 0 where multiplication is the usual multiplication of matrices up to rule r ¯ ~ = ~o(r) for any r E R and ~ E D(R). In our investigations we shall use vector space category methods. Details on vector space and subspace categories Can be found in [5] and [6]. The facts necessary to follow the below considerations can be also found in [2]. Wehave to consider different cases which may occur. First assume that B[R] is a tilted algebra of type ~m, ra >_ 1. Then B has to be a tilted algebra of type Amand according to [2, Proposition 3.5] we get that A belongs to some family A(u,w,v,t), u,v,w,t >_ 1, u +v+w+t=ra + Assumenowthat B[R] is a tilted algebra of type ~n, n _> 4. Again the case when B is a tilted algebra of type Dn, n _> 4, has been studied in [2] and it follows from [2, Proposition 3.5] that A or A°p has to be one of the algebras from the families B(2,v,2,t), v,t_> 1, v + t- 1 = n- 2, C(2,v,l,t), v _> 2, E 1, v +t- 1 = n-2. Hence we have to consider the case when B is a product of at least two tilted algebras of Dynkin type. It follows from [2, Lemma3.3] that we can start from the situation when the unique projective-injective A-moduleis sincere. Then A is a tilted algebra of type E, where E is obtained from the following quiver

k _> 0, by orienting edges. Hence, according to [4, Theorem1], A is of the form KQ(p,q,r,s)/I(p,q,r,s), p,q,r,s 0, p+q+r +s = n- 4, where Q(p,q,r,s) is

48

BobRiski

the following quiver

and the ideal I(p, q, r, s) is generated by the relations al" "" o~p~/l~pl ""Pr -/~ "’"/~q~14~4al’"a,,

~/1¢1 - ~/2¢’2, ~/3~3 - ~/4~4.

If we omit the assumption that the unique projective-injective A-module is sincere, then the knowledge of modules lying on the mouths of tubes in FAo, where Ao is one of the algebras above, leads to the conclusion that A is one of the algebras from the families D(u, v, w, t), u, v, w, > 0, u +v + w + t = n - 4. Let now B[R] be a tilted algebra of type ~,6. There are possible three situations: B can be a product of three tilted algebras of type A2, B can be a product of a tilted algebra of type A5 and a tilted algebra of type A1, and finally B can be a tilted algebra of type Es. In the first case the vector space category Hom(R, mod B) has to be the following category _ *=Homv(R,Z1) _ o=Hom.~(R,Zu) o--~om~(~,X2) _

¢~=Hom~(R,Za)

$~OmB

where R = X~ 6~ X~ $ X~. following category

Similarly

the dual category

Horn(rood B, R) is the

¯ =Hom~(X, ¯ ,=HomB (Y’, ~,=Homo

(X2,R)

¯=Hom~(Xs,R) HenceXI, X2, X3, ZI, Z2, Z3 axe injectiveB-modules and it followsthat A has to be the uniquealgebrafrom familyI. In the secondcase,when B is a productof two tiltedalgebrasof ~ypesA4 and A1, the vector space category Horn(R, mod B) is a full subcategory of the following category o=HomB(R,X~) ~ ~Hom~

Hom~ (-~,X~)=*

/

’

(R,Zt)

TameTilted Algebraswith AlmostRegularConnectingComponents

49

where R = X1 $ X2. Since the algebra B[R] is representation-infinite it follows that Horn(R, mod B) has to contain the objects HomB(R,X1), HomB(R, Horns(R, Z2), Hom~(R,Z3). Dually, the vector space category Horn(rood B, a full subcategory of the following category

¯ =HomB (X1 ,R) ¯ =Hom~ (YI,R)

¯

~"

~

and has to contain the objects Hom~ (X~, R), Horns (Y~, HomB(Y2,R), HomB(Y3,R). Since the functions f1,]2,]~ : (rB)0 -~ Z given f~(X) := dimKHoms(Y~,X) for i = 1, 2, 3 are additive on FB and take nonnegative values it follows that the modules Z~, Z~ and Zs are injective. Of course, the module X1 is also injective. Hence, the possible configurations (up to symmetry) of indecomposable injective B-modules are the following

X~

X~

X~

and dimKHomB(R, I) = 1 for each indecomposable injective B-module. Thus, easily follows that A or A°p is one of the algebras from the family 2. The last case of B being a tilted algebra of type IE6 has been studied in [2] and hence according to [2, Proposition 3.5] we get one of the algebras from the families B(2,v,3,t), v,t >_ 1, v + $- 1 = 3, C(u,v,w,~), u,v >_ 2, w,t >_ 1, (u + w - 1,v + t - 1) = (3,3), or their opposite algebras. Analogous considerations as above conducted in cases whenB[R] is tilted of type ~ or ~s give us the families 3-14 and B(u,v,w,t), u,v,w,t >_ 1, (u,w,v+t- 1) (2,3,4), (2,4,3), (2,3,5), (2,5,3), C(u,v,w,t), u,v >_ 2, w,t >_ 1, (u +w1, v+ t - 1) = (3, 4), (3, 5). Here we only list for each family the type B[R] andB, and if A is a tilted algebra of extended Euclidean type then also the type of A. If the algebra B is not connected then we list the types of blocks.

50

Bobidski

Family Family 3 Family 4 Family 5 Family 6 B(u,v,w,t), u,v,w,t > 1, (u,w,v + t- 1)= (2,3,4), (2,4,3) C(u,v,w,t), u,v >_ 2, w,t >_ (u+w- 1,v+t- 1) = (3,4) Family 7 Family 8 Family 9 Family 10 Family 11 Family 12 Family 13 Family 14 B(u,v,w,t), u,v,w,t >_ (u,w,v÷t- i) = (2,3,5), (2,5,3) C(u,v,w,t), u,v >_ 2, w,t >_ (u+w- 1,v+t- 1) = (3,5) 3

APPLICATION

Type of B A3, A3, A1

Type of

B[R] Type of

A

A7

As, A2 De, A1

K~ K~

~7

As, A2, A1 As AT, A1 ~)s A4, A4 Ds, A3 E6, A2

K8

Ks K8

KS

~8

TO SELFINJECTIVE

ALGEBRAS

An algebra A is called selfinjective if each projective A-moduleis injective. An important class of selfinjective algebras is formed by the selfinjective algebras of Euclidean type, that is algebras of the form [~/G, where/} is the repetitive category of a tilted algebra B of Euclidean type and G is an admissible (infinite cyclic) group of K-linear automorphismsof/} (for definitions of notions presented in this section we refer to [2], [7] and [9]). Wemay even assume that B is a domestic tubular extension of a tame concealed algebra. Precisely, for each tilted algebra B of Euclidean type there exists a domestic tubular extension B’ of a tame concealed algebra such that /} ~_ /}’. The analogous fact holds for domestic tubular coextensions of tame concealed algebras. It has been proved by Skowrofiski in [7] that a connected selfinjective algebra which admits a simply connected Galois covering is of domestic representation type if and only if it is a selfinjective algebra of Euclidean type. The connection between the Auslander-Reiten quivers F~/a of J~/G and of ~ described in [7] and the reflection procedure of constructing the repetitive category for domestic tubular extensions of tame concealed algebras investigated in [1] allow to classify selfinjective algebras of Euclidean type whose AuslanderReiten quivers admit almost regular nonperiodic components. Namely, we have the following theorem (see [2, Section 4] for arguments). THEOREM 2. Let A be a selfinjective algebra of Euclidean type. The Auslander-Reiten quiver F A of A admits an almost regular nonperiodic component if and

TameTilted Algebras with AlmostRegularConnectingComponents

51

only if A ~ [3/G, where B is the left end (respectively, right end) algebra of representation-infinite tame tilted algebra with almost regular connecting component and G is an admissible group of K-linear automorphisms of [3. Following [8] a subquiver C of FAis called generalized standard if for any two modules X and Y in C the infinite radical rad°°(X, Y) is zero. Wehave the following consequences of the above theorem and [9, Theorem 5.5, Corollary 5.6] (compare also [2, Theorems 2 and 3]). In the below corollaries ~’t} denotes the Nakayama automorphism. COROLLARY 3. Let A be a connected selfinjective algebra. The following conditions are equivalent. (i) A is of Euclidean type, F A has at least two nonperiodic components, and at least one of them is almost regular. (ii) F A admits an almost regular nonperiodic componentand a generalized standard left stable full translation subquiver of Euclidean type which is closed under predecessors in (iii) FA admits an almost regular nonperiodic componentand a generalized standard right stable full translation subquiver of Euclidean type which is closed under successors in F A. (iv) A _~/~/(~ouB), where B is the left end (respectively, right end) algebra of representation-infinite tame tilted algeb~va with almost regular connecting component and ~ is a positive automorphism of B. COROLLARY 4. Let A be a connected selfinjective algebra. The following conditions are equivalent. (i) A is of Euclidean type, F A has at least three nonperiodic components, and at least one of them is almost regular. (ii) A is tame, F A has at least one generalized standard almost regular nonperiodic component. (iii) FA contains a nonperiodic componentC such that A/ ann C is a representation-infinite tame tilted algebra with almost regular connecting component (iv) A =/~/(~ou/~), where B is the left end (respectively, right end) algebra of representation-infinite tame tilted algebra with almost regular connecting component and ¢p is a strictly positive automorphismof The arguments needed to prove the above results are similar to the ones presented in the proof of the mainresults of [2]. In [2] one can also find a characterization of selfinjective algebras of Euclidean type whose all nonperiodic components are almost regular. REFERENCES [1] I. Assem,J. Nehring and A. Skowrofiski, Domestic trivial extensions of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72. [2] G. Bobiriski and A. Skowrofiski, Selfinjective algebras of Euclidean type with almost regular nonperiodic Auslander-Reiten components, preprint, Torurl, 1999. [3] O. Kerner, Tilting wild algebras, J. LondonMath. Soc. 39 (1989), 29-47.

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Bobifiski

[4] J.A. de la Pefia, The families of two-parametric tame algebras with sincere directing modules, Canad. Math. Soc. Conf. Proc. 14 (1993), 361-392. [5] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Note.,~ in Math. 1099, Springer, 1984. [6] D. Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications 4, Gordon and Breach Science Publishers, 1992. [7] A. Skowrorlski, Selfinjective (1989), 177-199.

algebras of polynomial growth, Math. Ann. 285

[8] A. Skowrofiski, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), 517-543. [9] A. Skowrofiski and K. Yamagata, Galois coverings of selfinjective algebras by repetitive algebras, Trans. Amer. Math. Soc. 351 (1999), 715-734.

Reflexive

modules are not closed

under submodules

GABRIELLA D’ESTEDipartimento di Matematica, Universit~ di Milano, via Saldini 50, 20133 Milano, Italy, email: [email protected]

ABSTRACT Weshow that the two classes of reflexive modules with respect to a cotilting bimodule fail to be closed under submodules. More precisely, we show that any generalized Kroaecker algebra A of infinite dimension has the following property: AAA is a cotilting bimodule, and any faithful module Msuch that M is reflexive with respect to AAAhas a non reflexive socle. 1

INTRODUCTION

The first remark of Colpi in his paper [C] on cotiiting bimodules and their dualities says the following: "The main difference between our and Colby’s setting is that we are not assumingthe further hypothesis that the class of reflexive modulesis closed under submodules". The example presented in this note shows that the situation studied by Colpi in [C] is muchmore general than that considered by Colby in [Cbl] and [Cb2] for several reasons, concerning the shape and the size of both the rings and the A-reflexive modulesinvolved. (Before we recall all the useful definitions, we point out that A-reflexive module means W-reflexive module, in the sense of [AF], with respect to a cotilting bimodule W.) In the following we construct a cotilting bimodule RWssuch that even the most obvious A-reflexive left R-modules (resp. right S-modules) [AF, Propositions 20.13, 20.14 and Corollary 20.16], namely the summands of both RR and ~W(resp. Ss and Ws), may have a submodule which is not A-reflexive. More precisely, given any infinite cardinal d, we construct an algebra A of dimension d over an algebraically closed field K, with the following properties: " AAA

is a cotilting

bimodule (Lemma2.2).

54

D’Este Both the classes of A-reflexive modules fail to be closed under submodules (Theorem2.5 (ii)).

In our example it actually occurs that the A-reflexive modules are as few as possible, i.e. coincide with the finitely generated projectives modules(Lemma2.3). Moreover, the A-reflexive modules admitting only A-reflexive submodules are as small as possible, i.e. coincide with the A-reflexive modules of finite dimension over K (Lemma2.4). Before we describe the last part of the paper, we recall some definitions, and we fix the notation used in the sequel. First of all, we say that a left (resp. right) moduleWover a ring R is cotilting mo dule [CDT1] ifW satisfies the following conditions: 1. inj dimR(W)_< 2. Ext~(W~, W) = 0 for any cardinal 3. Ker HomR(-, W) V~ Ker Extla( -, W) = O. Next, we say that a faithfully balanced bimodule ~Ws is a cotilting bimodule [C] if both ~Wand Ws are cotilting modules. As usually, for any ring A, we denote by A-Mod(resp. Mod-A)the category of all left (resp. right) A-modules. Moreover, given a cotilting bimodule I~Ws, we simply denote by A both the contravariant functors Homn(-, W) : R Mod -~ Mod- S, Horns (-, W) :

Mod- S -~ R - Mod

In the following, for any left R-module(resp. right S-module) M, the evaluation morphism ~M: M -+ A2(M) is defined by the formula (~M(X))(~) ---- ~(x) x E M and ~ E A(M). If 5Mis an isomorphism, i.e. if Mis W-reflexive in the sense of [AF], we say that Mis A-reflexive. Finally, we simply denote by F both the contravariant functors Ext,(-,

W) : R Mod -- + Mod - S, Ext,(-, W)

: Mod- S -- > R - Mo

With this notation, we point out other surprising properties of our example. First of all, even indecomposable left (resp. right) modules with a very easy structure belong to Ker A2NKerF~. Indeed, for any positive integer n, we exhibit (Theorem 2.5 (iv)) an indecomposable module M, of dimension n overK, such that ¯ A(M) = ¯ F(M) and AF(M) are free modules of uncountable rank. As we shall see, these modules M are of the form X/Y, where X is an indecomposable projective module, hence a A-reflexive module, and Y is a semisimple projective module which is not A-reflexive. Hence, by just dealing with algebras and cotilting bimodules of infinite but countable dimension, even a simple module Mwhich is countably presented does not admit an exact sequence of the form

(+)

0 -~ r~(M)M -+ AS(M) -~0,

Reflexive Modules

55

Werecall that, by the Cotilting Theorem proved by Colpi [C, Theorem 6], every module which is the quotient of two A-reflexive modules admits an exact sequence as in (+). Wealso recall that the results obtained by Tonolo in [T] explain the relationships amongthe functors F2, A2 and the identity functor, that is the three functors in (+). More precisely, by IT, Theorem1.2], a derived functor has a "key role to relate" these three functors. Secondly, using a A-reflexive module whose socle is not A-reflexive, that is a A-reflexive modulewhich is not finitely cogenerated, we construct (Theorem 2.5 Off)) infinitely many pairwise non-isomorphic indecomposable modules X such that ¯ X is isomorphic to F2(X); * A(X) = 0 and X is the quotient of two indecomposable A-reflexive modules. Consequently, even a cotilting bimodule admitting only finitely manyindecomposable A-reflexive modules may admit infinitely many indecomposable F-reflexive modules in the sense of [C]. For a new homologicai definition of F-reflexive modules, we refer to [T], where Tonolo addressed and solved the problem of a good notion of F-reflexive modules with respect to the so-called weakly cotilting bimodules. Finally, the cotilting bimodule of our example suggests that the asymmetry between the dualities induced by A and F [C, Theorem 6] and IT, Corollary 2.9] does not depend only on how many indecomposable modules are involved. Indeed, the behaviour of F (resp. 2) i s a s b ad ( resp. a s g ood) as p ossible o n a ll f initely generated modules belonging to Ker A which are not finitely presented (see (a) (c) in Corollary 2.8). As a partial symmetry between A and F, we show that countably generated modules Mwhich are A-reflexive (resp. such that A(M) and Mis isomorphic to F2(M)) are just the finitely presented modules (Lemma 2.3; Corollaries 2.8 and 2.10). However,also by looking at finitely presented modules and by dealing with a regular cotilting bimodule AAASUChthat there exists an isomorphism f : A -~ A°p, the functor F seems to act as a kind of concealed reflection. More precisely, the functor F used in our example acts in an easy and geometric way on infinitely manyquotients of an indecomposable A-reflexive module (Remark 2.11 (a)). However, in the same example the action of F is much complicated even on the quotient of an indecomposable A-reflexive module with respect to a two-dimensionai A-reflexive submodule (Remark 2.11 (b)). 2

PROOFS

AND REMARKS

Throughout the paper, we always assume that K is an algebraically closed field and that d is an infinite cardinal. Moreover,we say that A is the generalized Kronecker algebra of dimension d over K (compare with [HU, page 182]) if A is the K-algebra given by the quiver depicted in Figure 1, where the arrows, say aj, from 1 to 2 are

56

D’Este

indexed by a set J of cardinality d. Hence, following the terminology of [R], A is the one-point extension of K by a vector space V of dimension d over K (i.e. A is

isomorphic

to

VK ’

v b and v E V, subject to the usual addition and multiplication of matrices). Finally, given a generalized Kronecker algebra A, we denote by el (resp. e2) the priraitive idempotent of A corresponding to the vertex 1 (resp. 2), and we denote by P, /5, 0 the following indecomposable modules: P = Ael,

Q = Ae2, /5 = elA,

O = e2A

Keepingall this notation, we recall some properties of direct products of projective modules used in the sequel. LEMMA 2.1. Let A be the generalized Kronecker algebra of infinite dimension d, let P (resp. O) be the indecomposablefaithful projective left (resp. right) A-module. If m is an infinite cardinal, then the following facts hold: (i)

pm (resp. m) admits a decomposition of the form X @ Y, where X is isomorphic to the direct sum of [Km[ copies of P (resp. O) and Y is semisimple projective module of dimension IAml over K.

(ii) pm (resp. m) i s f ree i f a nd only i f [ Km[ > _ d. Proof. The proof of [D, Lemmas2.1 and 2.2] shows that the left A-module P’~ satisfies (i) and (ii). On the other hand, there is an isomorphism °p f : A -4 A satisfying el ~ e2, e2 ~ el and (~j ~-~ c~j for any arrow a1 from 1 to 2. Since 0 is the right A-module obtained by means of f from the right A°P-module P [J, page 26], a dual argument shows that also the right A-module0m satisfies (i) and (ii). The next lemma shows that the modules P and 0 are subsPaces of codimension one of a cotilting bimodule. LEMMA 2.2. Let A be a generalized Kronecker algebra of infinite dimension. Then A is coherent and perfect on both sides and AA~ is a cotilting bimodule. Proof. An argument similar to that used in the first part of [CDT1, Example 5.3 (c)] (see also the correction [CDT2]of the last part of (c)) shows that A is coherent and left perfect and that P is a cotilting module. Since P is a summand of AA, we obviously have Ker HomA(-, AA) f)KerExt~A( -, AA) = O. Since A is left hereditary, hand, Lemma2.1 and a dual and that both 0 and AA are this implies that AAA is a

it follows that AA is a cotilting module. On the other proof show that A is left coherent and right perfect, cotilting modules. Since AAAis faithfully balanced, cotilting bimodule. []

It is easy to see that the regular bimodule constructed in Lemma2.2 admits as few as possible A-reflexive modules.

Reflexive Modules

57

LEMMA 2.3. Let A be a generalized Kronecker algebra of infinite dimension, and let M be an A-module. Then the following conditions are equivalent: (i) Mis projective and finitely

generated.

(ii) M is A-reflexive with respect to AAA. Proof. (i) ~ (ii). This is well-known[AF, Proposition 20.13 and Corollary 20.16]. (ii) ~ (i). Since A is hereditary on both sides, we deduce from Lemma2.2 and [C, Lemma2 (b)] that A is semiperfect on both sides and that any A-reflexive A-module is projective. By the characterization of projective modules over semiperfect rings [AF, Theorem27.11], this implies that any A-reflexive A-module is a direct sum of indecomposable projective modules. Moreover, we clearly have A(P) _/5 and A(Q) - ~. It follows that (1) A interchanges indecomposable faithful jective modules.

projective

modules and simple pro-

To end the proof, let L be an indecomposableprojective module, let m be aninfinite cardinal, and let X denote the direct sum of m copies of L. Then we clearly have (2) A(X) _~ ’n. Assume first L is simple. Then, putting (1) and (2) together, we deduce Lemma2.1 that A(X) has a non-zero free summand. Consequently, also A2(X) has a non-zero free summand, and so X is not isomorphic to A~(X). This means that (3) A semisimple projective A-moduleof infinite

dimension is not A-reflexive.

Suppose now L is faithful. Then, by (1), (2) and an obvious remark (see [D, Remark 2.3 (iii)]), A(X) is a semisimple projective module of uncountable dimension over K. This observation and (3) imply that A(X) is not A-reflexive. Therefore, by [AF, Proposition 20.14 (3)], X is not A-reflexive. Thus any reflexive moduleis finitely generated. This result completes the proof of the lemma. As the following lemmashows, the property of admitting only A-reflexive submodules maybe very restrictive. LEMMA 2.4. Let A be a generalized Kronecker algebra of infinite dimension, and let M be a A-reflexive module with respect to AAA. Then the following conditions are equivalent: (a) Every submodule L of M is A-reflexive. (b) dimKMis finite. (c) M is artinian. (d) Mis finitely

generated semisimple.

Proof. By Lemma2.3, it suffices to note that every faithful projective modulehas an infinite dimensional socle. []

58

D’Este

We are now ready to prove THEOREM 2.5. Let A be the gene~’alized Kronecker algebra of infinite dimension d over K. Then AAAiS a cotilting bimodule with the following properties: (i) There are only finitely morphism.

many indecomposable A-reflexive

modules, up to iso-

(ii) Both the classes of A-reflexive modules are not closed under submodules. (iii) There are infinitely many pairwise non-isomorphic indecomposable A-modules X such that A(X) = 0 and F2(X) is isomorphic to X.

6.)

For every cardinal c such that 1 < c < d, there is an indecomposable cyclic A-module Y such that diml~Y = c, A(Y) = 0 and F(Y) is a free module of uncountable rank.

Proof. Wefirst note that P and Q (resp. /5 and (~) are the only indecomposable projective left (resp. right) A-modules, up to isomorphism. Consequently, (i) (ii) follow from Lemmas2.2, 2.3 and 2.4. To prove (iii), fix an arrow aj from 1 to 2, and let X denote the left (resp. right) A-module P/Aaj (resp. O,/a~A). Then we clearly have A(X) = 0 and Ac~j _~ (resp. ~iA _~ /5). Hence, either a direct calculation, or an application of [C, Theorem6] shows that l~2(X) is isomorphic to X. Since the annihilator of X is the subspace generated by aj, it follows that the modules P/A~I (resp. O,/c~jA) are pairwise non-isomorphic. Thus also (iii) holds. Finally, take a cardinal c such that 1 < c < d. Since the arrows a1 with j E J are a base of socP and [JI = d, we can fix a subspace L of soc P such that L is generated by d arrows and dimKP/L = c. Next, let i : L -+ P denote the canonical inclusion, and let Y denote the module P/L. Then there is an exact sequence in Mod-Aof the form 0 ~ A(P) ~-~ A(L)

r(Y)--+

0.

(1)

For brevity, let F denote the module A(L), and let V1 and V~ denote the subspaces Fel and Fe2 respectively. Since L is isomorphic to a direct sum of d copies of Q, we have F _~ ~a. By Lemma2.1, this implies that F = A(L) is a free module of rank [Kd[.

(2)

Let T denote the submodule of F generated by V2, that is let T = V~A. Then T is a summandof F. Wealso note that F=TSU for

anysubspace

U of

V~

(3)

such that Vx = Tex (~ U. Fix any t E T. Since T = V2A = (Fe~)A, there exist finitely many elements fl,"" ,fn E F and al,’" ,an ~ e2A such that we may write t = f~al +...+f~an. Nowlet Wdenote the K-vector space generated by the subset {a~,... , a~}. Then Wis a left ideal of A of finite dimension over K. Hence our hypotheses on t and the

ReflexiveMffdules

59

/

struc -ure [AF, Proposition 4.4] of the right th~at/t(L) ~_ W.It follows that

A-module F = Homa(L, aAa) imply

dim/~ t(L) is finite for any t E T.

(4)

On the other hand, let g : L ~ A denote the canonical inclusion. Since g(x) = x xel for any x E L, we obtain g = gel e VI = Fel and dim/~ g(L) =

(5)

Putting (4) and (5) together, we conclude that the subspace of F generated g, which coincides with ImA(i), is a subspace of V~ such that T f~ ImA(i) = Therefore, we may choose a (semisimple) module U containing Im A(i) such that F has a decomposition of the form F = T $ U as in (3). Thus we deduce from (1) and (2) that F(Y) is a free right module of rank IKal, as claimed in (iv). assumptions on L guarantee that L is also a submodule of 0 with the property that dim/¢ OIL = c and A((~/L) = 0. Hence a dual proof shows that F(0/L) free left module of rank IKd[. This remark completes the proof of (iv). Before we point out an application of the previous results, concerning A-reflexive modules and generalized linearly compact modules, we recall some definitions and results. Following the terminology of [CF] suggested by [GGW],given a cotilting module nW, we say that a left R-module M is W-torsionless linearly compact if M ~KerF and, for any inverse system of morphisms {Px : M -+ Mx} with M~~ Ker F and Cokerp~ ~ Ker A for all A’s, we have Coker (~_~p~,) E Ker A. Let us recall two facts used in the sequel concerning A-reflexive modulesand torsionless linearly compact modules with respect to a cotilting bimodule W(see also [Mii] and [X, Theorem 4.1]). ¯ Every W-torsionless linearly compact module is A-reflexive [C, Proposition 10]. ¯ A A-reflexive module Mis W-torsionless linearly compactif and only if every submodule of A(M)is A-reflexive [CF, Theorem 1.8]. Surprisingly enough, a very easy property, namely having only infinite dimensional indecomposable summands, may characterize the A-reflexive modules which are torsionless linearly compact. COROLLARY 2.6. Let A be a generalized Kronecker algebra of infinite dimension, and let M be a A-reflexive module with respect to AAA. Then the following conditions are equivalent: (a) M is A-torsionless linearly compact. (b) Every indecomposable summandof M is faithful. (c) M does not have an indecomposable summandof finite

dimension over

Proof. By the proof of Lemma2.3, any A-reflexive left (resp. right) A-module M is isomorphic to pr @ QS (resp. r ~/ss) f or s ome natural n umbers r and s. Consequently, we have A(M) ~_ /5r ~ 0s (resp. A(M) _~ ps) . Moreover,

60

D’Este

by Lemma2.4, every submodule of A(M) is A-reflexive if and only if socA(M) is A-reflexive, that is if and only if s -- 0. This remark and the characterization of torsionless linearly compact modules given in [CF, Theorem 1.8] c6mplete the proof of the corollary. [] REMARK 2.7. Given a cotilting bimodule RWs, we know from [M, Proposition 1.6] that a nice property, i.e. the property that AF(M)= 0 for any left R-module M, implies that the class of A-reflexive S-modules is closed under submodules. On the other hand, F induces a duality between the modules X such that A(X) = and X is the quotient of two A-reflexive modules [C, Proposition 5 (d); Theorem 6 (c)]. By Lemma2.3, this implies that (,) ExtlA( ., A) induces a duality between the finitely presented modules belonging to Ker HornA(., A) for any generalized Kronecker algebra A of infinite dimension. Moreover, by condition (iv) of Theorem 2.5, a module in the image of AF may be extremely big. The next corollary shows that the behaviour of F and AF on 2. certain finitely generated modules is quite different from that of F COROLLARY 2.8. Let AAA be the regular cotilting bimodule over a generalized Kronecker algebra of infinite dimension, and let Mbe a finitely generated A-module such that A(M) = O. Then the following facts hold: (a} F(M)is finitely

generated if and only if Mis finitely

(b) If M is not finitely of uncountable rank. (c} Fg(M)is finitely

presented,

presented.

then F(M) and AP(M)have a free summand

presented.

Proof. Let Mbe a non-zero left A-module as in the hypotheses. Then there is an exact sequence of the form (1)

0--~

L -~

P" ~ M ~ 0,

where n is a positive integer and L is a su~moduleof socPn. AssumeMis finitely presented. Then the results of [C] mentioned in Remark2.7 guarantee that F(M) finitely presented. Nowsuppose Mis not finitely presented. Then L is isomorphic to the direct sum of infinitely many copies of Q. Hence, by (1) and Lemma2.1, there is an exact sequence of the form (2)

0 ~ A(Pn) -~ A(L) ---+

P(M) --+

where A(L) is a .projective module admitting a free summandof uncountable rank. Since A(Pn) _’2 pn, this implies that F(M) has a decomposition of the form X (3 with the following properties: (3) X is finitely

presented and A(X)

(4) ~ i s a projective mo dule ad mitting a

fr ee su mmand ofuncountable ran k.

Reflexive Modules

61

Hence we deduce from (4) that F(M) is not finitely generated, and that AF(M) has a free summandof uncountable rank. Thus (a) and (b) hold for any module satisfying the hypotheses of the corollary. Moreover, by (3) and (4), obviously have F2(M) _~ F(X). This remark and (a) guarantee that F2(M) finitely presented, as claimed in (c). This completes the proof for anyfinitely generated left A-module belonging to Ker A. A dual argument shows that (a), (b) and (c) hold also for any finitely generated right A-module belonging to Ker The corollary is proved. [] REMARK 2.9. As in Corollary 2.8, let A be a generalized Kronecker algebra, and let A and F be the contravariant functors induced by the cotilting bimodule AAA. Then the structure of projective A-modules (see the proof of Lemma2.3) guarantees that A carries finitely generated modules to finitely generated modules. This observation and Lemma2.3 imply that (*) AA and AA are finitely

cotilting

modules and Colby-modules in the sense of

[An]. Hence, by (*) and [An, Remark4.5], the existence of a finitely generated A-module Msuch that F(M) is not finitely generated (Theorem 2.5 (iv), Corollary 2.8 follows also from the fact that A is neither left nor right noetherian. The next partial result gives some information on the images under F and F~ of the non-finitely generated modules belonging to Ker A. COROLLARY 2.10. Let A be the generalized Kronecker algebra of infinite dimension d over K, and let M be an A-module such that A(M)= 0 and M is not finitely generated. If m is the smallest cardinality of a set of generators of M, then the following facts hold: (a) Any set of generators of F(M) has at least IKm[ elements. (b) Either F2(M)is finitely Proof. Let Mbe a left sequence of the form

generated, or FZ(M)is not countably generated.

A-module as in the hypotheses.

Then there is an exact

O ---> L -L~ L’ ---~ M ---~ 0,

(1)

where L’ is isomorphic to the direct sum ofm copies of P, while L is a submoduleof soc L’. Let l = dimKL. Since Mdoes not have a non-zero projective summandand the dimension of soc L’ is equal to din, it follows that l _> m. Hence L is isomorphic to the direct sum of infinitely many copies of Q. Thus, by (1) and Lemma2.1, there is an exact sequence in Mod-Aof the form (2)

0 ---+ A(L’) -~ A(L) ---~ F(M)

where A(L) has a free of A(L)el, it follows Therefore (a) holds. X @ Y, where A(X) =

(3)

summandof rank equal to IK~I. Since Im A(i) is a submodule that any set of generators of F(M)has a least ~] el ements. On the other hand, F(M) has a decomposition of the 0 and Y is projective. Thus we obviously have

r~(M)_~ r(x).

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D’Este

Assumefirst X is finitely antee that

presented. This hypothesis and (.) in Remark2.7 guar-

F(X) is finitely

(4)

presented.

Suppose now X is finitely generated, but not finitely (b) of Corollary 2.8 guarantees that (5)

presented.

Then condition

F(X) has a free summandof uncountable rank.

Suppose finally X is not finitely generated. In this case, by simply rePlacing Mby X in the first part of this proof, we see that (6)

F(X) is not countably generated.

By combining(3), (4), (5) and (6), we obtain (b). The proof is

[]

Weend with a remark on the behaviour of F on infinitely many very special finitely presented modules defined over a generalized Kronecker algebra. REMAP~K 2.11. Let A be a generalized Kronecker algebra of infinite dimension. As in the proof of Lemma2.1, let f : A ~ A"p be the isomorphism such that f(el) = e2, f(e~) = andf(aj ) = aj for a ny a rrowaj fro m 1to 2. SinceA-M °p (resp. A°~-Mod) in an obvious way [J, (resp. Mod-A)is isomorphic to Mod-A page 26], we may use f to obtain an isomorphism M ~ M’ between A-Mod and Mod-A(resp. Mod-Aand A-Mod). For any arrow aj, and let X denote the module P/Aaj (resp. ~,]ajA). Next, let i : Q ~ P (resp. i :/5 _.+ 0) denote the right (resp. left) multiplication by aj. Then there is an exact sequence of the form

0 a(P) a(Q) --, r(x) (resp. 0 --* a(O) a(P) r(x) Thus P(X) is isomorphic to Coker A(i), and so P(X) has a base of the {v,v~[i ~ j} such that ve2 = v, vaj = 0, vai = vi (resp. ely = v, ajv = O, air = vi) for any i ~ j. This means that F(X) is isomorphic to XL Hence, condition (iii) of Theorem2.5, we conclude that

(a) r(M)~_ M’ for

infinitely

many indecomposable modules M such that M ~_

r~(M) and A(M) = Finally, fix two different arrows aj and at, and let Y denote the module P/(Aaj Aat) (resp. O/(ajA ~ atA)). Then we obviously have (1) dim~¢ Y’ez = 1 (resp. dimK e~Y~ = 1). Now, let i : Q~Q-~ P (resp. i :/5~/5_~ ~) beamorphism such tlhat Irai is the submodule of P (resp. (~) generated aj andat. Thenwe have an ex act sequence of the form 0 -~ a(e) (resp.

~ a(Q ¯ Q) --~ r0")

0 ---+ A(Q) ~ A(P ~/5) ---+ P(Y)

Consequently, P(Y) is isomorphic to Coker A(i), and so we clearly

Reflexive Modules (2) dimK F(Y)e2 = 2 (resp.

63 dim~: elF(Y)

Therefore, by (1) and (2), F(Y) is not isomorphic to Y’. This remark and another application of [C, Theorem6] guarantee that (b) F(M) ;~ M’ for infinitely r2(M) and A(M) =

many indecomposable modules M such that M

ACKNOWLEDGEMENTS I wouldlike to thank the referee for his suggestions "to improve the readability" of my paper. He also pointed out that we may proceed as in tl~e proof of Lemma2.2 to obtain the following more general result: "Every ring A which is hereditary and perfect on both sides is a cotilting bimodule". In fact, by a well-known result of Chase [Theorem 3.3, Trans. Amer. Math. Soc. 97 (1960), 457-473], both AA and AA are product-complete modules in the sense of Krause-Saorin [Theorem 3.8, Proceedings of the Seattle Conference]. I take the opportunity to mention that Professor K.R. Fuller made a similar remark at the Ohio Algebra Conference (Athens, March 1999), during his conversations dualities with R. Colpi, F. Mantese, E. Gregorio, A. Tonolo and myself. REFERENCES [AF] ANDERSON F.W. - FULLERK.R., Rings ’and categories ed. GTM13, Springer-Verlag (1992). [An] ANGELERIHOGELL., Finitely (2000), 2147-2172.

cotilting

modules, Comm.Algebra 28 (4)

[Cbl] COLBY R.R., A generalization of Morita duality Comm.Algebra 17 (7) (1989), 1709-1722. [Cb2] COLBY R.P~., A cotilting M. Dekker (1993), 33-37.

of modules, 2nd

and the tilting

theorem,

theorem for rings, Methods in Module Theory 140,

[C] COLPIR., Cotilting bimodules and their dualities, ference Proceedings 1998, M. Dekker.

to appear in Murcia Con-

[CDT1] COLPIR. - D’ESTEG. - TONOLO A., Quasi-tilting equivalences, J. Algebra 191 (1997), 461-494.

modules and counter

[CDT2] COLPI R. - D’ESTE G. - TONOLOA., Corrigendum, (1998), 370-370. [CF] COLPIR. - FULLERK.R., Cotilting 192 (2) (2000), 275-291.

J. Algebra 206

modules and bimodules, Pacific J. Math.

[D] D’ESTEG., Free modules obtained by means of infinite appear in Ohio Conference Proceedings.

direct products,

to

[GGW] G~)MEZ PARDOJ.L. - GUIL ASENSIO P.A. - WISBAUERR., Morita dualities induced by the M-dual functors, Comm.Algebra 22 (1994) 5903-5934.

64

D’Este

[HU] HAPPEL D. - UNGEI~L., A family of infinite dimensional non self-extending bricks for wild hereditary algebras, CMSConference P~oceedings 1]L4 (1991), 181-189. [J]

JACOBSON N., Basic Algebra H, W.H. Freeman and C., San Francisco (1980).

F., Hereditary cotilting [M] MANTESE

modules, J. Algebra, to appear.

[Mti] MOLLEP~ B.J., Linear compactness and Morita duality, J. Algebra :16 (1970), 60-66. [R] RINGELC.M., Tame algebras 1099 (1984).

and integral

A., Generalizing Morita duality: IT] TONOLO bra, to appear. IX] XUEW., Rings with Morita duality,

quadratic

forms, Springer LMN

a homological approach, J. Alge-

Springer LMN11523 (1992).

Fibre sum functors

and the bimodule Ext

PETERDR~XLER Fakult~it fiir Mathematik, Universit~it D-33501 Bielefeld, Germany

Bielefeld,

POBox100131,

ABSTRACT Representations of the bimodule Ext,(-,-) and fibre sum functors both provide techniques for the investigation of modulecategories for finitedimensional algebras. Weclarify the relation between these two constructions.

1

INTRODUCTION

It is a classical technique in the representation theory of finite-dimensional algebras to consider the A-modules as extensions of modules over smaller subcategories :~ and T thus identifying the module category with the representation category of a bimodule Ext,(-,-) acting on :~ × T. If T = addS for a simple module S, then the category of representations of this bimodule can be identified with the subspace category of a vector space category (see [Rill). In [Drl] the fibre sum functor with respect to a module P is introduced. We will also use properties of the fibre sum construction which were established in [Dr2]. The fibre sum functor relates the category of A-modules with the category of representations of the bimodule HomA(-,-) acting on 7" × ~ for appropriate subcategories 7", ~. In case the endomorphism algebra of P is a field or more specially if P is a simple module, this leads to a vector space category as well. The aim of this note is to analyse the relation between these two reduction processes to vector space categories. After recalling the first reduction concept in the next section, in the final section we will clarify the relation completely. For simplicity we consider only finite-dimensional algebras over an algebraically closed field k which we assume to be basic. Weuse the term algebra for this concept. For notation and background we refer to [GR] and [Ri2].

66 2 REDUCTION

Dr§xler TO THE BIMODULE

Ext,(-,-)

2.1 Let us start out by recalling the concept of a bimodule. Following [GR] an aggregate is a k-additive category with finite-dimensional morphismspaces ~,~uch that each object is the direct sum of subobjects with local endomorphismalgebras. A typical example for an aggregate is the category A-modof finit~-tli]aaensional (left) modules over an algebra A bimodule over two aggregates ~- and 7" is a k-linear bifunctor H: .T × 7" -4 k-mod which is covariant in the second and contravariant in the first argument. The category rep(H) of representations of H has as objects the triples (X, h, where X E ~’, Y E T and h ~ H(X, Y). A morphism from (X, h, Y) to (X’, h~ is a pair (s,t) of morphisms s: X -4 X* in 9v and t: Y -4 Y’ in 7- such that H(X, t)(h) = H(s, Y’)(h’). The category rep(H) is again an aggregate. 2.2 The classical example for studying the category A-mod of an algebra A by representations of a bimodule is the following: Let ~" and T. be two subaggregates of A-mod and G : A-mod -4 A-mod a subfunctor of the identity functor such that G(X) ~ T and X/G(X) fo r all X in A-mod. We co nsi der the b imodule H = Ext,(-, -) acting on ~" x T and obtain a full functor R : A-rood -~ rep(H) by mapping the module X to its canonical exact sequence:

o -4 a(x) x x/a(x) For illustration we mention that the zero object in rep(H) is the triple (X, h, with X = Y = 0. In general, the functor R is neither dense nor faithful. Its kernel consists of the morphisms f : X -4 Y which factorise as ] =, ~yg~rx. Since this kernel is contained in the Jacobson radical of the aggregate A-mod, we see that A-modis representation equivalent to its image category inside rep(H). Thus have ’reduced’ the study of A-modto the study of its image. It turns out that in manycases representations of a bimodule are easier to handle than the module category itself. Let us provide someexamplesfor choices of ~’, 7- and G. Wefirst start with a full subaggregate T of A-mod. For an A-module X we define G(X) = GT(X) as the trace of 7- in X i.e. the sum over all images f(Z) for all Z in T and f ~ HOmA (Z, X). For 9v we choose a full subaggregate of A-mod containing all modules X/G(X). Another class of examples arises by considering an ideal I of A. Wetake 7- as the subcategory of A/J-modules and ~" as the subcategory of A/Imodules of A-rood where J is the left annihilator of I in A. The functor G = G~ is defined as GI(X) = IX. Dually, we can take T = A/I-mod, ~ = A/J-mod, and IG = G as the annihilator of I in X where J is the right annihilator of I in A (this means J = GI(I)). Note, that the lack of density of R is repaired if one assumes that the :pair (7", v) i s a torsion th eory orin other words G is a r adical sub functor of theidentity functor i.e. G(X/G(X)) for all A-mo dules X. I f i n a ddi tion to ( T, ~’) bein a torsion theory one also assumes that HomA (~’, T) = 0 or that (~’, 7") is also torsion theory, then R is faithful. Being a torsion class forces 7" to be extension closed. Therefore, if T = add S, then Ext~ (S, S) =

Fibre SumFunctors

67

2.3 Wewill look in more detail a the situation that H : bimodule such that T is an aggregate of the shape 7" = add S for some object S of T. Additionlly we assume T(S, S) ~ k. In this case rep(H) can be rewritten subspace category. Let us recall the relevant notation. A vector space category is a k-additive functor M: ~" ~ k-mod where ~" is an aggregate. Its subspace category/~(M) is an aggregate which has as objects the triples U = (U~,~/u, Uo) where U~ e k-mod, U0 e 9c and ~/u e Homk(U~,M(Uo)). MorphismsU ~ U’ in this category are pairs f = (f~, f0) such that f~ : U~ ~ U’~ is k-lineax, ]0:U0 -~ U~ is a morphismin 3c and M(fo)~/v If now H is a bimodule such that T = addS and T(S,S) ~- k as considered above, then H is completely determined by the contravariant functor M:= H(-,S): ~" --~ k-rood which we consider as covariant functor ~-op __~ k-rood. Moreover, any object of rep(H) lying in H(X, ’~) may b y t he Yoneda l emma be identified with a morphismin Hom.r(s,s)(T(S ’~, S),H(X, S)). This identification yields an antiequivalence rep(H) -~/~(M). Dually, one can transform rep(H) into a subspace category if ~" = add S. This happens e.g. for G~ introduced above for an ideal I satisfying A/J ~- k. This is used in [GNRSV]where the considered functor R usually will not be dense, but the image is calculated precisely. 3

FIBRE

SUM FUNCTORS

3.1 Let (T, v) be a to rsion th eory in A-m od. We define K:7 - to be thefull subaggregate of A-modwhose objects axe the modules V satisfying Ext,(V,7-) = The bimodule L acts on T × K: as HomA(--,--). Weput rep,non(L ) to be the full subcategory of rep(L) given by all monomorphismsh : X --~ V. Then the definition of K:T implies that the functor ~ : rep,~on(L) -~ A-modwhich sends h to its cokernelis full. As a torsion class 7" is closed under factor modules. Let us assume that 7" has a cover P i.e. T = fac P for some module P in A-mod. Then we can even assume that P is a minimal cover and therefore Ext~ (P, fac P) = 0. It follows from JAR, 1.4] (see also [Dr2, 2.2] that ~ is dense. The category rep,non(L ) seems to be haxd to understand. To improve the situation we assume additionally that also Ext,(P, sub P) = 0. Let C be the bimodule which acts as HomA(-,-) on addP x K:7-. Then the functor ¯ : rep(C) repmon(L) which sends f : U --~ V to the inclusion of Imf into V is full and dense. Altogether we have derived the following result from [Dr2] where we put Fp = which is said to be the fibre sum functor with respect to P. PROPOSITION. Suppose the S-module P satisfies Exth(P, subP) = 0 and Ext~ (P, fac P) = O. Then the fibre sum functor Fp is full and dense. It is calculated factoring through shownin [Dr3, 2.5] phism only finitely

in [Dr2, 1.4] that the kernel of Fp consists of the morphisms an object f : U -> V in rep(C) which is an epimorphism. there are interesting cases such that there exist up to isomormanyindecomposable objects of this shape in rep(C).

68

Dr~xler

3.2 Nowwe consider the special case that EndA(P) -~ k. Then we can identify rep(C) with/2(N) where N is the functor HomA(P,-) acting on /C7-. Namely, homomorphismf in HomA(P®k knl V) is mapped to its adjoint homomorphism ’~, HomA in Homk(k (P, V)). If in addition P = S is a simple module, then two functors relating A-rood with the subspace category of a vector space category were introduced namely Fs : /)(N) -~ A-mod and the antiequivalence R : A-mod -~ /)(M) from previous section. It is our final aim to calculate the composition RFs. Note, that R is already an equivalence whereas Fp in general cannot be an equivalence because Fp(k,O, O) = 0 = F(k, idk, S). Erasing these two objects makes also Fs into an equivalence. More precisely, we consider the full subcategory I(~- of whose objects do not admit a summandisomorphic to S and denote by N’ the restriction of N to K~-. Considering only/2(N’) excludes direct summandsof the form (k, idk, S). Unfortunately, it cancels also direct summandsof the form (0, 0, but, as we will see below, this is the price we have to pay to get satisfactory results. Furthermore, we replace/)(N’) by its full subcategory//(N’) having only objects U such that 7v is a monomorphism. In this way we get rid of direct summandsof the shape (k, 0, 0). Nowthe restriction Fs :/J(N’) ~ A-rood’ is equivalence where A-mod’ is the full subaggregate of modules X admitting now direct summand S. On the other hand, R maps A-mod’ onto/~(M) °p. Thus RFs becomes an antiequivalence L/(N’) -~/~(M). The best that can happen for such an anti equivalence of subspace categories is that it is induced by a suitable antiequivalence of the corresponding vector space categories. Wefirst provide an equivalence ]C~- -~ ~. LEMMA.Let G’ be the/unctor an equivalence IC~ -~ ~.

sending X in A-mod to X/G(X). Then G’ induces

Proof. For given V in K:’ the adjoint of the inclusion of G(V) into V is an object U = (k n, 7u, V) satisfying Fs(U) = G’(V). Therefore the density and fullness of Fs implies the required density and fullness of G’. That G’. acts faithfully on follows easily because there do not exist non-zero maps from/C~r to S. But note that G’(S) = 0. Here it pays out that we passed from K:7- to K:~ r. Furthermore we need an natural isomorphism between the involved functors. usual D = Homk(-, k) denotes the dual space functor.

As

LEMMA. There is a natural ~somorphism %0 : DHomA(S,-) -~ Ext~(G’(-),S) functors R:’7- -+ k-mod. Proof. Let us consider V in/C~-. The canonical exact sequence

of

o a(v)

v c’(v)

induces an exact sequence HomA(~rv ,S))

HomA(G’(V),

HomA (V, S) Extl (G’ (Y),

HomA(~v,S),~

HomA(G(V),S)

E~t~(~v,S)~

Ext~(Y, S) =

where HomA(~ry,S) is actually an isomorphism because HomA(V,S)ev = O. Hence we obtain an isomorphism HomA(G(V), ~- Ext~4(G’(V), S). G(V)is in T =

Fibre SumFunctors

69

addS, we obtain D HomA(G(V), ~- HomA(S, G(V)). Finally, Hom A(S, G(V)) ~HomA(S,V) by the definition of G. The composition of all these isomorphisms yields the desired natural isomorphism ~o. [] Using the two lemmas above and calculating

RFs we obtain:

THEOREM. If P = S is a simple projective A-module, then the functor 14(N~) --~ Lt( M) sending an object U=(U~, 7u, Uo) (D ~, ~vo-1 D ~,~,G~ (Uo) is an antiequivalence which is isomorphic to RFs. REFERENCES

[AR]

M. Auslander, I. Reiten, Applications of contravariantly finite gories, Adv. Math. 86 (1991), 111-152.

[C-B]

W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. (3) 56 (1988), 451-483.

[Drl]

P. Dr~Lxler, Lt-Fasersummenin darstellungsendlichen Algebren, J. Algebra 113 (1988), 430-437.

[Dr2]

P. DrLxler, On the density of fiber sum functors, Math. Z. 216 (1994), 645-656.

[Dr3]

P. DrLxler, Generalized one-point extensions, 645-667.

[Ddl]

Y.A. Drozd, Matrix problems and categories of matrices, Zap. Nauchn. Sem. LOMI28 (1972), 144-153

[Dd2]

Ju. A. Drozd: Tame and wild matrix problems, Lecture Notes in Math. 832 (1980), 242-258.

subcate-

Math. Ann. 304 (1996),

[GNRSV] P. Gabriel, L.A. Nazarova, A.V. Roiter, V.V. Sergejchuk, D. Vossieck, Tame and wild subspace problems, Ukr. Math. J. 45 (1993), 313-352. P. Gabriel, A.V. Roiter, Representations of finite-dimensional algebras, Encyclopedia of the Mathematical Sciences, Vol. 73, Algebra VIII, A.I. Kostrikin and I.V. Shafarevich (Eds.), Berlin, Heidelberg, NewYork, 1992.

[NR]

L.A. Nazarova, A.V. Roiter, Kategorielle Matrizenprobleme und die Brauer-Thrall-Vermutung, Mitt. Math. Sem. Giessen 115 (1975).

Jail]

C.M. Ringel, Report on the Brauer-Thrall conjectures, Math. 831 (1980), 104-136.

[Ri2]

C.M. Ringel, Tamealgebras and integral quadratic forms, Lecture Notes in Math. 1099 (1984).

Lecture Notes in

Smooth Automorphism

Group Schemes

DANIELR. FAR.KASDepartment of Mathematics, Virginia Polytechnic and State University, Blacksburg, VA24061

Institute

CHRISTOF GEISS Instituto de Matem~ticas, UNAM,Ciuadad Universitaria, 04510, Mexico D. F., Mexico

EDUARDO N. MAP~COS Departamento de Matem~itica, S~o Paulo, CP 66281, S~o Paulo, SP05389.970, Brasil

I.M.E.,

Universidade

C.P.

de

ABSTRACT Smoothness for the automorphism group scheme of a finite-dimensional algebra in positive characteristic can be interpreted as a property of the Hopf algebra representing the scheme. With this approach, it is proved that the scheme is smoothif and only if all derivations of the original finite-dimensional algebra are integrable. This criterion is applied to commutative monomialalgebras and used, as well, to establish a general Morita invariance theorem. The most naive way to understand a finite-dimensional associative algebra is to find a basis and analyze its multiplication table. In the modern incarnation, one considers the schemeof all associative n-dimensional algebras over the field k as a n ® kn, k’~). Then GL,~(k) acts on the k-rational subschemeof affine space Homk(k points of the scheme so that orbits can be interpreted as isomorphism classes of n-dimensional algebras. The stabilizer of a point can be identified with the automorphismgroup (scheme) of the corresponding algebra. The geometry at a point seems to behave particularly well when the automorphism group scheme is smooth. For example, Gabriel ([Ga], 2.4) proves that if the algebra A corresponds to the point # then 2 (A) i s i somorphic to the tangent space at ~ in the entire scheme modulo the tangent space at # in its GLn-orbit. This result requires smoothnessof the stabilizer, as first explicitly 71

72

Farkas, Geiss, andMarcos

pointed out in [Maz]. The automorphism group scheme is automatically smooth when char k = 0 by the classical characterization of cocommutative connected Hopf algebras. The situation in positive characteristic has been more mysterious. The main contribution of this paper is to provide a simple, user-friendly reformulation of smoothness. We prove that the automorphism group scheme of A is smooth if and only if every k-derivation of A is integrable. Here we mean that D is integrable if it is a member of a sequence of k-endomorphisms of A, D(°) = I, D(1) = D, D(2),D(3),... such that D(’n)(ab) = ~ D(i)(a)D(J)(b) i-bj=rn

for all a, b E A. The notion of integrability (which also appears in the literature under the name "higher derivations") is far from new although we believe that this application is novel. The proof of our criterion is essentially Hopf algebraic and found in the first section. The second section reviews knownproperties of integrable derivations. It also includes a generalization of the well known fact that a derivation of a finitedimensional algebra over a field of characteristic zero sends the Jacobson radical into itself. Next, a particular class of examples is studied. Using our criterion, we present a clumsy but algorithmically tractable description of those commutative monomial algebras whose automorphism group scheme is smooth. Weobtain both expected results (e.g., smoothness follows when relations "avoid" the characteristic) and bizarre examples. In the fourth and last section, we prove that the property of having a smooth automorphism group scheme is a Morita invariant. Indeed, it is shown more generally that integrable derivations contribute to a Morita invariant piece of the first Hochschild cohomology group. 1

SWEEDLER’S

THEOREM

Webegin by deriving a transparent, intrinsic condition on a finite-dimensional algebra which is equivalent to its having a smooth automorphism group scheme. Twodifferent proofs are presented. The first is a leisurely algebraic exposition which depends on classical Hopf algebra constructions. The second proof is short and geometric. This time the real work is hidden in several standard lemmas. Until further notice, we let H denote a commutativeaffine Hopf algebra over the field k. If H represents an afflne group scheme then the scheme is smooth precisely when H is reduced, i.e., when H has no nonzero nilpotent elements. It is well knownthat H is always reduced when chark = 0 ([WaD. In case the characteristic of k is positive and k is perfect, Sweedler ([Sw]) has found a characterization reduced Hopf algebras which we wish to apply. This result depends on the analysis of a certain k-coalgebra, the hyperalgebra, associated with H. Let eg be the augmentation map for H and let A~ be its kernel. Hyp(H) the subcoalgebra of the dual H° consisting of all linear functionals which vanish on

SmoothAutomorphismGroupSchemes

73

some power of A4. It is possible to prove that Hyp(H)is the irreducible component of H° containing e H ([Abe], p.198). Suppose C is a coalgebra with counit ec. Given d E C, an infinite sequence of divided powers lying over d is a sequence do, dl, d~,.., of elements in C such that Adn = ~ di ® dn-i for all n , ec(d,~) = 0 for n > 0 , and co(do) = i:-O

with dl = d. Equivalently, we mayregard an infinite sequence of divided powers in C as a coalgebra morphism from the coalgebra of divided powers B = kx(°) + kx(~) +... to C. Nowsuppose do, d~,.., is an infinite sequence of divided powers in Hyp(H). Notice that d0(1) = 1 and dn(1) = 0 for n > 0. Since do must be group-like, have do = ell. Moreover, consider any infinite sequence of divided powersdo, d~,... in H° such that do = ell. If a, b E A4 then d~ (ab) = do(a)dl (b) + d~ (a)do(b) Continuing by induction, we see that dn(A4n+~) = 0. Hence the sequence lies in Hyp(H). Thus we may identify the collection of all infinite divided powers in ttyp(H) with the subgroup (under convolution) of coalg(B, °) consisting o f those ~ with a(x(°)) = ell. Recall that an element a in a bialgebra is primitive when Aa = 1 ® a + a ® 1. Since 1 in H° is identified with ell, we see that any term dl belonging to an infinite sequence of divided powers in Hyp(H)must be primitive. (In this context, a linear functional d E H° with d(ab) = eH(a)d(b) + eH(b)d(a) for all a, b e H is also called an e-derivation.) THEOREM 1.1 ([Sw]). Assume H is an affine commutative Hopf algebra over a perfect field k of positive characteristic. Then H is reduced if and only if there is an infinite sequence of divided powers in Hyp(H)lying over each primitive element. In order to apply this theorem when H represents the automorphism group scheme of a finite-dimensional k-algebra A, we need to interpret infinite sequences of divided powers intrinsically for A. This will be done in a series of steps which are more or less standard. Webegin by reminding the reader that if B is the kcoalgebra of divided powers then B* can be identified with k[[t]] by sending f ~ B in" to E f(x(n)) LEMMA 1.1. coalg(B, °) _~ a lg(H, B *) as g roups. Proof. We have the obvious group homomorphism ¯ : coalg(B, H°) -~ alg(H, B*) given by ~(c~)(h)(b) a(b)(h) for h e H and b ~ B.We fir st arg ue tha t ¯ i s surjective. Indeed, let 0 6 alg(H, B*). Let 7rm : H --r k[[t]]/(t m) be the composition

74

Farkas,Geiss, an~dMarcos

of 0 with the obvious projection. The kernel of this algebra mapis a two-sided ideal I,~ of cofinite dimension in H. Let c,~ be the linear functional in H* which sends h E H to the coefficient of t m in 0(h). Then lm+i _CKercm, whence CmE °. I f a ~ coalg(B, °) i s d efined b y a(x (m)) =cmthe n ¢(a ) = 8 Next, we compute the kernel of ¢, {a : B ~ H° [ ~(a)

= ~B*eH}

where r/ denotes the unit. Since ~(a)(h) = ~a(x(m))(h)t. "~ we see that if a ~ Ker~ then a(x (°))=ell and a(x (’~))=0 for m_>l. Wehave described the identity

element for coalg(B, H°).

[]

For the remainder of this section, we shall assume that H represents the automorphism group scheme Aura of A. That is, if R is any commutative k-algebra then Aura (R) = alg(H, Of course, AurA(k) = Autk(A). The action of this automorphism group on A be described via an H-comodulealgebra structure on A: there is a coaction A:A-~ making A a left

H®A

H-comodule so that A(ab) = Ea(0)b(0) ® a(1)b(l)

and A(1)

for a, b ~ A. (See [Mo], section 4.1.) The explicit AUtk(A) sends ~7 to the automorphism ~ where

isomorphism from alg(H, k)

If R is any commutative k-algebra then the group AurA(R) is isomorphic AutR(R ®k A) under the extension of the comodule algebra action to R ® H R®A. Weare particularly interested in the case that R = B*. Since A is finitedimensional, we have B* ® A - k[[t]] ® A _~ A[[t]] . Again, since A is finite-dimensional, a k[[t]]-automorphism of A[[t]] is determined by its effect on elements of A. A k-algebra map ~ : A -~ A[[t]] is a higher derivation of A provided that for all a ~ A, the constant term of the power series 6(a), is simply a. Clearly, higher derivations of A are in one-to-one correspondencewith k[[t]]-automorphisms of A[[t]] which "preserve constant terms". Alternatively, we mayregard a higher derivation as a sequence of linear endomorphismsof A, say D(°) = I, D(~), D(z),..., such that D(n)(ab)

= E D(i)(a)D(~)(b) i+j=n

for all

c~A.)

a,b E A and n > O. (The point is to expand ~(c) Y’~.~=oD(n)(c)tn for

SmoothAutomorphismGroupSchemes

75

LEMMA 1.2. There is a one-to-one correspondence between infinite sequences of divided powers in Hyp(H) and higher derivations of A. The map sends eH = do,d1,.., to I = D(°),D(1),... where DCm)(a) = E d~n(a(o))a(1) for all a E A. Proof. By virtue of the previous lemmaand our discussion so far, there is a group isomorphism eoalg(B, g°) ~ Autk[[t]l(A[[t]]) The isomorphism sends a ~ coalg(B, °) t o t he a utomorphism

n. n i+j=n

In particular, this automorphismsends c ~ A to ~n[~ tn" a(x(n))(C(o))C(~)] Note that if a(x(°)) =eHthen the associated automorphismpreserves constants. Conversely, we argue that if f = a(x (°)) E H° and ~f(c(o))CO) for all c e A (i.e., the automorphismpreserves constants) then f = ell. But f is group-like, f ~ alg(H, k). The claim follows from our isomorphism alg(H, "~ Autk (A) The lemmais now a consequence of restricting the group isomorphism to coalgebra maps from B to H° which send x(°) to ell. [] It is easy to see that the D(1)-term of a higher derivation is always an ordinary derivation. Wesay that a derivation D ~ Derk (A) is integrable provided there exists a higher derivation D(°) = I, D(1), D(2),... such that (1) =D. THEOREM 1.2. Let A be a finite-dimensional k-algebra and assume that H represents the affine group scheme AurA. Every e-derivation of H has an infinite sequence of divided powers in Hyp(H)lying over it if and only if every derivation of A is integrable. Proof. It is easy to check directly that if d is an e-derivation of H then the linear endomorphism D of A given by D(a) = E d(a(o))a(~) is a derivation. It is well knownthat this mapfrom e-derivations to Derk (A) is isomorphism ([WaD. (This can also be seen by replacing k[[t]] with kit]It 2 in the arguments we have just presented.) Apply the lemma. [] If the characteristic of k is zero then it is a well knownconsequence of the Leibniz rule (see [Hu], p.8) that any derivation D can be integrated to the higher derivation 1 2 I,D,~D ,...,o.n, ~ ~D .... Thus integrability of derivations is only an issue whenchar k > 0, which brings us back to Sweedler’s Theorem. Wesummarize our discussion for this section.

76

Farkas, Geiss, and Marcos

COROLLARY 1.1. Assume that A is a finite-dimensional algebra over the perfect field k. The aj~ine group scheme AutA is smooth if and only if every k-derivation of A is integrable. Gerstenhaber observes (see [GS]) that if the second Hochschild cohomology group H2(A, A) vanishes then all derivations of A are integrable. Though this only a sufficient condition, it does suggest that a detailed study of H2 might pinpoint the precise obstruction. As promised, we outline a second proof. Again, assume that H is an affine cocommutative Hopf algebra over k with augmentation e. Set ~I = H/rad(H), so .~ is reduced. The tangent space at unity for the algebraic group schemeassociated to H can be described by using "dual numbers", which we identify with k[[t]]/(t~): TH = {a e alg(H,k[[t]]/(t

2)

l a(h) =

e(h) (mod t) for all h e H}

It is easy to see that a 6 THif and only if a - e is an e-derivation of H. Thus when H represents the automorphismgroup scheme of A, our earlier discussion identifies TH with Derk(A). The quotient map p : H -~ ~ induces an injective k-linear map dp* : T~I-+ TH by sending fl to f~ o p. The key technical lemma we need (see [WaD is that H is reduced if and only if dp* is an isomorphism. Our characterization can now be expressed in the following form. THEOREM 1.3. Assume that k is a perfect field. Then H is reduced if and only if for each ~ e THthere exists a k-algebra map& : H ~ k[[t]] with 2) 5(h)=a(h)

(modt

for all h 6 H. Proof. First suppose that each c~ 6 TH can be lifted to ~. Then ~ (rad(H)) because k[[t]] is an integral domain. Hence a (tad(H)) = 0. This says that a factors through a map in 7-~. Conversely, assume H is reduced. The local ring HKer(,) is regular, so the Cohen Structure Theoremimplies its completion C is isomorphic to a power series algebra k[[tl,... ,t,~]] with m the Krull dimension of H. (It is at this point that we use the assumption that k is perfect.) The natural ring homomorphism~ : H -÷ induces an isomorphism d~?* : Tc "4 TH. Suppose a 6 TH. Choose 7 6 We such that a = 7 o ~. Certainly 7 lifts to some ~. Then a lifts to ~ o ~. [] It is clear from the second argument that the restriction to perfect fields is only required for one direction. In general, if all derivations are integrable then the automorphism group scheme is smooth. Here is an application: any finitedimensional hereditary algebra has a smooth automorphism group scheme because its second Hochschild cohomologygroup vanishes ([Ha]). Wewill give an alternative proof at the end of the paper.

SmoothAutomorphismGroupSchemes 2

INTEGRABLE

77

DERIVATIONS

A derivation of a finite-dimensional algebra whose scalar field has characteristic zero always sends the radical into itself. Weshall see that integrable derivations extend this behavior. LEMMA 2.1. If S is a semiprime ring then so is S[[t]]. Proof. Wemust show that if a e S[[t]] is nonzero then ~S[[t]]a ~ O. But ass ~ O, as can be seen by looking at the lowest term of a. [] As a consequence, if R is any ring and I is a nilpotent ideal of R[[t]] then I C_ (prime rad (R))[[t]]. THEOREM 2.1. Let A be a finite-dimensional algebra. If the algebra map ¢: A -+ A[[t]] is a higher derivation then ¢(radA) C_ (radA)[[t]]. Proof. It suffices to prove that ¢(rad A) lies in a nilpotent ideal of A_[[t]]. Let denote the algebra generated by ¢(A) and t. (We do not ask that A be closed.) The condition that ¢(a) = a+"higher terms" implies that .~ is dense in A[[t]] with respect to the (t)-adic topology. Let n be the index of nilpotence for rad A and choose wj E rad A. Choose rj,sj ~ ~ for j -- 1,... ,n. Because ¢ is an algebra mapand t is central, (rl¢(Wl)81)(r2¢(w2)82)""

(rn¢(wn)sn)

Each memberof A[[t] is the limit of a sequence of elements in ~. Thus, by continuity, the identity above extends to all rj, si ~ A[[t]]. Weconclude that the ideal in A[[t]] generated by ¢(rad A) is nilpotent. COROLLARY 2.1. Let A be a finite-dimensional integrable then D(rad A) C_ tad

k-algebra.

If D ~ Derk(A) is

Proof. Let I = D(°), D = D(1), D(2),... be a higher derivation. theorem, D(m)(radA) rad A for all m. []

According to the

Weshall see, when we examine monomialalgebras, that it is possible for every derivation of a finite-dimensional algebra to leave the radical invariant even though its automorphism group scheme is not smooth. Nonetheless, the corollary does provide a useful test. THEOREM 2.2. Assume that k is a perfect field of characteristic p and G is a non-trivial finite p-group. Then the group algebra k[G~ never has a smooth.automorphism group scheme. Proof. Since GIG~ is not trivial, there is a nonzero additive character A ~Hom(G,k+). Define D k[ G] -+k[G] by lin early ext ending the func tion D(g) A(g)g with g E G.It is easy to seethat D is a deriv ation. Choose h ~ G with A(h) ~ 0. Then h - 1 lies in the augmentation ideal of k[G], which coincides with the radical. But D(h - 1) = A(h)h

78

Farkas, Geiss, and Marcos

so D(h - 1) is not in the radical. It is tempting to conjecture that k[G] does not have a smooth autoraorphism group scheme whenever p divides the order of G. However, we will see in a few moments that inner derivations are always integrable. Thus a "prerequisite" to the conjecture is the knowledgethat such group algebras possess outer derivations. The good news is that this weaker assertion is true ([FJL]). The bad new is that the only knownproof requires the classification of finite simple groups. Werecord some well knownproperties of integrable derivations for future use. (See, e.g., [Mat].) Let Z( ) denote the center of a ring. PROPOSITION 2.1. The integrable derivations Z(A)-submodule of all derivations.

o] the k:algebra A constitute

a

Proof. If D and E are integrable derivations then there exist ¢ and ¢, constant preserving k[[t]]-automorphisms of A[[t]], such that ¢(a)=a÷D(a)t+...

and

¢(a)=a+E(a)t+...

for all a ¯ A. Then the composition ¢¢ is an automorphism which preserves constants. Explicitly, if D(°),D (1) = D,D(2),... and E(°),E (~) = E,E(2),... are the corresponding higher derivations then we have constructed a new higher derivation whose mth term is ~i+j=m E(1)D(J). In particular, the m = 1 term is D + E. Thus the collection of integrable derivations is closed under addition. For any central ), E A, the sequence (1), ~2 D(2) ,... h°D(°) , A~D is also a higher derivation. PROPOSITION 2.2. Every inner derivation of the algebra A is integrable. Proof. Let a E A. Conjugation by the unit 1 - at is an algebra automorphism of A[[tl] and for any r ~ A, (1 - at)-~r(1 - at) = r ÷ (at - ra)t hi gher te rms . Thus .ada is integrable. It is well knownthat a diagonalizable derivation of a k-algebra A is equivalent to a grading of A by the additive group k+. Indeed, the eigenspaces of the derivation are the homogeneouscomponents for the grading. Such gradings can be difficult to deal with whenthe characteristic of k is positive; it would be nice to lift Z/(p)-gradings to Z-gradings. This goal is encoded in the following definition. Wesay that a higher derivation D(°), D(~),... is diagonalizable when the (m) are simultaneously diagonalizable k-endomorphismsof A. If ¢ : A -+ A[[t]] is the algebra mapversion of the higher derivation then diagonalizability means that there is a basis v~,... , v,~ of A so that ¢(vi) fi vi for so me f~¯ k [[ t]]. Mor eover, the fact that ¢ preserves constants tells us that f~ ¯/~l(k[[t]]), the multiplicative group units in k[[t]] with constant term 1. With very little additional work, we have

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PROPOSITION 2.3. There is a one-to-one correspondence between diagonalizable higher derivations of the finite-dimensional k-algebra A and Ltl (k[[t]])-gradings A. Observe that the group //l(k[[t]] is always torsion free, no matter what the characteristic of k is. Thusif a diagonalizable derivation of A lifts to a diagonalizable higher derivation then a k+-grading lifts to a grading by a torsion free abelian group. The converse is morevaluable. Since L/1 (kilt]I) is uncountable,it is abelian of infinite rank. As a consequence, every finitely generated torsion free abelian group embeds in L/~. Weconclude that if D is a diagonalizable derivation of A whosegradinglifts to a second grading via a (finitely generated) torsion free abelian group then the is integrable (and is the D(~)-term of a diagonalizable higher derivation). 3

COMMUTATIVE

MONOMIAL

ALGEBRAS

Weregard monomialalgebras as a rich source of elementary examples. Our study of this family of rings begins with a more or less computablecriterion for integrability in this case. Recall (cf. [FGGM]) that if I is an ideal of the polynomialalgebra k[X1,. ¯ ¯ Xn] then every derivation of n = k[X~,...,Xn]/I lifts to a derivation of k[X~,... , Xn] which stabilizes I. If I is a monomialideal then every derivation of R is a linear combination of images of such derivations with the special form m-~--° for some monomial m. OXj In this section, we will always assume that I has finite codimension in the polynomial algebra. THEOREM 3.1. Let I be a monomialideal of k[X, Y~, . . . Yn] and set R = k[X, Yl,...

, Yn]/I.

Assume that m is a monomialwhich does not involve X such that m-~x stabilizes I. Then the derivation D it induces on R is integrable if and only if for each monomial Xev E I, where ~ does not involve X, (;)Xe-JmJt/Elforj=O,

1,...,e.

Proof. Choose an automorphism ¢ of R[[t]] such that ¢ IR= I + Dt +... . Underline to denote the image of a polynomial in R. Suppose d < e. Since ¢(X) X + mt + ..., the coefficient of t a in ¢(X)e has the form (~)

"xe-dm---d

q- se-d-bl

sd

for some Sd ~ R. (The pigeon-hole principle is at work here: no more than d of the factors ¢(X) in ¢(X__)e can contribute a term rt i for i > 1.) Similarly, ~b(Z) = ahth with ao = Z. h>0

80

Farkas, Geiss, andMarcos

Hencefor j _< e, the coefficient of t j in ¢(X)~¢(_~)has the ~e-d+l

~=o \d]--

__

Xe-JmJy

sd)aj-d

+ xe-J+l

s

+ ~

for some s ~ R. On the other h~d,

=

=

0.

Since R is strongly graded by monomials, we see from the powers of ~ in our expression for 0 that

This proves one direction of the theorem. As to the converse, ~sume that for a set of generating relations have (~)X~-Jm~g = in R f orj = 0, .. ., e. C onsider the assi gnments

Xev ~ I we

¢(X) = ~ + mt and ¢(~i) for i = 1,... ,n. Since ¢(X)e¢(~) = 0, ¢ extends to an algebra map from R[t]]. It is easy to check that the coefficient of t in the expansion of ¢ agrees with D on the generators ~, ~ .... , ~n of R. Hence D is integrable. The previous theorem only handles images of m~ when X does not appear in m. Fortunately, the remaining "monomial"derivations are always integrable. THEOREM 3.2. Le$ I be a monomial ideal of k[X, ~,...

Yn] and set

R= k[X,Yx,..., Assume that m is a monomial which involves X such that m~ stabilizes the de,ration D it induces on R is integrable.

I. Then

Proof. According to Proposition 2.1, it suffices to show that the image D of X~ is integrable. Define ¢(~) = ~+ X~ and ¢(Y_j) = Y_j for j = 1,...n. If Xe~ monomial in I such that X does not appear in ~ then

=

(x

+ e

Thus ¢ extends to an algebra map from R to R[[t]].

Its t term agrees with D.

Weuse the previous two theorems to illustrate the metatheorem that an algebra whose relations do not interfere with the characteristic has a smooth autoxnorphism group. THEOREM 3.3. Assume that k is a field of characteristic mial ideal of k[X~ .... Xn] and set

p > O. Let I be a mono-

R= If no minimal monomialin I has positive degree in any X1 which is divisible then the automo~hism group scheme of R is smooth.

by p

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81

Proof. By virtue of the previous theorem and Proposition 2.1, we need only prove that if ra is a monomialwhich does not involve X8 and m~°OX~ stabilizes I then its image derivation of R is integrable. Weapply Theorem3.1. It suffices to show that if X~v is a monomialin I such that X8 does not appear in v then X~-JmJ~EI

for

j=0,...,e.

By induction, we are reduced to the case j --- 1. Since mo--~. stabilizes I, eX~-lvm ~ I. Weare done unless pie. Suppose this is the case. NowX~evis divisible by some minimal monomialrelation ft. But the X~-degree of # is either zero or a positive [] integer not divisible by p. In either event, we must have X~ v~ I. The hope is to look at the minimal monomials generating an ideal and immediately tell whether the corresponding monomial algebra has a smooth automorphism group scheme. Since we do not yet know how to do this, we offer a more modest result. THEOREM 3.4. Assume that k is a field of characteristic mial ideal of k[Xl,... Xn] and set

R=

p > O. Let I be a mono-

x,l/s.

Every derivation of R stabilizes the radical if and only if for each j there exists a minimal monomial#j ~ I such that the Xj-degree of ttj is not divisible by p. Proof. First assume that every minimal monomial in I has the form X~’a where a is a monomial not involving X~. Then

Thus b~x stabilizes I. Wesee that ~x induces a derivation of R which sends the image of X~, which is in the radical, to 1. Conversely, assume that I has minimal generators as described in the theorem. Wemust show that if D is derivation of k[X~,... , X~] and D(I) ~ th en D(X~) (X~,... , X~) for j = 1,... , n. Choose a minimal monomialX]fl ~ I such that does not involve X~ and p does not divide ]. fXf-’D(X~)~+

X~D(O)= D(X~)

Write D(X~) = c+ H where c e k and H e (X~,...

,Xn).

Then

If we write the second term ~ a nonredundant linear combination of monomiMs then each monomial which appears h~ length greater than the length of Each monominl in the support of the third term h~ X~-degree at le~t f. Thus

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Farkas, Geiss, andMarcos

the first monomial term cannot be in the support of ]X:-I~H + X[D(~3). The fact that I is a monomialideal implies now that

But X:-~/~ ~ I by minimality and f is not zero in k. Weconclude that c = 0, i.e.,

[]

D(x) (zl,..., zn).

As promised, we can now construct many examples of finite-dimensional algebras all of whose derivations stabilize the radical, but which do not have smooth automorphism group schemes. For example, suppose that k is a perfect field of characteristic p, that n ~ 2, and p < e(1) ~ e(2) ~--. e( n). Let I be the ideal of k[X~,...

Xn] generated by

X;(~),...,X~

(n),

and X~X~...X~.

Wefirst observe that if p is relatively prime to e(1) not have a smooth automorphism group scheme. Set check that w~ induces a derivation of R. We~gue integrable. Otherwise, we may apply theorem 3.1 to

x,x

...

x2

then R = k[X~,... , Xn]/I does w = X~ ... X~. It is e~y to that this derivation is not conclude that

-= [

e ¯

However,e(1) e(j) for al l j, so e(1)-l<e(j)

for all

Wehave created an element of I which is not divisible by any of the defining generators for I. If we assume, in addition, that all e(j) are relatively prime to p then Theorem 3.4 tells us that every derivation of R sends the radical into itsel£ The curious ~pect of this example is that when e(1) ~ 2p we can add the relation (X2 .-- 2 and the new algebra h~ a smooth automorphism group scheme. 4

MORITA

INVARIANCE

Weprove that having a smooth automorphism group scheme is a Morita invariant for finite-dimensional algebras over perfect fields. In fact, we establish a stronger result with no restriction on the scalar field. It is well knownthat Hochschild cohomologyis a Morita invariant. According to Proposition 2.2, for any finitedimensional algebra A it makes sense to define the subspace

~ (A)= integrable derivations of A / inner derivations of A fHH ~ (A). Weshow that fHH~ is a Morita invariant in the sense that if A and B in HH are Morita equivalent finite-dimensional algebras then there is an isomorphismt~om 1 (A) to 1 (B) which ca rries fH 1 (A)isomorphically to f 1 (B). Althou HH

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83

this subspace is the correct one for our strategy of proof, there are good reasons to "cointerpret" it. As a consequence of our theorem, the obstruction to integrability Derk (A) / integrable derivations of is a Morita invariant. To prove the result as originally stated, we need a particular explicit mapfrom I(A) to HH 1 (B). Throughout this section, we adopt the following set-up. A HH a finite-dimensional k-algebra and P is a left progenerator for A. Wemayassume that the algebra Morita equivalent to A has the form B = (EndA(P)) °p . (Henceforth, we ignore the presence of the opposite ring in the description of B. The concerned reader maymoveall of our functions from the left of their arguments to the right.) Let {(f/,Pi)

nomA(P,A) x P [ 1 < i <

be a projective basis for P. If D E Derk(A), denote by D* ~ Endk(P) the function given by D*(x) = ~ Dfi(x)pi i:l

A linear

map. ~ Endk(P) is said to be D-twisted provided that .(ap) = a.(p) + D(a)p

for all a ~ A and p E P. It is easy to check that D* is a D-twisted map. LEMMA4.1. (1) If.

Endk(P) is a D-t wisted map then ad. e Derk(B)

(2) If .1 and .2 are D-twisted then ad (.1 - .2) is an inner derivation of B. (3) If D is inner then ad D* is inner. Proof. To prove (1), it suffices to demonstrate that if/~ ~ B = Endn(P) . o ~ -~ o. e B. (It is a standard observation that ad. is a derivation of Endk(P) .) Suppose that a E A and p ~ P. ad.(/~)(ap) = .(/3(ap) = .(afd(p)) - [J(a.(p) + = a.(~(p)) + D(a)l~(p) - al~(t~(p))

=a adv(fl)(p). Hence ad ~(fl) is an A-module map.

84

Farkas, Geiss, andMarcos

As to (2), notice that ~1 - ~2 is an A-modulemap; that is, it is already some element ~, E B. Therefore ad (~1 - ~2) = ad ~/. Finally, suppose that D = ad w for some w E A. Let ),~ denote left multipScation by w, thought of as a function in Endk(P). For a E A and p ~ (D* - A~)(ap) = D*(ap) - w(ap) = aD* (p) ad(w)(a)p - w ap = aD* (p) + wap - awp - wap = a(D* - Aw)(p) We conclude that

D* - Aw 6 B. Hence adD* -adAw = ad (D*

is inner. But if ~ 6 B then ~ is A-linear. In other words, Awo ~ - ~ o A~ = 0. It follows that ad A~ = 0. Thus ad D* is inner. [] Parts one and three of the lemmatells to ad D* induces a linear map

us that the k-linear map which sends D

~ (A) -~ SI-I ~ (S) ¯ : HH THEOREM 4.1.

¢ is an isomorphism.

Proo]. Since both cohomologygroups are finite-dimensional and Morita equivalence is symmetric, it suffices to prove that the kernel of ¯ is zero. This amounts to showing that if ad D* is inner then D is inner. So assume that there is some 7 E B so that D*ofl-~oD*=7o~-~o7 for all/~ E B. Wefocus on those fl of the following type: $(x) = h(x)y for a fixed h 6 HomA(P,A) and a fixed y 6 P. In this case, (D* o Z - Z o D*)(x) = D*(h(x)y) - h(D*(x))y = h(x)D*(y) + D(h(x))y - h(D*(x))y = h(x)E Dfi(y)pi

+ D(h(x))y

- ~ Df~(x)h(p,)y.

By assumption, the last expression is also equal to h(x)~l(y) - h(7(x))y Weapply this equality in two situations. P~ecall that P is a progenerator. It satisfies a trace condition of the following form: there exist gt ~ HomA(P,A) and ~rt ~ P such that

1. t.~l

SmoothAutomorphismGroupSchemes

85

First set y = ~rt, apply gt to both sides of the equality, and then sum over t. h(x) i,t

i

= h(x) E gt(7(Trt))

- h(7(x)).

t

Rearranging terms and using A-linearity,

we obtain

(t) D(h(x)) = h(x)[ Egt(7(Trt))t

E Dfi(Trt)gt(P~)] i,t

- h(7(x)

- Dfi(x)pi). i

Next set h = gt, set x = ~rt, and then sum over all t. E Df,(y)p,

+ 0 - E Dfi(rtgt(pi)y

i

= 7(y) - E gt(7(rt))y.

i,t

t

Replace y with x and rearrange.

i

t

i,t

Finally, apply h to each side. (tt)

h(7(x)

- E Dfi(x)Pi) i

[E gt(’y(Trt) )x -- E D fi(Trt)gt(Pi)]h(x) t i,t Notice that the expression in square brackets is an element of A independent of h and x. Call this expression c and substitute (~) in the second long term on the right side of equality (~). D(h(x)) = h(z)c - ch(x) Since h and x in (t) are arbitrary, we can invoke the trace condition one last time to see that D(a) = ad (-c)(a) for all choices a The remainder of the argument is inspired by [GAS]. Weshall prove that if D Derk(A) is integrable then ad D* is integrable. Assumethat 0 = I + Dt + D(2)t2 + .-. is a k[[t]J-automorphism of A[[t]]. The kilt]I-vector space P[[t]] can be given two A[[t]]-module structures. The first is extended from the action of A on P in the

86

Farkas,Geiss, and Marcos

obvious way; we denote this module P[[t]]. A second action comes from the automorphism.If a e A[[t]] and p E P[[t]] define a,p = O(a)p. The second module will be written eP[[t]]. projective A[[t]]-modules.

Both modules are finitely

generated

LEMMA 4.2. P[[t]] _~ eP[[t]] as A[[t]]-modules. Proof. Since A is finite-dimensional, A[[t]] is semiperfect. Hencefinitely A[[t]]-modules have unique projective covers. The key is that 0 = I(mod t). This means that

generated

as A-modules. It is immediate that they are isomorphic as A[[t]]-modules. tA[[t]] C_rad A[[t]] so

But

P[[t]]/ (rad A[[t]])P[[t]] ~_ °P[[t]]/ (rad A[[t]])OP[[t]] as A[[t]]-modules. Their respective projective covers are P[[t]] and ~P[[t]].

[]

Let ¢ : P[[t]] -~ OP[[t]] be an A[[t]]-isomorphism guaranteed by the lemma. We will regard ¢ as a memberof Endk[[~]lP[[t]] which satisfies ¢(ap) for all a E A[[t]] and p ~ P[[t]]. p ~ P, we can write

Since ¢ is determined by its behavior on those

where Cj ~ Endk (P) for all j. Once again we use the fact that 0 _= I(mod t) to that ¢0 ~ EndA(P). By repeating this observation for the inverse of ¢ we conclude that ¢0 is an A-module isomorphism of P. Extend it in the obvious way to an A[[t]]-module isomorphismof P[[t]], and call the new map¢o as well. It is easy to check that ¢~-1¢: P[[t]] --~ eP[[t]] is also an A[[t]]-isomorphism. assume that ¢o -- I. LEMMA 4.3.

¢1 is a D-twisted

Thus we may replace

¢ with ¢~-1¢ and thereby

k-endomorphism of P.

Proof. We know that ¢(ap) -- 0(a)¢(p) whenever a E A and p ~ P. As a "power series",

this means

+ ¢2(ap)t 2 +... = (a + D(a)t D(2)t 2 a u... )( p + ¢l(p)t

ap+¢l(ap)t

Readoff the coefficient of t.

+

¢2(P)2 "{-""" )

¯

[]

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87

LEMMA 4.4 ([GAS]). If we identify B[[t]] with EndA[[t]](P[[t]]) then ¢ o s o ¢-1 6 S[[t]] ]or all s e B[[t]]. Proof. For a e A[[t]] and p e Rift]f, we have (¢ o s o ¢-l)(ap) = (¢ s)(¢-l(ap))

=(¢ o s) =(¢ os)(¢-l(e-l(a), = (¢ o s)(~-l(a)¢-l(p))

=¢ (~-l(a)(s¢-~)(p)) ---- ~(~9-1(a))(¢o s o ¢-l)(p)

THEOREM4.2. ~ (:HH1 (A)) = J’HH~ (B) Proof. By symmetryand finiteodimensionality,

it suffices to show that

¯ (fHH1 (A)) _C ~ (B) According to the previous lemma, the map s ~ ¢ o s o¢-1 defines an automorphism ~ ¯ AutkII,ll (Btttl]) Since this automorphismis determined by its behavior on B, it has a power series description,

2 +... ~=&+~t+~t

Computing ~’(~) = ¢ o ~ o ¢-1 when ~ ¯ B, we obtain

~(Z)= Z+ (¢1o ~ - Zo ¢~)t Thus if0 = I and ~ = ad ¢1. It follows from Lemma4.1 that ad ¢1 ¯ Derk (B) and that this derivation is integrable (to ~’). The same lemmasays that ad ¢1 -- ad D* 1 (B). But inner derivations are always integrable, and integrable as a memberof HH derivations constitute a vector space. Therefore ad D* is integrable. [] As promised earlier, we provide a second proof that hereditary algebras over an algebraically closed field have a smooth automorphism group scheme. THEOREM 4.3. All derivations of a finite-dimensional algebraically closed field are integrable.

hereditary algebra over an

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Farkas, Geiss, and Marcos

Proof. A theorem due to Gabriel (JARS]) states that one of these hereditary algebra is Morita equivalent to a path algebra kF whose underlying graph F has no oriented cycles. The main result of this section tells us that we may take A = kF. It is well knownthat every derivation of kF is the sum of an inner derivation and a derivation which vanishes on all vertices. (A stronger version of this observation can be found in [FGGM],Theorem 4.1). A derivation of the second kind is determined by its behavior on arrows. Notice that if D vanishes on vertices and a is an arrow from vertex e to vertex f then

D(a) = D(ea/) It follows that every derivation of the second kind is a linear combinationof derivations rn~-~a where m is a path which shares the same origin and the same terminus as a. (Here m~-~asends a to m and all other arrows to zero.) Weare reduced to proving that if D = m~then D is integrable. It is at this stage that we take advantage of the hypothesis that F has no oriented cycles by noting that D(A)D(A) = Indeed, if y E D(A) then any path in the support of y has a subpath which begins at the origin of a a~d ends at the terminus of a. No two paths like this can be concatenated. It follows immediately that I + Dt is an automorphism of A[[t]]. [] REFERENCES [Abe] E. Abe, Hopf Algebras, Cambridge Univ. Press, Cambridge, 1977. [FGGM]D. R. Farkas, Ch. Geiss, E. L. Green, E. N. Marcos, Diagonalizable derivations of finite-dimensional algebras I, Israel Journal, to appear. [FJL] P. Fleischmann, I. Janiszczak, and W. Lempken, Finite groups have nonSchur centralizers, Manuscr. Math. 80 (1993), 213-224. [Ga] P. Gabriel, Finite representation type is open, in Representations o] Algebras, Springer Lect. Notes 488, Berlin, 1975, 132-155. [GS] M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, in Deformation Theory of Algebras and Structures and Applications, NATOASI Ser. C 247, Kluwer, Dordrecht, 1988, 11-264. [GAS] F. Guil-Asensio and M. Saorin, On automorphism groups induced by bimodules, Archiv. der Math Basel, to appear. [Ha] D. Happel, Hochschild cohomologyof finite dimensional algebras, in Sgminaire d’Alg~bre, Springer Lect. Notes 1404, Berlin, 1984, 108-126. [Hu] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag Grad. Texts in Math. 9, NewYork, 1980. [Mat] H. Matsumura, Integrable derivations,

Nagoya Math. J. 87 (1982), 227-245.

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[Maz] G. Mazzola, The algebraic and geometric classification of associative algebras of dimension five, Manuscr. Math. 27 (1997), 81-101. [Mo] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS82 Amer. Math. Soc., Providence, 1993. [Sw]M. E. Sweedler, Hopf algebras with one group-like element, Trans. AMS127 (1967), 515-26. [Ta] M. Takeuchi, Tangent coalgebras and hyperalgebras I, Japanese Jour. of Math. 42 (1974), 1-143. [Wa] W. C. Waterhouse, Introduction to A]fine Group Schemes, Springer-Verlag Grad. Texts in Math. 66, NewYork, 1979.

A combinatorial characterization of hereditary categories containing simple objects DIETERHAPPELFakult~t ffir Mathematik, Technische Universit~t D-09107 Chemnitz, GERMANY IDUNREITENDepartment of Mathematical Sciences, Science and Technology, 7491 Trondheim, Norway

Chemnitz,

Norwegian University

of

ABSTRACT Let 7/be a connected hereditary abelian category with finite dimensional homomorphismand extension spaces. If 7/ contains a tilting object then 7/ has almost split sequences and ff 7/ is not a module category this yields an equivalence ~’. Weconsider here ~--invariant additive functions on 7/and show as a main result that 7/contains simple objects iff there exists a nonnegative additive T--invariant function.

Let k be an algebraically closed field and 7/ a connected hereditary abelian k-category with finite dimensional homomorphismand extension spaces. Assume that 7/has a tilting object, that is, an object T such that Ext~ (T, T) = 0 and such that Homn(T, X) = 0 = Ext~(T,X) = 0 implies X = 0 (see [HRS],[H],[HR]). It was proved in [HR] that if 7/ has a simple object and 7/ is not equivalent to rood A for a finite dimensional hereditary k-aigebra A, then 7/must be derived equivalent to a category coh :K of coherent sheaves on some weighted projective line X in the sense of Geigle-Lenzing [GL1]. However,not all hereditary categories derived equivalent to some coh ]K have a simple object. In this paper we give a combinatorial characterization of the hereditary abelian k-categories 7/with tilting object, which are not equivalent to modA for a finite dimensional hereditary kalgebra A, and which have some simple object. When7/is a hereditary abelian k-category with tilting object, then 7/has almost split sequences [HRS]. If 7/is not equivalent to rood A for some finite dimensional hereditary k-algebra A, then 7/has no nonzero projective object (see [H] ). There is then an equivalence T: 7/ ~ 7/ such that for each indecomposable object C in 7/ 91

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there is an almost split sequence 0 -~ TC--> B -~ C -~ 0 [HRS]. A function l: 7/~ No (the nonnegative integers) is additive if for each exact sequence 0 -~ A -~ B -~ C -~ 0 we have l(B) = l(A) +l(C). Whenwe have suc:h an additive function, there is induced a function ~: K0(7/) -> Z, where Z denotes the integers and Ko(7/) denotes the Grothendieck group of 7/(modulo exact sequences). Under our assumptions Ko(7/) is free abelian of finite rank [HRS]. Assume that our 7~ is not equivalent to some mod A for a finite dimensional hereditary k-algebra A. A function l: 7/-~ N0 is r-invariant if l(C) = l(TC) for each C in ~/. Our main result is that ~/has some simple object if and only if there is a nonnegative additive T-invariant function on 7/. Note that if 7/is modA for a finite dimensional hereditary k-algebra A, then there can be no nonzero ~-invariant additive function I on 7/. For if P is projective, then l(P) = si nce rP= 0and for ever y C th er e is a n e xact sequ ence 0 -~ P -~ P’ -~ C ~ 0 with P and P’ projective. Wewould like to thank the referee for helpful comments. 1

COHERENT

SHEAVES

AND MODULE

CATEGORIES

In this section we recall somebackground material on the category coh 3[ of coherent sheaves on the weighted projective line X, and on hereditary categories derived equivalent to coh :K, from the point of view of existence of simple objects and of nonnegative additive r-invariant functions. Through this we give motivating examples for the main result of this paper. Wealso discuss hereditary categories derived equivalent (but not equivalent) to mod A for some finite dimensional hereditary k-algebra A. Let p = (p~,... ,p~)(t >_ 3) be a weight sequence, that is, a sequence of positive integers and ~ = (A3,’" ,At) a sequence of pairwise distinct nonzero elements the field k. Let :K = X(p,_~) be the associated weighted projective line, and cob the corresponding category of coherent sheaves. This category has simple objects, and here simple objects give naturally rise to additive functions, as we nowshow. Let S be a simple object in a homogenoustube of coh :K, so that TS ~_ S. Then we have the rank function r =< -, S >-- dimk Horn(-, S) - dimk Extl( -, S) on cob X (see [GL1] ). This function is clearly additive and T-invariant, and is also nonnegative. It takes value 0 on the objects of finite length, and is positive on each indecomposableobject of infinite length. With a weight sequence _p = (p~, ... , p,) is associated the number h = (t- 2)p~,~=~ p/pi, where p is the least commonmultiple of p~,... ,pt. Whenh ~ 0, that is, cob :K is not of tubular type, the hereditary categories 7~ derived equivalent to coh :K are obtained by "tilting" with respect to a split torsion pair (T, 5r), where 7- consists of objects of finite length (see [LS, I-I]). The rank function r on coh gives rise to a function .~ on K0(coh :K) Ko(Db(coh :K)) -~K0(7/). In thi s way we obtain an additive T-invariant function on 7/when 7/is not equivalent to some modA for a hereditary k-algebra A. This function is nonnegative since r takes value 0 on 7", and hence also on 7-[-1]. In the tubular case, that is, h = 0, we have families of tubes C~ indexed by Q+ ~ {oo}, where Q+ denotes the positive rational numbers. If a ~ b there is a map from an object in the family Ca to an object in the family Cb if and only

HereditaryCategoriesContainingSimpleObjects

93

if a < b. Let (7-, ~’) be a split torsion pair where the torsion class 7- consists the union of families Ca where a > s for a fixed irrational number s. Tilting with respect to (7-, :Y) we obtain a hereditary category 1/having no simple object. For it is knownfrom [HR] that the simple objects must lie "furthest to the right" or "furthest to the left". As a consequenceof our main result it will follow that there is no nonzero nonnegative additive ~’-invariant function on 1/. When1/ is derived equivalent (but not equivalent) to mod A for some wild indecomposable finite dimensional hereditary k-algebra A, we know that 1/has no simple object since there are no tubes. For the structure of such 1/, see [H]. We nowshow that in this case there is also no additive function of the desired type. PROPOSITION 1.1. Let 1/ be a connected hereditary category derived equivalent (but not equivalent) to rood A for a finite dimensional wild hereditary k-algebra Then there is no nonzero nonnegative additive ~--invariant function on 1/. ¯ Proof. We know that 1/ has a component of type Z~, where ~ is a connected quiver which is not Dynkin or extended Dynkin. Assume to the contrary that there is somenonzero nonnegative additive T-invariant function l on 1/-/. There are two different types of such t/. In one case, when the component Z~ is on the left hand side, it is easy to see that there is for each C in 1/on exact sequences 0 -+ B1 ~ B0 ~ C -~ 0 where the indecomposable summands of B1 and B0 belong to Z~. In the second case there is for each C in t/ an exact sequence 0 -~ C -~ Bo -~ B1 ~ 0 where the indecomposable summandsof B0 and B1 belong to Z~. Hence I is nonzero also on Z~. Since l is ~--invariant, there is induced a nonzero nonnegative function on the underlying graph A of ~. Weclaim that this function is positive. For if not, let x be a vertex with l(x) = O, having a neighbour y with l(y) > 0. Then the additivity formula at x gives a contradiction. Since A is a wild graph, this is impossible. Hence there is no nonzero nonnegative additive ~’-invariant function on 1/. [] 2

THE MAIN

RESULT

The aim of this section is to prove our main result. Wealso show that a nonzero nonnegative T-invariant function must be positive on a central class of objects, which will be used for proving in which sense such functions are unique. Recall that an object E in 1/ is exceptional if Ext,(E, E) = 0. The object E is torsionable if it is the factor of a direct sumof copies of sometilting object. Wedenote by E± the perpendicular category, whose objects are the C in 7-/with Hom(E,C) = 0 = Ext~ (E, C). Wehave the following main result. THEOREM 2.1. Let 1/ be a connected hereditary abelian k-category with tilting object, where k is an algebraically closed field, and assumethat 1/ is not equivalent to rood A for a finite dimensional hereditary k-algebra A. There is some nonzero nonnegative additive ~’-invariant ]unction l on 1/ if and only 1/ has some simple object. Proof. Assumethat 1/is not equivalent to modA for a finite dimensional hereditary k-algebra A.

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Happeland Reiten

Assumefirst that 7/has some simple object. Then there is a split torsion pair (T, r) i n 7/ w ith a ll o bjects i n j c o f f inite l ength, s uch t hat when we t ilt w ith respect to this pair we obtain some category coh X [HP~, Section 3, Th 6.1]. Then (~’(1), 7") is a split torsion pair for coh X. Since the rank r is 0 r, it f oll ows that the induced r-invariant additive function on 7/is also nonzero and nonnegative. Assume now that 7/ has some desired function l, but no simple object. By Proposition 1.1 it follows that 7/ is not derived equivalent to mod A for a wild finite dimensional hereditary k-algebra A. Further 7/is not derived equivalent to modA where A is tame hereditary, since 7/would then have a simple object because we are in the situation h 7t 0 discussed in section 1. If A was hereditary of finite type, then any 7/ derived equivalent to modA would have a nonzero projective object, which is impossible by the assumption that 7/ is not equivalent to any modA where A is a finite dimensional hereditary k-algebra (see [HRS]). Hence 7-/ is not equivalent to modA for any finite dimensional hereditary k-algebra A. Then there must be someobject of infinite length in 7/. For otherwise all objects are of finite length, and since there are no nonzero projectives, the AR-quiverfor 7/is a union of tubes (see [L]), with no nonzero maps between objects in different tubes. Since 7/is connected, there is only one tube. The rank of the tube is the rank of Ko(7/). This contradicts the fact that 7/has a tilting object, since a tube of rank t does not contain an exceptional object which is the direct sum of t nonisomorphic exceptional objects. Since 7/has some object of infinite length, there is by [HI a tilting object of infinite length, and hence some indecomposable torsionable exceptional object E of infinite length. Wechoose E such that l(E) = is minimal. As pointed out before, the perpendicular category E± is equivalent to modH for some finite dimensional (basic) hereditary k-algebra H. Consider the almost split sequence 0 -4 rE -4 M-4 E -4 0. Since 7/is not equivalent to a hereditary modulecategory, it follows that Mis a sincere H-module [HI. Let now n be the rank of the Grothendieck group Ko(7/). The algebras Endn(T)°p where T is a tilting object are the quasitilted algebras [HRS]. A quasitilted algebra has no oriented cycles [HRS], and hence must be hereditary if there are at most two simple modules. This contradicts the assumption on 7/, so that we must have n > 3. Since T = H @ E is a tilting object, Ko(modH) has rank n Let St,... ,Sn-1 be the nonisomorphic simple H-modules. We have [M] = ~=~ ti[Si] in Ko(modH), where all ti axe nonzero since Mis sincere. Each is exceptional, and Si is torsionable since T = H @E is a tilting object. Since by assumption 7/has no simple object, each S.i has infinite length. Then we have l(S~) >_ bythedefi nition of a . We n ow obtain the inequality 2a _ > ~z_,a~V’’~-I ~=l _ t~) (n - 1)a. It follows that n = Since n = 3, we must have t~ = 1 = t2, so that [M] = [S~] + [$2] in Ko(7/). The quiver of H is then

where m _> 1 denotes the number of arrows from the vertex v~ at 1 to the vertex v2 at 2. The one-point extension H[M]is a quasitilted algebra derived equivalent

HereditaryCategoriesContainingSimpleObjects

95

to 7/[HR], and it is not tilted since 7/is not derived equivalent to some modA for finite dimensional hereditary k-algebra A. Then it follows from [HRS] that Mis indecomposable. For the algebra H[M]we have the quiver

with m arrows from vl to v~, and the space of relations of paths from voto v2 has dimension m - 1. Weview this algebra as the one-point coextension [N]H1, where H1 is the path algebra of the quiver 0 ¯ ------* ¯ 1 . Since the indecomposable injective H[M]-moduleassociated with the vertex v2 has dimension vector (lml), the Hi-module N has dimension vector (lm). Since there is no arrow from vertex vo to vertex v2, the indecomposable summandsof N must be one copy of k ~ k, together with m - 1 copies of 0 -+ k. Since the modules are on a slice, it follows from [HRS] that [N]H is tilted, and this gives a contradiction. [] Nowwe show that a nonzero nonnegative ~--invariant on some class of exceptional objects.

function must be positive

PROPOSITION 2.2. Assume that the hereditary abelian k-category 7/ with tilting object is not equivalent to rood A for some finite dimensional hereditary k-algebra A. Let l be a nonzero nonnegative ~--invariant additive function on 7t. If E is an indecomposabletorsionable exceptional object of infinite length, then

l(E)> Proof. Assumeto the contrary that E is indecomposable torsionable exceptional and l(E) = O. Since E has infinite length and is torsionable exceptional, we have E± = mod H for some finite dimensional hereditary k-algebra H [HR]. Consider the almost split sequence 0 -~ ~-E -+ M--~ E -+ 0. Then we know that Mis in E± = modH (see [HR]). Since l is v-invariant, we have l(~-E) = As already mentioned it follows that Mis a sincere H-module[HI. The restriction llEX: EJ- --~ No is still additive. Wehave l(M) = l(E) + l(rE) and s ince l _> 0, we have l(S) = for ea ch si mple composition fa ctor S ofM. Since M i s a sincere H-module, it follows that l(S) = for ea ch si mple H-module, and he nce liE ± = O. Weknow that T = H $ E is a tilting object in 7/[HR], and hence gives rise to a basis for Ko(7/). Then we get 1 = 0, which is a contradiction. Weend the section by pointing out that when we have a nonzero nonnegative ~--invariant additive function, then it is essentially unique. PROPOSITION 2.3. Let 7/ be a hereditary abelian k-category with tilting object not equivalent to modA for a finite dimensional hereditary k-algebra A, and having some nonzero nonnegative additive T-invariant function l. Then any other function is a (rational) multiple of Proof. It follows from Theorem2.1 that 7/has a simple object, and is hence derived equivalent to some coh X.

96

Happeland Reiten

Weknowthat 7/is obtained from coh X by tilting with respect to a split torsion pair (T,~-), where 7- consists of objects of finite length. Let l be a nonnegative 7-invariant additive function on 7~. Wefirst show that I is zero on the objects of finite length. Let S be a simple object with Hom(S, X) = 0 for each X indecomposable of infinite length, and assume l(S) > 0. There is some indecomposable Y of infinite length with (Y, S) 0. Using the lifting property for almost split sequences, as in [HR], we get an epimorphism fn : Y -+ An, where An is uniserial of length n with S on the top. Since l(T) > 0 for each simple composition factor T of An, we get that l(A,~) > l(Y). Hencewe get/(ker fn) < 0, which is a contradiction. Similarly, if S is a simple object with Horn(X, S) = 0 for each X indecomposable of infinite length, there is some indecomposable object Y of infinite length with Hom(S, Y) ~ 0. Then we get monomorphismBn ~ Y, where Bn is uniserial of length n with socle S, and for n large enough we get the contradiction that l(Y/Bn) < It follows from the above that if l is a r-invaxiant additive function which is nonnegative on 7/, which is derived equivalent to coh :K, then the induced function on coh :K is also nonnegative on coh X. Hence it is sufficient to consider coh Wehave the rank function r, which is positive on indecomposableobjects of infinite length. Let C be indecomposable of infinite length and let E be an exceptional object of rank 1. Then for some i there is a nonzero map g: E -~ ~-iC [LP], which must be a monomorphismsince E has rank 1. Hence we have l(C) = l(~-~C) >_/(E), so that l(C) > 0 by Proposition 2.2. Let A be indecomposable of infinite length, with l(A) = a > 0 minimal. If r(A) > 1, we have an exact sequence 0 -~ A1 -~ A ~ A2 -~ 0 with r(A1) > 0 r(A2) > 0. Since then l(A1) > 0 and/(A2) > 0, we get a contradiction, showing that r(A) = Assume r(B) = 1 and l(B) > a. With A chosen as above, there is for some i a nonzero map g: A -~ riB, which must be a monomorphismsince A has rank 1. Then l(viB/A) > 0 and r(riB/A) = so that ~’i B/A hasfini te leng th. This gives a contradiction, and hence l(B) = It now follows that l = a. r on coh :K, and hence on 7/, and we are done. [] 3

POSITIVE

ADDITIVE

FuNcTIONS

When7/ = mod A for a finite dimensional hereditary k-algebra A, we have a positive additive function given by ordinary length. However, no other hereditary k-category with tilting object has this property, as we now show. Note that we do not assume that our functions are r-invariant. PROPOSITION 3.1. Let 7/ be a hereditary abelian k-category with finite dimensional homomorphismand extension spaces, and having a tilting object. Then there is a positive additive function on 7~ if and only if 7~ is equivalent to rood A for some finite dimensional hereditary k-algebra A. Proof. Assumethat 7/is not equivalent to modA for some finite dimensional hereditary k-algebra A, and assume that there is some positive additive function l on 7/. Assumefirst that 7/has some object of finite length. Then 7/is derived equivalent to some category coh X. If S is a simple object with Hom(S,X) = 0 for each

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97

indecomposable object X of infinite length, then choose Y indecomposable of infinite length with Horn(Y, S) ~ 0. As in section 2 we have an epimorphismY --~ An, where An is uniserial of length n and with top S. For n large enough we have l(An) = n >/(Y), which gives a contradiction. If there is no such simple object then there is some simple object T with Horn(X, T) = 0 for each indecomposable object X of infinite length. In a similar way as above we get a contradiction. [] We can now assume that 74 has no simple object. Let X be indecomposable with l(X) = minimal. Le t Y bea p ro per ind ecomposable sub object of X. Then we have l(Y) = a, and hence l(X/Y) = O, which is a contradiction. REFERENCES [GL1] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensionalalgebras, in: Singularities, representations of algebras and vector bundles, Springer Lecture Notes 1273 (1987), 265-297. [GL2] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. of Alg. 144 (1991), 339-389. [HI D. Happel, Quasitilted algebras, Adv. in Proc.ICRA VIII (Trondheim), CMS Conf.proc., Vol. 23, Algebras and modules I (1998), 55-83. [HR] D. Happel and I. Reiten, Hereditary categories with tilting Zeitschr.232 (1999) 559-588.

object , Math.

[HRS] D. Happel, I. Reiten and S. O. Smal¢, Tilting in abelian categories and quasitilted algebras, Mem.Amer. Math. Soc. 575 (1996). [L] H. Lenzing, Hereditary noetherian categories with a tilting Proc. AMS125 (1997), 1893-1901.

complex, Adv. in

[LP] H. Lenzing and J. A. de la Pena, Wild canonical algebras, Math.Z. 224 (1997) no.3, 403-425. [LS] H. Lenzing and A. Skowronski, Quasitilted algebras of canonical type, Colloquium Mathematicum, Vol 71 (1996), 161-181.

Symmetric Quasi-schurian

Algebras

OCTAVIOMENDOZA HERN/i, NDEZDepartamento de Matem~tica, Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina, E-mail: [email protected] 1

ABSTRACT Let k denote an algebraically closed field. Wesay that a finite dimensionalk-algebra A is quasi-schurian, if it satisfies the following two conditions: QSl) dim~Homh(P, Q) _< 1 if P, Q are not isomorphic indecomposable projective A-modules. QS2) dimkEndA(P) = 2 for each indecomposable projective A-module P. An important class of quasi-schurian algebras is the trivial extensions of finite representation type. In this paper, we give necessary and sufficient conditions for a given quasischurian algebra h to be weakly-symmetric or symmetric. These conditions are given in a combinatorial approach using a graph GS(A) associated to A, and function CA: Ch(GS(A)) wher e Ch(GS(A)) is t he set of c hai ns of t he grap h GS(A). Finally we give some connections between symmetric quasi-schurian algebras and trivial extensions of algebras. 1

INTRODUCTION

Throughoutthis paper, we let k denote a fixed algebraically closed field. By algebra is always meant a finite dimensional associative k-algebra with an identity, which we assume moreover to be basic and connected, and by module is meant a finitely generated left A-module. Let A be a schurian triangular algebra. It is well knownthat the trivial extension T(A) of A satisfies dimkHOmT(A)(P,Q) _( 1 and dim~EndT(A)(P) = 2 where P, Q are non isomorphic indecomposable projective T(A)-modules. In this way, are interested in the class of algebras A satisfying the above property. Thus, we say that an algebra A is quasi-schurian if it satisfies the following two conditions: Supported by a fellowship from CONICET,Argentina. ~raat from CONICET, Argentina.

The author gratefully

acknowledges a

I00

E[erndndez

QS1) dimkHomh(P,Q ) _< 1 if P, Q are not isomorphic indecompo.’~able projective A-modules. QS2) dimkEndA(P) = 2 for each indecomposable projective A-module P.. The aim of this paper is both to give necessary and sufficient conditions for a given quasi-schurian algebra to be weakly-symmetric or symmetric, and to say whena symmetric quasi-schurian algebra arises from a trivial extension of a schurian triangular algebra. Let A = kQA/I where QAis the ordinary quiver associated with A and I is an admissible ideal. If ~f is a path in the quiver QAwe will denote by -6 the sub quiver of QAhaving as vertices and arrows those which belong to 6, this _6 is called the support of 6. Let C be an oriented cycle. Each vertex j in the support C of C determines a cycle with origin j which we call C(j). Finally we denote by ~ the congruence class "), + I in A = kQA/I. In section 3 we prove the following theorems THEOREM.Let A = kQA/I be a quasi-schurian conditions are equivalent

I)

algebra.

Then the following

A is weakly-symmetric. For every non zero path 7 there exists a path ¢Y such that ~7 is a non zero minimal oriented cycle.

III)

For each non zero f in HomA(P , Q) the induced morphism HOmA(Q,f) HomA(Q,P) -r EndA(Q) is nonzero , if P andQ are in decomposable non isomorphic projective A-modules.

IV) A satisfies

the following conditions

a) If a minimal oriented cycle C is non zero, then C(t~ ~ 0 for each vertex t in the support C_ of C. b) Let {C._L~,C_~2... ,Cm} be the set of supports correspondingto the non zero oriented cycles. Then Q A = THEOREM. Let A = kQA/I be a quasi-schurian weakly-symmetric algebra. Let {C1,C2... , Crn} be the set of supports of the non zero minimal oriented cycles. The following statements are equivalent: I) A is a symmetric algebra. II) There are non zero elements al,... , am in the field k such that, for each i and j with (2)0 ~ (~’~)o th e foll owing condition hold s

e (t)

vt e (2)0

In section 4 we give a combinatorial approach to the above last theorem using a graph GS(A) associated to A, and a function CA Ch(GS(A)) ~ wher e Ch(GS(A)) is the set of chains of the graph GS(A). In this way, the existence of the non zero constants ax,... ,am which are required in the last theorem, is

SymmetricQuasi-SchurianAlgebras

101

very closely related with the structure of the graph GS(A) and with the function CA: Ch(GS(A)) -+ In fac t, we prove tha t the quasi-schurian Weak lySymmetric k-algebra A is symmetric if either the graph GS(A) is a tree or satisfies CA(C) = 1 for each minimal cycle C in GS(A) with at least three tices. In section 5 we give a connexion between symmetric quasi-schurian algebras and trivial extensions of algebras, which we state next. THEOREM. Let A be basic connected finite dimensional k-algebra. The following statements are equivalent 1) There exists a schurian basic triangular algebra ~ such t hat A"~T(A’). 2) A is symmetric quasi-schurian, and there exists a set C(A) consisting of exactly one arrow in each non zero minimal oriented cycle, such that Qc has non oriented cycles, where Qc is the quiver obtained from QAby deleting the arrows in

C(A). If these conditions hold, then A’ ~_ A]Ie where Ie is the ideal 9enerated by C(A) in A. In the case that Q is an oriented tree and A = T(kQ) we can always choose a set C(A) as in 2) in the theorem. Moreover, we prove that for any such choice factor algebra A/Ic is iterated tilted of type Q. This is a useful approach to obtain iterated tilted algebras of a given tree class. 2

PRELIMINARIES

It is well knownthat each basic finite dimensional algebra A over an algebraically closed field k is isomorphic to k-algebra kQ/I where Q is the finite quiver associated with A and I is an admissible ideal of the path algebra kQ. Let Q be a quiver. Wewill denote by Q0 the set of vertices and by Q1 the set of arrows of Q. Given an arrow c~ E Q1, we say it starts at the vertex o(c~) and ends e(c~). A path in the quiver Q is either an oriented sequence of arrows p = an"" with e(a~) = o(at+~) for < t < n,or the symbol ei f or i E Q0. We call the p aths e~ trivial paths and we define o(el) = e(ei). For a nontrivial path p = c~,, ... al define o(p) = o(~) and e(p) e( an). If ~ i s a p at h in Q, we will den ote by -6 the support of 5 in Q. Thus, _6 is a sub quiver of Q having as vertices and arrows those which belong to 6. A nontrivial path p is said to be an oriented cycle if o(p) = e(p). Let C = an~-x"’ctz~l be an oriented cycle in Q. We will call C minimal oriented cycle if n = 1 or all the vertices o(a~),o(a2),... ,o(a,~) are different case n > 1. Let j be a vertex in the support ~ of C, then the arrows of~_ determine a cycle with origin j, whichwecall C(j). That is, C(j) = at-1 ¯ ¯ ¯ a~cqc~n¯ where j = o(c~r) is the origin Let A be a finite dimensional k-algebra, we denote by rood(A) the category of finitely generated left-A modules, by Q^ the ordinary quiver associated with A, by S(a) the simple A-module corresponding to the vertex a in (Q^)0, by P(a) the projective cover, and by I(a) the injective envelope of S(a). Let 7 be a path in QA" By ~ we denote the congruence class 7 + I in A = kQA/I. Wewill say that the path 7 is zero if ~ = ~.

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102

DEFINITION: An algebra A is called quasi-schurian, if it satisfies the following two conditions: QS1) dimkHomA(P, Q) < 1 if P, Q are non isomorphic indecomposable projective A-modules. QS2) dimkEndA(P) = 2 for each indecomposable projective A-module. An important class of quasi-schurian algebras consists of the trivial extensions of Caftan type D, with D a Dynkin quiver. These algebras are closely related with the iterated tilted algebras of Dynkintype D, see [1],[2]. Moregenerally, consider a schurian algebra A such that QAhas no oriented cycles. Then the trivial extension T(A) of A will be quasi-schurian. 2.1 Symmetric algebras. duality

Let A be a k-algebra.

We denote by DA the usual

Ham~(-,k) mod(A) -~ mod(A°~). The ) algebra A is called symmetric if there exists an isomorphism ~o : A -% DA(A as A - A bimodules. It is well knownthat A is symmetric if and only if there is a non-degenerate A-balanced symmetric k-bilinear mapping 0 : A × A -~ k, see [4]. Wewill point out the following equivalent version of the above property. PROPOSITION 1. Let F be a finite dimensional k-algebra and f E Dr(F). Then there exists a F - F bimodule isomorphism ~ : F -% Dr(F) such that ~(1) = f and only if f satisfies: a) For each 71 , 72 ~ F we have that 727~ = 0 is equivalent to 7~F72 C_ Kerr. ~) 7172 -- 7271 ~ Kerr for every 7~ , 72 ~ F. Proof. straightforward calculations.

[]

REMARKS: 1) The condition c~) may be changed by one of the following conditions cd) If 7F C_ Kerr, then 7 = 0. a’t) If F7 _C Kerr, then 7 = 0. 2) Let {el,’" , en} be a complete family of orthogonal idempotents in F. Then the condition c~) implies that

i) f(ejre~) =for i # j. ii f(e~Fe~) ~ for ea ch i. 2.2 The Supplement Property

for quasi-schurian

algebras.

DEFINITION:Let A = kQA/I be a quasi-schurian algebra. We will say that A satisfies the Supplement Property if for every non zero path 7 there exists a non zero minimal oriented cycle ¢ such that

1)0(7)o(C). 2) All the arrows in 7 lie in the support _C of the cycle C. The path 6 such that 67 = C is called the supplement of 7 in the cycle C.

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E. FernAndezand M.I. Platzeck proved that this property holds for the trivial extension T(A) of a schurian algebra A (see [3]). LEMMA 2. Let A be a quasi-schurian algebra and ~ a nontrivial C is an oriented cycle then ~ = ~-~ = O.

path in kQA. If

Proof. Suppose that C--~ ~ 0. Then we will prove that the set {~, ~-~} is linearly independent over k. This gives a contradiction since A is quasi-schurian. Let a~ + bC--~ = 0 where a and b lie in k. If a ~t 0 then (1 + ba-l-~)~ = O. But ba-~-~ lies in the radical of A and so 1 + ba-V~ is invertible in A. Thus ~ = 0, a contradiction. So, a must be zero. This means that bC5 = 0 which also gives that b = 0. Then, the set {~, ~--~} is linearly independent. [] 3

MAIN

RESULTS

Let A be a finite dimensional k-algebra. Recall that A is called weakly-symmetric if for any indecomposable projective A-moduleP we have that soc(P) ~_ top(P). It can be proven (see [4]) that a weakly-symmetricalgebra is self-injective. Moreover, symmetric implies weakly-symmetric. In case A is a quasi-schurian algebra, we give in this section an answer to the following questions. 1) Whenis A weakly-symmetric?. 2) Whenis A symmetric?. The Supplement Property which was defined above for quasi-schurian algebras is very closely related with these questions, as we will see in this section. THEOREM 3. Let A = kQA/I be a quasi-schurian conditions are equivalent

algebra.

Then the following

I) A is weakly-symmetric. II) A satisfies III)

the Supplement Property.

For each non zero f in HomA(P,Q) the induced morphism HomA(Q, f) : HomA(Q, P) -~ EndA(Q) is non zero, composable non isomorphic projective A-modules.

IV) A satisfies

if P and Q are

the following conditions

a) If a minimal oriented cycle C is non zero, then C(t) ~ 0 for each vertex t in the support C_ of C. b) Let {C__kx,C2... ,Cm} be the set of supports corresponding to the non zero oriented cycles. Then Q A = Before proving the theorem, we will need the following result. LEMMA 4. Let A = kQA/I be a finite dimensional k-algebra, let i be a vertex in QA and 7 a non trivial path in QA, nonzero in A. If soc(P(i)) ~_ S(i) ~ E soc(P(i)), then 7 is a cycle with origin at the vertex Proof. Assumethat soc(P(i)) ~_ S(i) and ~ lies in soc(P(i)). Let j = e(7). Then ~ ~ I(j). But kff = soc(P(i)) ~_ S(i), hence k~ ~_ S(i). But ~ is in I(j), then k~ = soc(I(j)) ~- S(j). This means that S(i) ~_ S(j) and hence i = j. []

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REMARK: Werecall that, if Mis a A module then the socle of M is equal to the right annihilator of tad(A) in M(see [4]). This property will be used in the proof. Proof. of Theorem 3: I) =~ II) Assume that A is weakly-symmetric. Let 7 = arar-l"’al be a non zero path such that o(7) ~ e(7). Therefore ff soc(P(o(7))) : in deed, if thi s is not the case, then Lemma4 would imply that o(7) = e(7), a contradiction. Then there exists an arrow fl such that/~7 is non zero. So, multiplying 7 by the necessary number of arrows ~3x,... ,~3m, we may assume that the non zero path 5 =/~-~-~-~"""/~7 is an oriented cycle or ~ lies in the socle of P(o(5)). Hence the assertion is now a consequence of Lemma2 and Lemma4. II) =:~ I) Assumethat A satisfies the Supplement Property. Let i be a vertex in QA and "~ a non zero path in A such that ~ ~ soc(P(i)). By the Supplement Property, there exists a non zero minimal oriented cycle and such that o(C) = i. If 7 # C, then there is an arrow/~ in C such that ~ # Hence ~ does not lie in soc(P(i)), giving a contradiction. Thus, 7 = C and hence soc(P(i)) = k-~. So, the socle of P(i) is isomorphic to the simple S(i). II) ~ III) III) is just a restatement of II). II) =~ IV) a): Let o(al). Since ~ ~ g we have that the path ~/-- an’" ai+~ai is non zero. Then by the supplement property there is a path ~ such that ~, is a non zero minimal oriented cycle. Since the paths ~ and ai_~ ... a~ have the same starting and ending vertices we obtain that ~ = aai_~.., a~ where a ~ k - (0}. Then ~ ~ ~ = a~--~ and hence b): Each arrow of QAis non zero in A. Hence by the Supplement Property we get that Q A = IV) =~ II) Let 7 be a non zero path. By b) and Lemma2 we get that 7 belongs to some non zero minimal oriented cycle C. Thus the Supplement Property holds since by a) we have that C(o(7)) ~ 0. COROLLARY 5. Let A = kQA/I be a quasi-schurian weakly-symmetric algebra. Then the ordinary quiver QA is the union of all non zero minimal oriented cycles. The other main result in this section is the following theorem. THEOREM 6. Let A = kQA/I be a quasi-schurian weakly-symmetric algebra. Let {.C~, C2"" , ~m}be the set of supports of the non zero minimal oriented cycles. The following statements are equivalent: I) A is a symmetric algebra. II) There are non zero elements a~,... ,am in the field k such that, for each i and j with (~_!)o ~ (e.~j)o ¢ ~ the following condition holds

Wewill need the next lemmato give a proof of this theorem.

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LEMMA 7. Let A = kQA/I be a symmetric k-algebra. Let ~ : A -~ D(A) be an isomorphism o] A - A bimodules and f = ~o(1). Then the following conditions hold for every non zero minimal oriented cycle C. a) If dimkEndA(P(i)) = 2 where o(C) = i, then f(-~) b) f(C(j)) = f(-~) for every j e Proof. b): Follows from/~) in Proposition 1 since 7172 - 7271 Kerr fo r ev ery 71 , 72 E A. a): By b) above it is sufficient to prove that f(~) ~ 0. Since dimkEndA(P(i)) 2 we get that {~,~} is a k-basis of EndA(P(i)) and ~2 = 0. We know that ~ ~ 0. Then by Proposition 1 it follows that there exists A e A such that f(~A~) ~ 0. In particular 0 ~ A~ EndA(P~), and weget that A~ = r~ + s~ where r, s e k. Then ~A~ = r~ + s~~ = rE and this means that f(~)

~ 0 since

0 ~ f(~A~7) = f(r-~).

REMARK: Let f : A ~ k be as in Lemma7, and C be a non zero minimal oriented cycle. It is clear by Lemma7 that f(C(i)) = f(C(j)) for all vertices i,j in __C. Hence f can be defined on the support C as follows, fix a vertex j in _C and let f(C__) = f(C(j)). In this way, we say that f is constant and non zero on __C. Proof. of Theorem 6: I) =~ II) : Assumethat A is a symmetric algebra. Let ~o : A ---> D(A) be isomorphism of A - A bimodules and f = ~(1). To obtain the nonzero constants al,.’. , a,~ we can use the above remark and define ai = f(~i). II) =~ I) : The idea of the proof is to construct a linear functional f : A -+ such that the properties c~’),/~) in Proposition 1 hold. Let us start with the linear functional F : kQA ~ k defined on the basis of the paths in QA as follows: F(7) = ai if there are i and t such that 7 = Ci(t), and zero otherwise. Then II) implies that ~ = F(7)(F(7’))-l~ 7, for nonzero cycles 7 and 7’ with the same origin. The next step is to check that F : kQA ~ k factors through the canonical ~t epimorphism 7r : kQA -+ A, that is, that I C_ KerF. Let 7 = ~=1 c~7~ ~ I be a linear combination of paths 7~ starting at the vertex a and ending at the vertex b for 1 < i < n. Wemay assume that a = b and 7~ is a non zero oriented cycle for i = 1,... ,n. Since ~ = (F(7~)/F(71)) ~-~ 2, 3, ... ,n, we get that 0 = ~ = ~=1 c~ = (~n=l ciF(’~i)/F(71))~Y. But ~ ~ 0. So ~-~n__1 c~F(Ti)/F(’h) and then 7 = ~’~i~ ci(Ti - (F(%)/F(7~)) 71) , therefore F(7) = 0. Hence there exists f:A-~ksuchthatf=Fm Wewill prove that a’) holds, that is A1A_C Kerr implies A1 = 0. Assume that A~ = ~’]~i=1 ci7~ be such that AIA _C Kerr where 7~ is a path in QAfor j = 1,2,... ,n. Observe that A1 = ~=1AI~-] ,AI~A _C A1AC Kerr. Hence it is enough to prove a’) only for each AI~ ; that is, for all linear combination of paths starting at the vertex i. Then we mayassume without loose of generality that i = 1 and o(7~) = 1 forj = 1,2,... ,n. Let {bl,... , b~} be the set of end points of the paths 7~, for j = 1, 2,-.. , n. Let Aj = {i : e(7~) = bj}. Then we can write A1 = ~.=~ ~-~i~A~ c~7. Let us prove that ~eA~ C~ = 0. Assume that ~ is a supplement to the paths {% : This path exists since A is quasi-schurian and the Supplement Property holds. Fix

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an index il in A1, then ~-~ = dj~ for some dj E k and each j E A1- {il}. Multiplying both sides of the above equality by V and applying f we get dj = ](~Tv)/f(~-~Tv,~). Now,by Lemma2 and the fact that f -- F~r we obtain ](~-~) = , and this implies for all i ~ Ai with j > 1. Hence 0 = f(AlP ) = EieAl cif(~-~) that Y~ieA1 C~7, = (Y’~eAI c~d~)~7"£~= (~,~eA, C~f(~))~-r£~/f(7--i~) = 0. Wepoint out that the equality ~,ieAj ci~ = 0 for j = 2, 3,... , r can be obtained in an analogous way. Hence ~ = 0, as we wanted. Wewill prove that f~) holds, that is A~,k~- ,k2,ki ~ Kerr for all A~,,~2 ~ A . Let ,~1 =- ~ c~ and )~. = ~~ d~, where 7~, ~ are paths for each i and j. Assumethat ~75i ~ 0, hence "),i5~. lies in a non zero minimal oriented cycle C such that o(-~i(fi) -- o(~). If 7i5~ -- C we obtain that the supports of 7~._ coincide. Hence F(7~5~)_= F(51,/i) and this implies that f(~) f( 5j-~). Incase e(v~5~) ~ e(C) we_have¢I~7 = 0 and also__F(%hj)._= Therefore, ~hj ~ 0 implies that f(~5j) =._.f(51~). In the same way it can be proved that ~5i = 0 implies that f(~-~hj) Hence the assertion follows. [] 4

A COMBINATORIAL

APPROACH

TO THEOREM

6

Let A -- kQA/I be a weakly-symmetric and quasi-schurian k-algebra. Weassociate to A a graph GS(A). The construction is as follows. Let {Ct,C~"" ,~m) be set of supports of the non zero minimal oriented cycles. Then the set of vertices of GS(A) is (1, 2,... ,m} and the edges are determined as follows a) If m= 1, the set of edges is empty. b) If m> 1, there is only one edge with vertices {i, j} in case (Ci)0 C~(Cj)0 and i 7~ j. It is not difficult parallel edges.

to see that GS(A)is a connected graph without loops and non

NOTATION: A chain C in GS(A) joining the vertices v~ and vk is a sequence of vertices and edges v~A~v~A~.., vk-~Ak-lVk where for each i the edge Ai has vertices vi, v~+~. Wesay that the length of C is k- 1. Let B -- WlBlW2B2... Wn-lBn-lwn be another chain in GS(A). Wewill say that the composition A o B is defined va = w~ and we let A o B be the chain v~ Alv2A2 " "vk-~ Ak-~v~B~w2B~...

wn-~ Bn-~w,~.

A chain v~A~v2A2... Vk-~Ak-~va is called reduced if vi_~ ~ vi+~ for each i = 2, 3,... , k - 1. The set of all chains in GS(A)is denoted by Ch(GS(A)). Usually we shall only be interested in reduced chains, and unless the contrary is explicitly stated, we shall assume that all chains under discussion are reduced. A cycle C in GS(A) is a chain of the form v~A~v2A~...v~_~Ak_~v~. If the vertices v~, v~,.. ¯ , v~_~are all different, then the chain C is called a minimalcycle. Weobserve that a minimal cycle has at least three vertices. Let C be the chain v~A~v~A2.. "Vk-~Ak-~Vk in GS(A). We denote by C the support of C which is defined as the subgraph of GS(A) with vertices v~,... Vk and edges A~,... , Ak_~. Given a minimal cycle C = VlAlV2A2"" Vk-IAk-lVl in

SymmetricQuasi-SchurianAlgebras

107

GS(A), we denote by C(vi) the cycle viAivi+l.., vk-iAk-~VlA~V2.. "Vi-l.Ai-lVi, for 1
~ = ~i~(t) Vt~ (2)0 n (2)0. Now, we can define a map CA: Ch(GS(A)) -~ in the foll owing way

Wepoint out that, if C1 and C2 are chains such that their composition is defined, then we have that CA(C~o C~) = CA(C~)¢A(C~). Let D be the chain v~A~v~A2.., vk-~A~-ivk. We denote by D-~ the chain -1 oD) = 1 and hence vkAk-~Vk--~ "’" v2A~v~. In this way CA(DoD-~)= CA(D -~) -~ . CA(D = CA(D) Let C be a minimal cycle in GS(A)with support __C. It is clear that CA(C(vi)) CA(C(vj))for each vi, v1 in _C. HenceCAcan be defined on __Cas follows, fix a vertex v in __C and let ~bA(C ) = CA(C(v)). The existence of the non zero constants al,a~,... ,am which are required in Theorem6 for A to be symmetric is very closely related with the structure of the graph GS(A) and with the function CACh(GS(A)) ~ k ca n b e se en in th e next theorem. THEOREM 8. Let A be a quasi-schurian weakly-symmetric k-algebra, and {C~, C~... , Ca} be the set of supports of the non zero minimaloriented cycles. Suppose that the non zero constants Aij (t) above defined do not depend on the common vertices t e (C_A)on (C_~)o. Then a) If the graph GS(A)is a tree, then A is symmetric. b) Suppose that GS(A) is not a tree. Then A is symmetric if and only if the function CA: Ch(GS(A)) --~ k satisfies CA(C) = fo r ea ch mi nimal cy cle C in GS(A). Proof. a): Assumethat GS(A)is a tree. Let us prove that in this case the required non zero constants always exist. Since GS(A)is a tree we have that for each vertex j ~ 1 in the graph GS(A) there exists only one nontrivial chain Dj in GS(A) joining the vertex j with the vertex 1. Hence we can define a~,a2,... , am in the following way, let a~ = 1 and a~. = CA(D~.) if j ~ 1. The next step is to prove that Aij = aia’~ 1 in case (~i)o n (_Cj)o~ and i ~ j. LetA beth e e dge with vertices i and j. Since GS(A) is a tree we have that Di = (iAj) o Dj and hence ai = CA(Di) CA((iAj) o Dj) = dPA(iAj)¢A(DI) = Thus, Aij = ala~ ~ as we wanted.

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b):(=~) assumethat A is symmetric,that is the non zero constants al, a,.~.,... , of Theorem6 exist. Let C = vlA~v2A3.., vk-lAk-lVl be a minimal cycle in GS(A). Then CA(C) -~ avl

av2 av2 av3 ¯ ¯ "ark_

1 avll

(¢=) Assume that CA(C) = 1 for each minimal cycle C in GS(A). Wewill the next Lemma. LEMMA: If Bj and Dj are two chains in GS(A) joining the vertex j with the vertex 1 where j ~ 1. Then CA(B~.) Proof. If Bj o D~-1 is a minimal cycle, then by hypothesis we have 1 = ¢~(Bj D~1) = CA (Bj)¢A (D~) -1. Hence CA(B~) = CA(D~). Assumethat B~ o D~-; is not a minimal cycle. Then it is not difficult to see that Bj and Dj have a decomposition Bi = F1 0 F2, Dj -- G1 o G2 in such way that F1 o G~"1 is a minimal cycle in GS(A). Hence Bj 0 1 = F;o F2 0 G~1o G~"~ implies CA(Bj o D~- 1 ) = CA(F; 0 G~- ~) CA(F2 0 G~ 1). But we knowby construction that CA(F1o G~";) = 1. Then ~bA(Bj o D~-1) = ~bA(F2 o G~~) but now, the length of the chain F2 o G~; is smaller than the length.of Bi o D~-1. Hence by induction we can obtain that CA(Bj o D~-1) = 1 and conclude that CA(BJ) = CA(Di). Now,using this lemmait is possible to define the required constants. Let Dj be a chain in GS(A) joining the vertices j and 1 where j ~ 1. Then we define al = and a~ = CA(D~)if j ¢ 1. By the above lemma we have that the constant aj is well defined. Nowwe will check that )t~j = a~a-~~ in case (C_~)o f3 (C_j)o ¢ 0 i ¢ j. Let A be the edge with vertices i and j, let Dj be a chain joining the vertex j with the vertex 1. Then the chain B~ = (iAj) ~ D~ is a chain in GS(A) joining the vertices i and 1. Hence a~ = This implies that )t~ = a~a~~ [] From the above theorem we obtain the following corollaries. Let A be a quasi-schurian and weakly-symmetric k-algebra, (~1,’"" C-m} be the set of supports of the non zero minimal oriented cycles and A~j (t) ~ k be the family of non zero constants such that C~(t) ~j(t)~(t) for t COROLLARY 9. If the graph GS(A) associated to the quasi-schurian weaklysymmetric k-algebra A is a tree, then the following conditions are equivalent I) A is a symmetric algebra.

II) ~ = ~ (t) e (¢~)o ¢3 (CA )o, i # COROLLARY 10. Suppose that the graph GS(A) associated to the quasi-schurian and weakly-symmetric k-algebra A is not a tree. Then the following conditions are equivalent. I) A is a symmetric algebra. II) X~j = )t~j(t) (C~)~ (Cj)o i ~ j, and the f unction CA : Ch(GS(A)) k defined above satisfies CA(C)= fo r ea ch mi nimal cy cle C. EXAMPLE: Let A be the factor algebra of the path algebra kQ for Q the quiver

SymmetricQuasi-SchurianAlgebras

modulotheidealI =< GIG0--aG4Ot3,

109

O~0G2G4~ Ot30~2~i,

OZ6G0, G6GhG6~ GhG3~ GIGh~

{0}. A is quasi-schurian and moreover weakly-symmetric since ~his algebra satisfies the Supplemen~ Proper~y (see Theorem 3). Wewill prove tha~ A is symmetric if and only if abc= 1. Let C~ = ~;ao~, C2 = a4a3~2 and C~ = a~a~. Then {C~,C2,Ca} is the set of supports of the non zero minimal oriented cycles. Hence the graph GS(A) 1

A1

2

Let us now compute the family of non zero constants A~j(t) E k, such that C~(t) = A~j(t)Cj(t) for t E (C._2i)0 I-I (Cj)0. In this case C1(0) = CqaoC~2,C2(0) ~4~3~2, C1(2) = ~2~1~o, and C~(2) = a2c~4a~. Using the relations given in ideal I we have C--~(0) = a~-~(0), C--~(2) = a~-~(2). Hence = A~(0) = A~(2=a. In an analogous way we obtain that A~ = A~3(1) = -1, A~ =A~3(3) = Consider the following minimal cycle C in GS(A), C = 1A~2A23A31. Then CA(C)= A~2A~sA3~ =abc, and {_C} is the set of supports of the minimal cycles in GS(A). So, by Corollary 10 we get A is symmetric if and only if abc = 1. 5

A CONNECXION

WITH TRIVIAL

EXTENSIONS

OF ALGEBRAS

Let A be a quasi-schurian weakly-symmetric algebra, {C_i,.. "C_m}be the set of supports of the non zero minimal oriented cycles. In case it is possible to select exactly one arrow in each of the C_C_i’s, we fix such a choice, and denote by C(A) the set consisting of the chosen arrows. Thus, C(A) is a set of arrows of QAsuch that C(A) ¢q (~i)~ has only one arrow for each i = 1, 2,... , m. The ideal generated by C(A) will be denoted by Ic. Moreover, C(A) induces a sub quiver Qc of QAas follows (Qc)o (QA)o an (Qc)= (QA)I - C(A)

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Hern~ndez

Let ~ be an arrow in QA’Wewill denote by Suppl(~) the set of supplements of /~. Thus, a path 5 lies in Suppl(~) if and only if 5f~ is a non zero minimal orieated cycle. Our aim in this section is to give a proof and someconsequences of the following result. THEOREM 11. Let A be a quasi-schurian symmetric algebra. If there exists a choice of arrows C(A) as above then A/Ic is a schurian algebra, and moreover A ~_ T(A/Ic) where T(A/Ic) is the trivial extension o/A/Ic. In the proof of this theorem we will need the following lemmas. LEMMA 12. Let A be a quasi-schurian weakly-symmetric algebra. If there exists a choice of arrows C(A) as above, then A/Ic is a schurian algebra and Qc is the ordinary quiver associated with A/ Ic. Proof. Assume that A = kQA/I where I is an admissible ideal. Let ~ : kQc A/Ic be defined as follows ~((i) = ~r(~) where 7r : A/Ic is th e c anonical epimorphism. Weget that rad(A/I¢) = radh/Ic and A/radA ~_ (A/Ic)/rad(A/Ic) since Ic C_ radA and ~ is an epimorphism. Hence A/Ic is basic and (~(i) : (Qc)o} is a complete family of orthogonal primitive idempotents of A/Ic. So, to obtain that Qc = QA/Ic it is enough to prove that {~(a) : c~ (Qc)l} is a k -basis of rad(A/Ic)/rad2(A/Ic). First, observe that a ~ C(A) implies that ~(c~) is zero and also does not lie in rad2(A/Ic). Therefore ~((~) : a ¯ (Qc)l} is a k-basis of rad(A/Ic)/rad 2(h/Ic) since rad(A/Ic) = radA/Ic. Finally let us prove that A/Ic is schurian. Using the k-module isomorphisms 7r(~-~) (A/Ic)~r(~) ~_ ~-~A~/’g]~Ic~for all i, j. It is enoughto prove that dimk~Ic~ 1 for each i, since A is quasi-schurian. But this follows easily because each non zero oriented cycle contains an arrow from ~(A). DEFINITION:Let F be a finite dimensional k-algebra. Wesay that maximal in case x ¢ 0 and xw = wx = 0 for all w E radF.

x ¯ F is

REMARK Let F = Qr/I with I an admissible ideal. Let 5 be a non zero path in Qr. Then ~i is maximalin F if and only if (is = a5 = 0 for all arrows a ~ Qr. LEMMA 13. Let A = QA/I be a quasi-schurian weakly-symmetric algebra with I an admissible ideal. If there exists a choice of arrows C(A) as above and QA has no loops then the next statements hold, where ~r : A -+ A/Ic is the canonical epimorphism. a) e-~Ice-~ = 0 for all ~ ¯ C(A). Therefore ~r(~) ~ 0 for any supplement of an arrow ~ in b) Let ~ be a path in QA" Then ~(~) is maximal if and only if there exists ~ C(A) such that 7 ¯ Suppl(~). c) LetC(A) = (~1,~2,"" if~r}. Then the set (~r(5i) : ~ ¯ Suppl(~), 1 < i is a k-basis of the vector space generated by all the maximalpaths in A/Ic. Proof. a): Suppose that 7 is a non zero path and ~ ¯ e-~Ice-~ for some f~ ¯ Let (f be a supplement of 7, which exists since A is weakly-symmetric (see Theorem 3). Now, we may assume that ~, contains an arrow ~’ of C(A) because ~ Ic andA is quasi-schurian. But f~ and (i have the same starting and ending vertices. Hence

SymmetricQuasi-SchurianAlgebras

111

(f = f~ and the non zero oriented cycle ]~7 has two arrows in C(A), a contradiction. So, ~ = 0 and therefore eo(-~Ice-~ = O. b): Let 7 be a path in QA" Assume that ~r(V) is maximal. So, V ~ 0. Let be a supplement of 3’. Then (f contains an arrow f~ in (:(A) since Ic and ~3" i s a non zero minimal oriented cycle. Nowwe will prove that ~ = f~ using the fact that n(3’) is maximal. Considering a decomposition of ¢I as 71f~72 we obtain by that 0 ~ ~r(~2---~-~) -- ~r(~-~)~r(~)r(~) and therefore 3’1,3’2 are trivial paths maximality of ~r(V). Hence ~ = ~ and this means that 3’ Suppl(B). Assumenow that 3" E Suppl(~3) with ~ in ~(A), and let us prove that ~r(V) maximal. From a) we obtain that ~r(V) ~ 0. Let a be an arrow in QA such that ~r(~)~r(V) ~ 0. So ~ ~ 0 and therefore there exists a supplement # of aT. But and pa have the same starting an ending vertices. Then there is c in k - {0} such that ~ = c~--5 since A is quasi-schurian, a contradiction because I is an admissible ideal. Hence ~r(~)~r(V) = 0 for all arrow a e QA"In an analogous way we obtain that ~r(~)~r(~) = 0 for all arrow a ~ c): The set {~r(~i) 1 < i < r} generates all the maximal paths since b) holds and A/Ic is schurian by Lemma12. Let ~’~ir__l ai~r(~) = 0 with as ~ k. Then ~i=x ai~i e Ic, and using a) we obtain aj~ --- eo(~)(~i=~ ai~)~-e(~,) = 0 since A is quasi-schurian and f~i,/~j do not have the same starting and ending vertices for i ~ j. So, a~. -- 0 for j = 1,2,... ,r. Therefore {~r(~) 1 < i < r} is a linearly independent set and hence a k-basis since we knewthat it generates all the maximal paths. [] LEMMA 14. Let A = kQ A / I be a quasi-schurian weakly-symmetric algebra with I an admissible ideal. If QAhas a loop then A "" k[x]/ < x~ > . Proof. Let n be the number of vertices of QA" If n = 1 then QAhas only one loop since A is quasi-schurian and I is an admissible ideal. Therefore in this case A ~_ k[x]/< x2 > since A is quasi-schurian. Assumen > 1 and let a be a 10op. As before we have only one loop at the vertex o(a). Let ~3 be another arrow starting o(a) and let J be a supplement of ~ (see Theorem3). Hence there exists c E k- {0} such that ~ = cJf~ since A is quasi-schurian, a contradiction because I is admissible. So, n has to be 1 and in this case we have already proved the lemma. [] Before giving a proof of Theorem11, let us recall from [3] (see also in [7],[8]) the description of the ordinary quiver and relations of the trivial extension T(A) of a schuria~ algebra A. Let A = kQA/I be a schurian algebra with I an admissible ideal, andpl,p2,. ¯ ¯ , pe be paths in QAsuch that {p-~, ~-~,. ¯ ¯ , ~} is a k-basis of the vector space generated by all the maximal paths in A. Then the vertices of QT(A) are the vertices of QA and (QT(A))I---- (QA)I(J (~p~,/~p~,... , f~p, }, where~, is an arrowstarting e(pi), ending at o(pi) and not belonging to QAfor i = 1,2,... ,t. Weobserve that all arrows of QT(A) are in oriented cycles. An oriented cycle C in QT(A) is called elementary if there exists a vertex j in C with C(j) = q~ for some i = 1, 2,... , t and some path q maximal in A with the same starting and ending vertices as pl. Wecan describe now the relations of T(A) given in [3]. THEOREM 15. ([3]) Let A = kQA/I be a schurian algebra with I an admissible ideal. Let IT(A) be the ideal of kQT(A)generated by the following relations:

112

Herndndez i) The composition of n + 1 arrows in an elementary oriented cycle of lenght n. ii} The composition of arrows not belongin# to a same elementary oriented cy-

cle. iii} The elements q - bq~ where q, q~ are paths in QT(A) having the same ending and starting vertices, and such that one of the following conditions holds. a} -~ = bq-7 with b E k - {0} and q, q’ paths in QA. b} There is a path ~ in QA such that uq = at-1 "" "~zo~pi~,~" .ar+~ar and t .Ott s-x . "’a~a~t~p~am’" s+~ 8 are elementary cycles. Then b is defined by b = a~/az for non zero a~,az ~ k with a,~...(~ = a~ and !a,~...a~ = a~. Then the ideal IT(a) is admissible and T(A) "~ kQT(A)/IT(A}. 1]qt __. Ott

Nowrecall the well known isomorphism of T(A) - T(A) bimodules ¢ : T(A) D (T(A)) given by ¢ (xx, f~) (xz, f~) = f~ (xz) + f2 (x~), and consider the F = ¢(1, 0) T( A) -- + k. This fun ctional is handy to describe the constansts of iii) in the above theorem. Westart by giving an explicit description of it. Let {ffi-,~-~, ¯ ¯ ¯ ,Pt,Pt+l,pt+2,"" ,~s} be a basis of A, extending the chosen basis {p~,p~,’" ,PT} of the vector space generated by all the maximal paths in A, we will denote by {~*,~-ff*,... ,~-~s*} the basis of D(A) dual to {~-~,~,... ,~s}. Let ~ : kQT(A) -+ T(A) be the map defined by ¢(a) = (~,0) for (Qahand ¢(f~p~) = (0,~-~*). It is not difficult to see that the induced linear functional kQr(A) ~ defined by thecomposition F¢ s ati sfies the foll owing condition: f is constant and non zero on the support _C of each minimal oriented cycle C non zero in T(A) (see Lemma7 and its remark). Moreover, for the elementary oriented cycle q/3~ we have that ~ = f(q/~pi)~. Therefore the condition iii) in Theorem 15 can be changed by iii)’ The elements ~ -f(~t/)(f(Jg.~))-~, where ~,~ are paths QT(A) having the same ending and starting vertices and such that there exists a path t~ with ~, and ~ elementary oriented cycles. This last condition will be used in the proof of Theorem11. Proof. of Theorem 11: Let A = QA/I where I is an admissible ideal and denote by ~ the congruence class 7 + I in A. If QAhas a loop we get by Lemma14 that A/Ic "~ k and hence A _~ T(A/Ic). Assumethat QAhas no loops and let ~(h) = {/3~,/3z,... ,/3t}. Let A A/Ic, then by Lemma12 we have that A is schurian and its ordinary quiver is Qc. Now, for each i = 1, 2,... , t we choose a path p~ in Suppl(/3~). So, By Lemma13 the set {Tr(p-~) 1 < i < t} is a k-basis of the vector space generated by all the maximal paths in A where 7r : A --r A = A/Ic is the canonical epimorphism. By Theorem15 we have that Q,T(A) = QAUC(A)= QA"Hence ¢ kQ, T( A) -- ~ h where ¢( 7) = ff an epimorphism of k-algebras. So, we have to check that Ker¢ = IT(A) to obtain T(A) -~ A. Wewill need the following Lemmas: LEMMA (A): Let C be an oriented minimal Cycle in QT(A). Then C is non zero T(A) if and only if it is non zero in Proof. (=~) : Assume C is non zero in T(A). Then by Theorem 15 there is an elementary oriented cycle q/3i such that C(j) = q/~i for somevertex j in C. Theretbre

SymmetricQuasi-SchurianAlgebras

113

C(j) is non zero in A and by Lemma7, C is non zero in A. (~) : Suppose that C is non zero in A. Then by the definition of C(A) we that C contains an arrow from C(A). So, C is an elementary oriented cycle and therefore it is non zero in T(A). Let {C_I,... C_m}be the set of supports of the non zero minimal oriented cycles in A. Since A is symmetric we have by Lemma7 (see also its remark) a linear functional ~o : A ~ k non zero on the supports _C_i for all i = 1, 2,.-. , m. Makinga change of variables/~i ~ ai~ with adequate a~ E k - {0} for all i = 1, 2,... , t we may assume that ~(p-~) = 1 for all i = 1,2,... ,t. Now,this functional satisfies the following property. LEMMA (B): Let q be a path in QA such that ~r(~) caw(~) with c aE k- {0}and some i = 1,2,... ,t. Then Cq = Proof. Since__ ~- cq~ E____Ker~r= Ic we get by Lemma13 a) that ~ = %~. Therefore cq = ~p(q~) since ~p(p~fl~) = 1. REMARK: Let C be a minimal oriented cycle non zero in T(A). Then this lemma give us that ~(-~) = f(C) where f : kQT(A) --~ is the line ar func tional defi ned above. Let us prove that Ker¢ D_ IT(A) : By Lemma(A)and Lemma2 we obtain that the relations i) and ii) in Theorem15 are zero in A. Using Lemma(B)and its remark we obtain that the above relations iii)’ are zero in A because it is quasi-schurian. So, Ker¢ D IT(A). Finally, we will check that Ker¢ C IT(a) : Let 7 E Ker¢. Then 3’ is zero in A and therefore 7 is zero in T(A) since A is a sub algebra of T(A). So, 7 E IT(A) because k(~T(A)/IT(A) ~ T(A). Hence Ker¢ C IT(A). Nowit is easy to obtain the main result of this section. THEOREM 16. Let A be basic connected finite dimensional k-algebra. The following statements are equivalent 1) There exists a schurian basic triangular algebra ~ such t hat A-~T(A~). 2} A is symmetric quasi-schurian, and we can choose a set C(A) such that the quiver Qc has non oriented cycles. If these conditions hold, then A’ ~_ A/Ic where Ic is the ideal generated by C(A) in A. Proof. Follows from Theorems 11 and 15. Another application of Theorem11 is the following result. THEOREM 17. Let Q be a quiver without oriented cycles, and A an iterated tilted algebra of type Q. /f P = T(A) is quasi-schurian then, for each choice C(F) above and such that the quiver Qc has non oriented cycles, F/Ic is an iterated tilted algebra of type Q. Proof. It follows immediately from Theorem 11 and the next lemma.

[]

114

l~Iern~ndez

LEMMA 18. Let Q be a quiver without oriented cycles, and A an iterated tilted algebra of type Q. Let A~ be a basic finite dimensional k-algebra. If T(A) ~ T(A’) and A’ has finite global dimension then A’ is iterated tilted of type Q.

Proof. The proof is based on knownresults about derived categories and repetitive algebras (see [2],[5] and [6]). Since T(A) -~ T(A’) we get that the repetitive algebra /~ of A is isomorphic to the repetitive algebra/~ of A~. In particular, we obtain that the triangulated category modAis triangle equivalent to mod~~. Since A and A~ have finite global dimension we have the diagram Db(A) -~ mod£ ~ mode’ ~ ’) Db(A where ~ denotes a triangle equivalence. Thus Db(A) is triangle equivalent Db(A’), and therefore A’ is an iterated tilted algebra of type Q (see [5] or [6]).

Weget nowa useful approach to obtain iterated tilted algebras of a given tree class, generalizing an analogous result proven in [3] for Dynkinquivers.

COROLLARY 19. Let Q be an oriented tree and F -~ T(kQ). For each choice as above with Qc without oriented cycles we have that F/Ic is an iterated tilted algebra of type Q.

Proof. It follows immediately from Theorem 17.

EXAMPLE: Let Q be the following

oriented

tree

Or4

~3

and F = T(kQ) be the trivial extension of kQ. Considering the maximal paths Pl = 0~30t20~1, P2 = o~40tl, andp3 = c~5~1 we obtain by Theorem15 that Qr is

115

SymmetricQuasi-SchurianAlgebras

0~4

~ ¯

~ ¯

~5

Let C(F)

= {0~3,0~4,~5}

,

and A = F/Ic. By Lemma12 we get that QAis

So, by Corollary 19 we have that A ..,_ kQA/ iterated tilted algebra of type Q.

< ~::~20~l~p~,

O~20~l~p;

~ > is

an

ACKNOWLEDGMENTS I would like to thank Prof. Maria In~s Platzeck for her many, very helpful comments and suggestions, and for a careful reading of this paper. Finally, I also thank the referee for the commentsabout the paper. REFERENCES [1] I.Assem,D.Happel,O.Roldfin. "Representation-finite bras". J.Pure Appl. Algebra 33(1984). 235-242. [2] D.Hughes,J.Wachbusch. "Trivial extensions of tilted Math. Soc. 46(3)(1983) 347-364. [3] E.FernAndez. Ph.D. Thesis "Extensiones triviales adas", 1999.

trivial

extension alge-

algebras" Proc. London

y filgebras inclinadas iter-

[4] F.W.Anderson, K.R.Fuller. "Ring and categories of modules". Second edition. Springer-Verlag 1992.

Herndndez

116

[5]I.Assem, "Tilting

Theory"- an introduction. Publ., vol 26, part 1 (1990).

Topics in algebra. Banach Center

D.Happel "Triangulated Categories in the representation mensional algebras". Cambridge Unit Press (1988).

[7]E. Fernandez, M.I.

Platzeck "Presentations of trivial of S. Brenner". Preprint (2000).

Theory of finite

di-

extensions and a theorem

[8] H. Asashiba "The derived Equivalence Classification of Representation-Finite Selfinjective Algebras". Journal of Algebra 214, 182-221 (1999).

On lattices at the ends of connected components of the Auslander-Reiten quiver ALFREDO JONESCentro de Matem~tica, Facultad tevideo, Uruguay, E-mail: [email protected]

de Ciencias,

Igu£ 4225, Mon-

ABSTRACT Let R be a complete discrete value~tion ring with radical T/~r and residue field R/~rR of characteristic p dividing the order of a finite group G. We show that a virtually irreducible RG-lattice L with exponent ~ra lies at the end of its Auslander Reiten componentif and only if L/rca-IL is indecomposable. 1

INTRODUCTION

Let G be a finite group, p a prime that divides IGI, the order of G, and R a complete rank one valuation ring of characteristic zero with maximal ideal Rr such that p is the rational prime with Rp C_ R~r. Weconsider RG-lattices, that is finitely generated RG-modulesthat are free as R-modules. In [2] it was shown that an absolutely irreducible RG-lattice which is indecomposablemodule~r lies at the end of a connected component of the stable Auslander Reiten quiver. In this note we extend this result giving a necessary and sufficient condition for a virtually irreducible lattice to lie at the end of its componentof the stable Auslander Reiten quiver. Absolutely irreducible lattices, as well as absolutely indecomposablelattices L with rank, rk L, prime to p, are special cases of virtually irreducible lattices. If RG is of infinite type there exist infinitely many indecomposables of rank prime to p, therefore if these are absolutely indecomposable, there exist infinitely many virtually irreducible lattices. 2

DEFINITIONS

AND NOTATIONS

For any indecomposable non projective A(L) of L is of the form

RG-lattice L, the almost split

.A(L) :0 --~ ~ L --+ M(L) ~ L ---+ 117

sequence

118

Jones

where f~ is the Heller operator. The lattice L lies at the end of its component of the stable Auslander Reiten quiver when the projective free part of M(L) indecomposable. Set HomRa(L, L) Hom(L, L) -- Proj Hom/ta(L, where ProjHomRa(L,L) is the submodule of those homorphisms which factor through a projective lattice. Then, as shown in [4], Horn(L, L) has a simple socle as a moduleover itself and A(L) is a pull-back of a projective cover of L along a generator of this socle. The exponent of a lattice L, as defined in [1], is expL = ~ra if RTra is the annihilator of the torsion module Hom(L,L), thus expL is the least power of such that multiplication of L by exp L factors through a projective lattice. If RIGI = R~~ then 0 < a < n, where a = 0 for L projective. A lattice L with exp L = ~ra is virtually irreducible if it is absolutely indecomposable and a-1 soc Horn(L, L) = Hora(L, L)~r This condition remark that if R(exp L)(rk L) we refer to [1], 3

THE

is equivalent to the following: if exp M(L) = b th en b < a. We L is virtually irreducible then exp L is a power of 7r because then = RIGI. For these and other results on virtually irreducible lattices [4] and [5].

THEOREM

Let v denote the p-adic valuation of R. LEMMA. If L is virtually irreducible then for every lattice summand ofM(L), v(rkX) > v(rkL).

X which is a direct

Proof. Let expL = rc a and RIGI = ~rn, then since L is virtually

irreducible,

v(rk L) = v(IGI) v(expL) : n On the other hand, if Tr is the trace function, from [1, Proposition 4.2] we know that for any RG-lattice X R(exp X) Tr(HomRa(X, X)) bThus if exp X = 7~ R(rk X) C_ Tr(Homna(X, X)) ~- ~ Therefore v(rk X) _> n - b, but b < THEOREM. If L is a virtually irreducible lattice with expL = r~, a ~ 2, then the projective free part of M(L) is indecomposableif and only if -~ = L/~ra-l L is indecomposable.

OnLattices at the Endsof Components

119

Proof. The condition is clearly necessary because since 7r a-x idLE socHom(L,L), M(L) is the kernel of a projective cover of the module ~ ([1, Theorem2.4]). Assumenow ~ indecomposable. As is well known the Auslander R.eiten sequence of L decomposes module 7r a-1, so M(L) ~- L @f~-£. Therefore if M(L) decomposes it must have an indecomposable direct summmandX such that X - L. But then rk X = rk L, and this is a contradiction with the lemma. [] COP~OLARY. If a virtually irreducible lattice L is indecomposable module 7r then the projective free part of M(L)is indecomposable. Proof. It suffices to remark that if exp L = ~r then the projective cover of L gives the Auslander Reiten sequence for L so the result also holds in this case. [] REFERENCES [1] J. F. Carlson, A. Jones. An exponential property of lattices J. London Math. Soc. 39 (1989), 467-479.

over group rings.

[2] A. Jones, S. Kawata, G. O. Michler. On exponents and Auslander Reiten components of irreducible lattices. Archiv der Mathematik, to appear. [3] R. KnSrr. Virtually irreducible lattices. 99-132.

Proc. LondonMath. Soc. 59 (1989),

[4] K. W. Roggenkamp.The construction of almost split sequences for integral group rings and orders. Comm.Algebra 5 (1977), 1363-1373. [5] J.Th~venaz. Duality in G-algebras. Math. Z. 200 (1988), 47-85.

Factorisations gebras

of morphisms for wild hereditary

al-

OTTOKERNER Mathematisches Institut, Heinrich-Heine-Universit~t, Universit~itsstra/~e 1, D-40225 Diisseldorf, Germany,E-mail: kerner~mx.cs.uni-duesseldorf.de

ABSTRACT Let H be a connected wild hereditary path-algebra. It will be shown that morphisms between modules, contained in the different parts of the module category H-rood have strong factorisation properties.

If A is a finite dimensional connected tame hereditary algebra, X a preprojective, respectively Y a preinjective, moduleand T a regular component, that is a regular tube in the Auslander-Reiten quiver F(A) of A, then each homomorphismf : X Y factorises through add T, the additive closure of T. It is the aim of the paper, to showthat muchstronger factorisation properties hold, if H is wild hereditary. Let H = k Q be a connected wild hereditary path-algebra, over somefield k. This means that Q is a finite connected quiver without oriented cycles which is neither of Dynkin nor of Euclidean type. Since H is hereditary the Auslander-Reiten translations T = ~’H ~ D Ext,(-, H), respectively r- = ~-~ ~- Ext,(D-, H), are left exact, respectively right exact, functors, where D = Hom~(-,k). Recall that H-moduleX is called preprojective, respectively preinjective, if "~mX, respectively ~- X, for all T-’~X, is zero for m >> 0. A moduleX is called regular, if Tin(V-reX) integers m. Wesay that a morphism f : X ~ Y ]actorises through a module M, if there exist morphisms fl : X --~ M and f~ : M -~ Y such that f = flf~. The main result of the paper is: THEOREM. Let H = kQ be a finite dimensional connected wild hereditary algebra, X1 a preprojective, X2 a regular and X3 a preinjective module. I] R ~ 0 is regular, then one has. (a) Each homomorphism.f : X~ --~ X2 ]actorises through r-mR ]or m >> O. (b) Each homomorphismg : X2 -+ X3 ]actorises through rmR for m >> O. 121

122

Kerner

(c) Each homomorphismh : X1 -~ X3 factorises

through ~-mRfor tin] >> 0.

For the proof of the theorem, a result of Lukas[9] is essential. It says that for any two nonzero regular H-modules X and Y there exist monos X --+ traY, respectively epis T-raX ~ Y, for rn >> 0. F. Lukas used infinite dimensional H-modulesfor his proof. A proof of this result, without infinite dimensional modules, was sketched in [8, 6.5]. For the convenienceof the reader, this proof will be presented in section 1. It was shown in [8, 6.4] that for any two nonzero preprojective, respectively preinjective, modules X and Y, there exist monos X -~ r-mY, respectively epis rraX -~ Y, for m >> 0. These results on the existence of monos, respectively epis, can be extended to the case, that the modules X and Y are in essentially different parts of the category H-modof finite dimensional left H-modules. COROLLARY. Let X~ ~ 0 be a preprojective, X2 ~ 0 a regular preinjective module. Then one has. (a) There exists a monoX~ -+ rraX~, for [m[ >> 0. (b) There exist monos Xi --> rmXa, (i = 1, 2) for m >> 0. (a’) There exists an epi vraX: ~ X3, for ]mI >> 0. (b’) There exist epis ~’-mX~-~ X~, (i = 2, 3) for m >> 0.

and X3 ~ 0 a

Since H = k Q is a path-algebra, the category H-modis equivalent to the (.’ategory of finite dimensional k-linear representations of Q, and we will not distinguish between these categories. Morphismswill be written opposite to the scalars. For general results on the representation theory of finite dimensional algebras I refer to [1, 11], for standard results on wild hereditary algebras one mayconsult [5]. 1

MONOS AND EPIS

BETWEEN

REGULAR

MODULES

1.1 Let H be a connected wild hereditary algebra and X, Y be nonzero regular modules. It was shown by Baer [2] that HomH(X,~’"~Y) ~ for m >>0. On the other hand HomH(T’~X,Y) = 0 for m >> 0 [3]. Denote by (-,-) : ~ x ~ -~ ~ the homological bilinear form, see [11]. Then we have (diraX, dira~-’~Y) = dim HomH(X,TraY) -dim Ext,(X, 7"~Y). Consequently, for m >> 0, we get (dimX, dim~-raY) = dimHomH(X, TraY), since Ext,(X, TraY) ------- D HomH(~’"~Y, ~’X) = 0 for m>> 0. It follows from the spectral properties of the Coxeter transformation, that (dimX, dimTraY) grows exponentially in m, see for example[10]. This implies the well known LEMMA.dimHomH(X, vmY) >> 0 ]or m >> 0. 1.2

The main result of this section is the following.

PROPOSITION. Let H = kQ be connected wild hereditary and let X, Y be nonzero regular H-modules. Then there exists a mono f : X -~ vmy, respectively an. epi g : r-raX ~ Y, for m >> 0. This result first was shown by Lukas [9, 2.3] using infinite dimensional Hmodules. The proof given here has already been indicated in [8]. By duality it is enough to show the first part. The proof is based on the following two lemmas.

Factorisations of Morphisms for WildHereditaryAlgebras 1.3 LEMMA.Let X,Y be nonzero regular H-modules. TroY, respectively generated by r-mY, for m >> 0. For a proof see [9, 2.2] or [5, 10.7].

123 Then X is cogenerated by

1.4 Call an indecomposable regular H-module E additively elementary, respectively elementary, if each short exact sequence 0 -~ U ~ Er -~ V --~ 0 with U, V regular and r _> 1, respectively r = 1, splits, see [7, 6]. Since the Auslander-Reiten translation ~- defines an equivalence on the category H -reg of regular H-modules, E is (additively) elementary, if and only if so is ~mE, for any integer m. Call a linear map f : X -+ Y right minimal, if no indecomposable direct summandof X is in the kernel Ker f of f, see [1]. LEMMA. For an indecomposable regular H-module E there are equivalent. (a) E is additively elementary. (b) Let R be regular and f : Er --¢ R right minimal. Then Kerf is preprojective. (c) Let R be regular and f : Er -~ R right minimal. Then vmf : ~’mEr -~ ~’mR is injective for m >> 0. Proof. The implications from [7, 1.2].

(c)~(b)~(a) are clear,

the implication (a)=~(c) [~

1.5 If H is connected wild hereditary and Mis indecomposable preprojective or preinjective, then Ext~/.(M, M) = 0 and HomH(M,M) = k, hence q/~(diraM) (dimM, dimM) = 1. Consequently an indecomposable module X with qH (dimX) < 1 is regular. IfH = kQ and Q has two vertices 1,2 and r _> 3 arrows al,... ,at from 1 to 2, then qH((X, y)) ---- (X y)2_ (r -- 2)xy. Let fo r ex ample E be t heindecomposable representation with dimE = (1, 1) and linear maps E(cq) id : k ~ k and E((~i) = 0 for i > 1. Then E is regular, but in addition it is additively elementary. Indeed, let I be the ideal of H, generated by a2,... , ar and/~ be the full subcategory of H-rood, consisting of modules annihilated by I. Since K: is closed under submodules and factors, contains E and is isomorphic to H’-mod, where H’ is the path-algebra of the Dynkinquiver A2, it immediately follows that E is additively elementary in H-mod. If E’ is any indecomposable regular H-module with dirnE’= (1, 1), then there exists an automorphism a of H, such that Et = cE. Consequently Et also is additively elementary. 1.6

The proof of 1.2 now will be given. It is done in two steps. (a) For each connected wild hereditary algebra H = kQ there exist additively elementary modules. This was shown in [4], if the quiver Q has n > 2 vertices. If Q has two vertices, the moduleE considered in 1.5 is additively elementary. Take an additively elementary H-module E. Then there exists an integer mo such that for all integers m with m _> mothe following hold: (i) X is cogenerated by TreE, see 1.3. Let g,~ : X --~ ~-mErbe a mono,for somer >_ 1.

124

Kerner

(ii) If R is regular and h TraE -+1~ is nonzero, the n h i s inj ective, see [7, 1.3] or 1.4. (b) Take s _> 0, such that t = dimHomH(E,TsY) >_ r, see 1.1, and let fl,...f~ be a basis of HomH(E, ~.sy). Then f~ = (fl,... , f~)~ : ~ --~ T sY i s r ight minimal. Choose ml _> mo, such that (a) T’n ’ :~-ml E~ -->vra~+sYis i njective, see 1.4. ’~° E,r "~E) ~0 [ 2] (fl) HomH Let h : ~-ra°E -~ Trn~Ebe nonzero, hence injective, by (ii). Then h ~ ... ~ h : ~-rao Er _~ ~.mi Er is injective, too. Since r < t, there exists a ~. mono e : Trn~ Er --> Tral E The mapgrao(h ~9 . . . ~ h)e(Tml ~) :X -+~.ml+sy is inj ective. [] 2

PROOF

OF

THE

THEOREM

2.1 Besides proposition 1.2, the proof of the theorem, respectively the corollary, is based on the following lemma. LEMMA. If 0 ~ I is preinjective, (a)

respectively 0 ~ P is preprojective,

then

TraI contains a regular nonzero submodule, respectively r-raP contains a regular nonzero factor module, for m >> 0.

(b) There exists a short exact sequence 0 --~ R~ -~ R2 -+ I -+ O, respectively 0 --~ P -~ R~ -~ R2 -+ O, with Ri regular. Proof. By duality, it suffices to showthe first parts. (a) Let X ¢ 0 be a regular module. There exists an mo > 0, such that for all m _> mothe following hold (see for example [5]): (i) HomH(X,rraI) ~ (ii) dim ~-raI > dim (iii) If Y is indecomposable HomH(Y,TmI) = O.

preinjective

with

dim Y < dim X, then

Let f : X -~ ~-mI be a nonzero morphism. ] is not surjective, by (ii). Let R the image of f. By (iii) R has no nonzero preinjective direct summand,hence it regular. (b) Consider first the case where the quiverQ has at least 3 vertices. Let be a regular H-tilting module [12]. Then Ext,(T, I) = 0, so I is generated T. Let f~,... ,]~ be a basis of Homg(T,I) and K be the kernel of the surjection f = (fl,. ¯ ¯ , fr) ~ : Tr -~ I. Application of HomH(T, -) to the short exact sequence 0 -+ K --> Tr --~ I --~ 0 shows Ext~/(T, K) = 0. Consequently K is generated by hence it has no nonzero preprojective direct summandoSince it also is a submodule of the regular moduleTr, it is regular, too. If Q has two vertices, let S(2) be the simple projective module and S(1) be simple injective module. It is enough, to show the assertion for I indecomposable preinjective. Denoteby E(i) the injective hulls of S(i).

Factorisations of Morphisms for WildHereditaryAlgebras

125

Consider first the preinjective modules r~E(1): Let H’ be the Kroneckeralgebra and S’(1) simple injective in H’-mod. Consider in H’-mod the nonsplit short exact sequence ~: O ~ M ~ E’ ~ S’(1) -+ where E’ is the injective hull of the simple module S’(2) in H’-mod and M indecomposable regular with dimM= (1, 1). Since H’ is a factor algebra of there exists a full exact embedding H’ -mod -4 H -mod and ~ can be considered as short exact sequence in H-mod, by this embedding. In H-modthe modules M and E’ are regular, since qg(dimM) = 2 - r < 0 and qH(dimE’) = 5 - 2r < 0, see 1.5. Application of r~/then gives ~’ : O--+ rbM --+ r~E’ -4 r~1E(1) -4 and T~M,respectively ~-~E’, are regular. Consider now r~E(2): If fm: treE(l) -4 v’~-IE(2), for m _> 1 is an irreducible map, then fm is surjective with kernel K,~. It follows for example from [13] that Kmis a brick with dim Ext~(K,n, K,~) = r- I, hence it is indecomposable regular. Consider the following commutative and exact diagram in H-mod 0

0

~

(m 0 -4

r~M

~

r~E’

~,

K

-4

v~E’

~ ~-~_/-1E(2)

K,~

T/T/E(1)

--~ -4

0

0 Since the category of regular H-modulesis closed under extensions, K(m) is regular. Therefore the second row of the diagram, respectively ~/, showassertion (b), Q has 2 vertices. 2.2 The proof of (a) now will be given, (b) follows from duality. The proof divided into 3 steps. (A) Let P # 0 be projective (possibly decomposable), X2, respectively R be regular and s >_ 0. Then each homomorphismf: r-sP -~ X~ factorises through v-mR, for m >> 0. Indeed, consider rsf: P -4 rsX2. By 1.1 there exists an epi for m >> 0. Since P is projective, Tsf factorises through g~, that is r~f = g,g~, where gl : P -4 r-mR¯ Application of T-~ gives f = T-s(Tsf) (r -Sgl)(r-sg2). (B) Let XI -~ ~--s,p~. with Pi # 0 projective and let f = (f~, "’’ f~)~ ~ ~i=1 X%where fi : r-8iPi "-> X2. Then f factorises through r-mRt, for m >> 0: By (A) there exists an m0, such that for all m _> m0 the maps fi factorise through r-mR. Let fi = glg~ i i be such a factorisation. Define gl g~ : X~ --+ r-’nR ~ and g2 = (g~,... ,g~)~ T- ’~R~ ~ X2. Then f = g~g2 is ~. factorisation through r-mR

126

Kerner

(C) By 1.2 there exists an epi h: T-rR -+ t f or s ome r> 0. LetK beth e kernel of h. Then K = K0 ¯ K1, with Ko preprojective and K1 regular. Take an integer mo> 0, such that for all m > m0 and all preprojective modules P, Ext}_/(X~, r-raP) = O. Application of r -’~ to the short exact sequence 0 -~ K --4 r-rR --~% R* -4 0 gives 0 -~ r-inK --+ r-m-rR r_~_+h r_mR~ -> 0 Weget ExtOl(X1, r-rnKl) = O, since ~’-mK~ is regular and Ext~(X~, ~--mKo) = for m _> mo, by the choice of too. Therefore, the map (X~, r-mh) : HomH(X~,r-m-rR) -+ HomH(X1,r-’nR t) is surjective, for m _> rno. Combining(B) and (C) gives the assertion Finally, (c) will be shown. Using (a), we will prove that h : X~ -, X3 factorises through r-mR for m >> 0. The other part of the statement (c) is dual. By 2.1 there exists a short exact sequence 0 -+ R1 ---> R2 --~ X3 -4 0, with Ri regular. Since Ext~z(X~,R~ ) = 0, the map (Xl,g) HomH(X1,R2) -> HomH(X~,X3)is surjective, that is h = ]g with ] : X~ --> R~.. By (a) the map factorises through ~’-"~R for m >> 0 and so does h. [] 2.3 REMARK. The proof tells a little bit more, than in the theorem was stated. For example, given X~,X3 and R as in the theorem. Let s < t be nonnegative integers. Then there exists an integer re(s, t) such that for any preprojective module ~i=~~"~ ~-s~., ~ with Pi projective and s~ < s~...s~ _ < s, each homomorphism f : ~=t ¯ ~ ~ X~ (i = 2, 3) factorises through r-~R for m re(s, t) 3

PROOF

OF THE

COROLLARY

(a) Let 0 --> X~ /-~ R~ ~ R2 --~ 0 be a short exact sequence with Ri regular, see 2.1(b). By 1.2 there exists a mono h : R~ -> ~’mX2, for m >> 0, hence fh : XI -> rmX2 is injective. By the theorem, ] factorises through ~--mX2 for m>> 0, that is f = f~f2 with f~ : X~--> r-taXi. Since f is injective, so is fl. (b) By 2.1(a), for s >> 0 there exists a monoe : R -+ rsX~, for some regular module 0 ~ R. By (a) there exists a monog~ : X~ ~ rmR, for m >> 0, by 1.2 there exists monog~ : X2 -> rmR, for m>> 0. Since r is a left exact functor, ~-"~e is injective, for m > 0. Consequently the maps gi(rme) : Xi --> T"~+~X3are injective, for m>>0. (a’) and (b’) are showndually. REFERENCES I. REITENANDS. SMALO.Representation theory of artin [1] M. AUSLANDER, algebras. Cambridge Studies in Advanced Mathematics, 1994

[2] D. BAron. Homological properties of wild hereditary algebras. In: V. Dlab, P. Gabriel and G. Michler (eds.) Representation theory I, Springer Lect. Notes in Math. 1177 (1986), 1-12.

Factorisations of Morphisms for WildHereditaryAlgebras

127

[3] O. KEI~NEP~.Tilting wild algebras. J. LondonMath. Soc. 39 (1989), 29-47. [4] O. KEI~NEI¢.Elementary stones.

Comm.Algebra 22 (1994), 1797-1806.

[5] O. KERNER.Representations of wild quivers. In: R. Bautista, R. Martinez Villa and J. A. de la Pefia (eds), Representation theory of algebras and related topics, CMSConf. Proc. 19 (1996), 65-107. [6] O. KERNER.Minimal approximations, orbital elementary modules and orbit algebras of regular modules. J. Algebra 217 (1999), 528-554 ANDF. LUKAS.Elementary modules. Math. Z. 223 (1996), [7] 0. KERNER 434. [8]

421-

O. KERNER. ANDM. TAKANE. Monoorbits, epi orbits and elementary vertices of representation infinite quivers. Comm.Algebra 25 (1997), 51-77.

[9] F. LUKAS.Infinite dimensional modules over wild hereditary algebras. J. London Math. Soc. 44 (1991) 401-419. Spectral properties of Coxeter transfor[10] J. A. DE LA PEI~A ANDM. TAKANE. mations. Arch. Math. 55 (1990), 120-134. [11] C. M. RINGEL.Tamealgebras and integral quadratic forms. Lecture Notes in Math. 1099, Springer, Berlin, 1984. [12] C. M. RINGEL.The regular components of the Auslander-Reiten tilted algebra. Chinese Ann. Math. Ser. B 9 (1988), 1-18. [13] L. UNGER.On wild tilted 542-550.

quiver of a

algebras which are squids Arch. Math. 55 (1990),

A note on concealed-canonical

Artin algebras

DIRKKUSSINFachbereich 17 Mathematik, Universit~it born Germany, Email: [email protected]

Paderborn, D-33095 Pader-

ZYGMUNT POGORZALY Faculty of Mathematics and Informatics, nicus University, ul. Chopina 12/18, 87-100 Torufi Poland, Email: [email protected]

Nicholas

Coper-

ABSTRACT In this article some omnipresence condition is given which assures that a derived-canonical algebra is already concealed-canonical. The proof exploits the theory of coherent sheaves over exceptional curves.

1

INTRODUCTION

Throughoutthis article let k be an arbitrary field, and A be a finite dimensional kalgebra. Weshall use the term modulefor a finitely generated right A-module. The category of (finitely generated right) A-modules is denoted by mod(A). Moreover, the derived category of bounded complexesof A-modules(see [4]) will be denoted Db(A). Wecall A derived-canonical, if there is a canonical algebra A (in the sense of Ringel/Crawley-Boevey [16]) such that Db(A) _~ Db(h) as triangulated categories. If moreoverA is of tubular type, then we call A derived-tubular. Note that a derivedcanonical algebra is connected since its derived category is. The Grothendieck group of rood(A) will be denoted by K0(A), the Coxeter transformation on Ko(A) Recall from [16] that for a canonical algebra A the module category rood(A) trisected into rood+ (A) V modo(A)V mod_(A), where mod0(A)is a stable ing tubular family, and there are no non-zero morphismsgoing from right to left. Recall from [11] that a k-algebra A is called concealed-canonical (almost concealedcanonical, resp.), if for somecanonical algebra A there exists a tilting modulelying 129

130

Kussinand Pogorzaly

in mod+(A) (in mod+(A) V mod0(A), resp.) and whose endomorphism algebra isomorphic to A. If additionally A is of tubular type, then we call A a tubular algebra. Concealed-canonical algebras (in particular: tubular and canonical algebras) were studied by several authors (see for example[6, 9, 11, 13, 14, 16, 17], also [1, 2] and [5, 10, 15]). It is well-knownthat the class of concealed-canonical algebras is not closed under derived equivalence. The aim of this note is to present a condition under which it follows that a derived-canonical algebra is concealed-canonical. The essential property will be the existence of some omnipresent indecomposable module. The notion of omnipresence was also successfully used in a similar context in [14, 17]. Recall that an A-module M is called omnipresent, if each simple A-module occurs as a composition factor of M. Moreover, an Auslander-Reiten component is called regular, if it contains neither a projective nor an injective module, and it is called semi-regular, if it does not contain at the same time a projective and an injective module. The main result of this note is the following THEOREM. Let A be a finite dimensional k-algebra over a field following conditions (1) and (2) are equivalent

k.

Then the

(1) (a) A is derived-canonical, (b) there is an omnipresent indecomposable M mod(A), such th at (i) the class [M] E K0(A)has finite ~-period. (ii) M lies in some regular Auslander-Reiten component in mod(A). (2) A is concealed-canonical. REMARKS. (1) As the proof of the theorem will show, condition (b) can be placed by the following condition: (b’) There is a (finite) family of indecomposables Mi ~ rood(A) (i e that their direct sum is omnipresent, and such that all Mi (i ~ I) lie in regular components in mod(A) and in the same tubular family in Db(A). (2) The almost concealed-canonical algebra A over an algebraically closed field, which is given as path algebra of the quiver 1 ¯

x

2 :~o

z

3 ~¯

Y with relation zx = O, shows, that in condition (ii) regularity cannot be replaced semi-regularity. Namely, A can be realized as endomorphismalgebra of a tilting sheaf over the weighted projective line of weight type (1, 2) (see [3]). The indecomposable projective A-module P(3) is omnipresent, lying in a semi-regular tube A. If we restrict

to the tubular case we have a stronger result.

COROLLARY. Let A be a finite dimensional k-algebra following conditions are equivalent

over a field

k. Then the

Concealed-Canonical Artin Algebras

131

(1) A is derived-tubular, and there is an omnipresent indecomposable M E mod(A) lying in some semi-regular Auslander-Reiten component in mod(A). (2) A is tubular. REMARK. (3) Let k be algebraically the quiver

closed and A be the poset algebra given

// 4~7 with all 6 possible commutativity relations. Then A is derived-canonical (of tubular type (3, 3, 3)), but not tubular (see [12]). The indecomposableprojective injective A-module P(8) = I(1) is omnipresent lying in a component in mod(A) is not semi-regular. Thus, semi-regularity of the component in the corollary is indispensable. Note, that in the theorem and in the corollary the implication (2) ==~ (1) is trivial. In the proof of our result we shall use the coherent sheaves technique approach to the representation theory [3, 7]. This approach makes our proof rather simple. The following characterization of concealed-canonical algebras from [9] is of great importance for our proof: A is concealed-canonical if and only if there exists an exceptional curve :K (see [7]) - that is, a weighted projective line if k algebraically closed - and a torsion-free tilting object in the category coh(iK) coherent sheaves whose endomorphismalgebra is isomorphic to A. 2

THE

DERIVED

CATEGORY

OF

A CANONICAL

ALGEBRA

Let A be a canonical k-algebra over the field k (compare [16]). By [16] mod(A) contains a stable separating tubular family modo(A),which is a coproduct of uniserial connected length categories L/z (called stable tubes). By the construction of [9] there is a small k-category 7/, which is abelian, hereditary (that is, Ext,(-,-) for all i _> 2), noetherian, locally-finite (that is, all Horn and Ext1 spaces are of finite dimension over k), containing no non-zero projective object and admitting a torsion-free tilting object with endomorphismalgebra isomorphic to A. Each indecomposableobject in 7/is either in 7/0, the full subcategory of objects of finite length (so-called torsion objects), or in 7/+, the full subcategory formed the torsion-free objects, which do not contain any non-zero torsion subobject. The relation HomT~(7/0,7/+) = 0 holds. Moreover, 7/0 = mod0(A). There is an auto-equivalence T : 7/ ~ 7/, called Auslander-Reiten translation, such that Serre duality holds naturally in X, Y Ext~ (X, Y) _’z D Homn(Y, -rX),

132

Kussinand Pogorzaly

where D denotes the duality Homk(-,k). Moreover, 7/ admits almo.,~t split sequences, and for indecomposable end term X in such a sequence the starting term is given by ~-X (see [9, Thin. 6.1]). The category 74 is also denoted by coh(:~), and :K equipped with coh(:~:) called exceptional curve [7]. By tilting theory the categories coh()£) and rood(A) derived-equivalent, Db(:K) = Db(A), in particular also have isomorphic Grothendieck groups: Ko()[) = Ko(A). For each object X in 7/denote by [X] the class in K0(X). Wethen have [~-X] = O[X]. Since 7/is hereditary, we have 7) := Db(X) = add( U T/[n]), s

where the 7/In] are (disjoint) copies of 7/; for each X E 7/ the copy in 7/In] denoted by X[n]. Each indecomposable object in T) is of the form X[n] for some (indecomposable) X E 7/and some n ~ Z. For all X, Y 6 7/and all m, n ~ Z have Homz~(X[m],Y[n]) = Ext~t-m(X, in particular, if m > n or n > m + 1, then Homv(X[m],Y[u]) = The Auslander-Reiten translation ~- extends canonically to an auto-equivalence ~- : :D --+ T) (which we denote by the same symbol). 3

PROOF

OF THE RESULTS

Assumethat condition (1) from the theorem holds, and that Db(A) = Db(A), A is canonical, and let :K and 7/be as above. The proof has three steps: First step: The omnipresent indecomposable M ~ mod(A) lies in 7/0In] for some n E Z. Without loss of generality, we assume n = 0. Second step: Realize A as (endomorphismalgebra of) a tilting complex T in 7:). By omnipresence, we immediately see that T ~ 7/o[-1] U 7/. Third step: Wehave to show, that (using regularity) actually T ~ 7/+, that is, A can be realized as (endomorphismalgebra of) a torsion-free tilting object in and hence is concealed-canonical (see [9]). The second step is clear. For the first: Weassume M6 7/. For non-tubular E and for non-zero M~ 7/+ it follows as in [8, Prop. 4.5], that [M] has no finite O-period. Thus, ME 7/0, and Mlies in a stable tube T of finite rank. Observe, that in the tubular case, Mlies in a stable tube in any case (since ind ~H consists entirely of stable tubes, compare [6]), not necessarily in 7/0, but after a possible change of the chosen separating tubular family modo(A) (and thus changing compare [6, Prop. 7]) we can assume ME 7/0. It remains to prove the third step. Weassume more generally, that M"lies in a semi-regular componentC of A. Then C contains either no projective or no injective A-module. Case 1. ~ contains no projective. Let P be an indecomposable direct summand of the tilting complex T, which is an indecomposable projective A = End(T)module. Assume that P E 7/0. By omnipresence, HomA(P,M) it 0, and by orthogonality of the stable tubes, P also lies in the tube 7". By assumption, P and Mlie in different Auslander-Reiten components of A, therefore Rad~(P, M) it

Concealed-Canonical Artin Algebras

133

and then also Rad~(P, M) ~ 0, which gives a contradiction since P and Mlie the same stable tube T, which is standard ([15]). Therefore, no indecomposable summandof T lies in 7/0, hence T E 74o[-1] U 74+ and therefore A is dual to an almost concealed-canonical algebra. Case 2. The component C contains no injective. Assumemoreover, that there is an indecomposable projective A-module P lying in 740[-1]. Then consider the corresponding injective A-module I = vP[1]. By omnipresence, HomA(M,I) ~ and by proceeding as above we see that T E 74+ tJ 740, and thus A is almost concealed-canonical. Nowby [11], if C is regular, or if A is of tubular type, it follows, that A is concealed-canonical. This proves the theorem and the corollary. ACKNOWLEDGEMENT This note was written during a stay of the second named author at the University of Paderborn. He would like to express his gratitude to HelmutLenzing for his hospitality. He also acknowledgesa partial support of the Polish Scientific Grant KBN 2 PO3A012 14. Both authors would like to thank Helmut Lenzing for stimulating discussions on the subject. REFERENCES [1] M. Barot, Representation-finite 74 (2000), no. 2, 89-94.

derived tubular algebras, Arch. Math. (Basel)

[2] M. Barot and J. A. de la Pefia, Derived tubular strongly simply connected algebras, Proc. Amer. Math. Soc. 127 (1999), no. 3, 647-655. [3] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras, Singularities, Representation of Algebras and Vector Bundles (Lambrecht 1985) (Berlin-Heidelberg-New York), Lecture Notes in Math., vol. 1273, Springer-Verlag, 1987, pp. 265-297. [4] D. Happel, Triangulated categories in the representation theory o] finite dimensional algebras, London Math. Soc. Lecture Note Series, no. 119, Cambridge University Press, 1988. [5] D. Happel and C. M. Ringel, The derived category of a tubular algebra, Representation Theory I. Finite Dimensional Algebras (Ottawa 1984) (BerlinHeidelberg-New York), Lecture Notes in Math., vol. 1177, Springer-Verlag, 1986, pp. 156-180. [6] D. Kussin, Non-isomorphic derived-equivalent tubular curves and their associated tubular algebras, J. Algebra 226 (2000), 436-450. [7] H. Lenzing, Representations of finite dimensional algebras and singularity theory, Trends in ring theory. Proceedings of a conference at Miskolc, Hungary, July 15-20, 1996 (V. Dlab et al., ed.), CMSConf. Proc., vol. 22, Amer. Math. Soc., Providence, R. I., 1998, pp. 71-97.

134

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[8] H. Lenzing and J. A. de la Pefia, Wild canonical algebras, Math. Z. 224 (1997), 403-425. [9] ~ , Concealed-canonical algebras and separating London Math. Soc. 78 (1999), no. 3, 513-540.

tubular families,

Proc.

[10] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, Representations of Algebras (Ottawa 1992) (V. Dlab and H. Lenzing, eds.), CMSConf. Proc., vol. 14, Amer. Math. Soc., Providence, R. I., 1993, pp. 313-337. [11] ~, Tilting sheaves and concealed-canonical algebras, Representation Theory of Algebras (Cocoyoc, 1994) (R. Bautista, R. Mart/nez-Villa, and J. de la Pefia, eds.), CMSConf. Proc., vol. 18, Amer. Math. Soc., Providence, R. I., 1996, pp. 455-473. [12] H. Lenzing and I. Reiten, Additive functions for quivers with relations, Colloq. Math. 82 (1999), no. 1, 85-103. [13] H. Meltzer, Auslander-Reiten components for concealed-canonical algebras, Colloq. Math. 71 (1996), no. 2, 183-202. [14] I. Reiten and A. Skowrofiski, Sincere stable tubes, Preprint 99-011, Bielefeld, 1999. [15] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, Berlin-Heidelberg-New York, 1984. [16] ~ , The canonical algebras, Topics in Algebra, Banach Center Publ., no. 26, 1990, with an appendix by William Crawley-Boevey, pp. 407-432. [17] A. Skowrofiski, On omnipresent tubular families of modules, Representation Theory of Algebras (Cocoyoc, 1994) (R. Bautista, R. Martinez-Villa, and J. de la Pefia, eds.), CMSConf. Proc., vol. 18, Amer. Math. Soc., Providence, P~. I., 1996, pp. 641-657.

Koszul algebras

and the Gorenstein

condition

1 Instituto ROBERTO MARTINEZ-VILLA de Matem~ticas, Universidad Nacional Autbnoma de M~xico, M~xico 04510, D.F., M~xico. e-mall: [email protected]

ABSTRACT Non commutative versions of regular algebras appear naturally in representation theory as the Yonedaalgebras of selfinjective Koszul algebras, they have been studied in [4], [10], [11]. Here we extend these notions by considering algebras such that someof the simple satisfy the Gorenstein condition [1], [9], [12]. Whenthey are Koszul, we study the corresponding Yoneda algebras, examples of such algebras will be the Auslander algebra, the preprojective algebra and selfinjective algebras of radical cube zero of infinite representation type. Wewill prove that by taking tensor products we can construct new algebras satisfying the Gorenstein condition. 1

NOTATION

AND KNOWN RESULTS

Wewill consider graded quiver algebras over an algebraically closed field K, this is: positively graded K-algebras A = $ Ai such that A0 = K × K... × K, where i>0 K is a field and for all i we have dimKAi < ~ and for all i, j there is an equality A~A~= A~+~. Weknow[6] such algebras are isomorphic to algebras of the form KQ/I, where Q is a finite quiver and I is an homogeneousideal of KQin the grading given by path length and I is contained in JU where J is the ideal generated by the arrows. Given a Z-graded module M = {M~}~z we denote by M[n] the nth-shift defined by M[n]~ = Mn+i. Weconsider the category /.f.modA of locally finite Z-graded modules M {M~}~ez, such that dim/~ M~< c~. 1 Part of this paper was written during my visit to Universidad de Murcia on December 1998 and part during my visit to Northeastern University on July 1999. I thank both Manolo Saorin and Alex Martzinkovsky for their kind hospitality, for exchanging ideas and for their encouragement, to the mentioned universities for funding. 135

136

Martlnez.,Villa

Weknow by [6], there exists a duality D : /.f.modh -> l.f.modho~ given by D(M)j = Hom~:(M_j, The category of graded A-modules and degree zero maps, HomA(M,N)o, will be denoted Gr Mod^, and by Mod~ the category of graded modules and maps Hom^(M, N) = ~ Hom^(M, N)i, with Homh(M, N)n the degree n maps. We have isomorphisms: Homh (M[-n], N)o~-HomA (M, N[n])o~Hom^ (M, N)n. In a similar way the k extensions of degree zero, ExtkA(-, ?)o are defined as the derived functors of Horn^(-, ?)o. Wedefine Ext~ (M, N)n = Ext~ (M, N[n])0 and Ext,(M, N) is the graded vector space: Ext,(M, N) = ~ Ext,(M, N)n. n_>o Werecall the following definitions and results concerning Koszul algebras [3], [~], DEFINITION. Let A = KQ]I be a graded quiver algebra. We say that a graded A-module M is Koszul if Mhas a graded projective resolution: ... -~ Pat-n] ~ Pn_~[-n + 1] -}... P~[-1] -+ P0[0] --} M-} 0, with each Pj[-j] finitely generated with all generators in degree j. Wesay that A is Koszul if all graded simple are Koszul. THEOREM 1.1. Let A = KQ/I be a Koszul algebra. Then the following are true:

statements

i) The algebra A is quadratic, this is: I is generated by linear combinations of paths of length 2. ii)

Let V2 = (KQ)~ be the vector space generated by all paths of length 2 and (,) : V2 V~. -~K t hebili near form defin ed by (~ /3, /3’a’) = 1 i f ~ = ~ and ~ = f~ and 0 otherwise. Let L2 be the orthogonal [2~ of the vector space I~ = I f~ (KQ)2. Denote by KQ°p the quiver algebra of the opposite quiver and by L the ideal generated by L~.. Then the Yoneda algebra F = ~ Ext~(ho, Ao) is Koszul and F ~- KQ°P/L. k>_O

iii) Let K^ and Kr°~ be the full subcategories o] Gr Mod~and Gr Modro~ consisting of Koszul A and F°~-modules, respectively. Then the functor F(M) ~ Extkh (M, Ao) is a duality from K^ to Kro~ satisfying F(JkM) = f~kF(M), k>_O where f~} denotes the kth syzygy. 2

KOSZUL

DUALITY

AND EXTENSIONS

All the algebras in this section will be Koszul, we will study the relations between the extension groups of two Koszul modules and the corresponding extension groups under Koszul duality. The main result is contained in the following: THEOREM 2.1. Let A = KQ/I be a Koszul algebra and M and N two Koszul modules. Then for any pair of integers k and l, with k >_ O, the following two statements are true:

KoszulAlgebrasand the GorensteinCondition

137

i) I] Ext~ (M, N[/])o ~ 0, then k >_ -l. ii) If k >_ -l, then there exists a vector space isomorphism: ’ Ext,(M, g[/])o

~ Ext~+o~(F(Y)[l], f(M))o.

Proof. i) AssumeExtrA(M, N[/])o # 0. Wehave an exact sequence: 0 -~ ilk(M) --~ Pk-l[-k 1]-~ flk -l(M) --> with Pk-~[-k + 1] the projective cover of f~k-~(M). Then we have an exact sequence: 0 ~ Homh(f~k-l(M), g[/])o

~ gorn^(Pk_~[-k + 1], g[/])o

-~ Hom^(f~(M), Y[l])o -+ Ext,(M, g[l])o

-+

-~

Since Ext,(M, N[/])o ¢ 0, also Horn^(~k(M), g[/])o ~ 0. The module ~k(M) is generated in degree k and NIl] in degree -l. k > -l.

It follows

ii) Weconsider first the case k = 0. There is a short exact sequence: 0 --~ JtN[l] -~ NIl] -~ N[l]/JIN[l] -~ O. It induces an exact sequence: 0 -+ gom^ (M, JtY[l])o

-~ Homh(M,

g[/])o -+ gomA (M, Y[l]/Jlg[1])o

Since HomA(M,N[l]/g~N[l])o = th e ve ctor sp aces HomA(M, fl N[l])o and HomA(M, N[/])o are isomorphic. The modules M and J~N[l] are both generated in degree zero and Koszul. It follows there exists isomorphisms: * gomh(M, N[/])o

~- goma(M, J~N[l])o ~- Homro~(fl~F(N)[l],

F(M))o.

As before, we have an exact sequence: 0-+ fl~F(N)[/] ~ Pt_~ [1] -~ fl~-~ F(N)[/] ~ 0, with P~_~[1] the projective cover of fl~-~F(N)[l]. Hence there exists an exact sequence: 0 ~ gomro~ (~-IF(Y[l]),

f(M))0 gomro~(P~_~ [1 ], F(M))o -~

-~ gomr°~(fl~F(N)[1], F(M))o -~ Ext~ro~ (F(N[l]), F(M))o --~ Since /~_~[1] is generated in degree -1 and F(M) in degree zero the maps: Homro,(P~-~ [1], F(M))o = and th ere ex ists an iso morphism: **

Homro~(~2tF(Y)[l],

F(M))o ~- Ext~ro, (f(g)[l],

f(M))o,

by using ¯ and ** we get the result for k = 0. Consider the case k >_ 1 and k = -l. The exact sequence 0 -~ fl~M -~ Pa_~ [-k + 1] -~ flk-~ M-~ 0 induces an exact sequence: 0 --~ gomA([~k-~M,N[l])o -~ gom~(Pk_~[-k 1],N[/])o -~

Martinez-Villa

138 -’~ Homh(~kM, N[/])o

~ Ex~(M, N[/])o

The module P~-l[-k ÷ 1] is projective generated in degree k = -l. As above, Homh(~kM, N[l])o ~- Ext~(M,g[-k])o. ated in degree k the duality F induces gomrop ( f(N)[-k], jk F(M))o. The exact sequence: 0 ~ jkF(M) -~ an exact sequence: 0 -+ Homrop (F(N)[-k],

~

generated in degree k - 1 and N[l] is HomA(P~_~[-k ÷ 1],N[/])o = 0 and Since both ~kM and N[--k] generan isomorphism: Homh(~M,N[-k])o -~ F(M) -~ F(M)/J~F(M) ind uc es

gkF(M))o -~ Homrop(F(Y)[-k],

-~ gomro~ (F(g)[-k],

F(M)/J~F(M))o.

It is clear Homrop (F(g)[-k], F(M)/jkF(M))o Then it follows Ext,(M, Y[-k])o ~ Homro~(F(N)[-k], F(M))o. It remains to consider the case k _> 1 and k > -l. Assumek = 1 and l _> 0. The exact sequence: 0 -~ JIN[l] --~ N[l] --~ N[l]/J~N[l] -~ 0 induces an exact sequence: 0 ~ HomA(M,J~N[l])o -~ Hom^(M, Y[/])o -~ Hom^(M, N[l]/JlY[l])o. Since Homh( M, N[l]/ fl N[l])o = th ere is an iso morphism: * Homh(M, flg[l])o ~- Hom^(M, g[/])0. Applying the duality F we obtain an isomorphism: Hom^(M, fl N[l])o ~- Homro~(12’f(N)[~], f(M))o. There is an isomorphism: . ¯ Hornro~ (~F(N)[I], F(M))o ~- Ext~o~ (F(N)[l], F(M))o. Using * and ** we obtain the result for k = 1. Assume k > 1 and k > -l. The exact sequence: 0 -+ J~N[l] --~ N[l] -~ N[l]/J~N[l] --~ 0 induces an exact sequence: --~ Sxtk~-~ (M, g[l]/ ’+k-’ N[/])o - ~ Extk~ ( M, J’+~-~ N[/])o ~ -~ Ext,(M, N[/])o -~ Ext,(M, N[l]/ fl+~-~ Y[l])o The exact sequences: 0 -+ ~k-~M -~ Pk-~[-k ÷ 2] -~ ~k-~M --~ 0 and 0 -~ fikM --~ Pk_~[-k+ 1] --~ fi~-~M -~ 0, with Pk-2[-k ÷ 2], Pk-~[--k + 1] projectives generated in degrees k - 2 and k - 1, respectively, induce exact sequences: 0 --~ gom/~(~k-2M, N[l]/Jl+~-~N[l])o --~ gorn^(Pk_2[-k 2] , N[l]/J~+k-~N[l])o --~ gomA(~k-~ M,N[l]/J~+k-lN[l])o -~ Ext~-~(M,N[l]/ Jz+a-~ N[l])o --~ 0 and 0 --~ Hom~.(fl~:-~M,Y[l]/J~+~-Ig[l])o --~ Hom~(P~_~[-k1], N[ll/g~+~-~N[l])o _.~ Hom~(~kM, N[l]/jl+k-~ N[l])o .-~ Ext~h (M, N[l]/Jt+~-~ N[l])o --~ Since Homh(~k-~M, N[l]/J~+k-~N[l])o = and th ere is an equality:: Hom~( ~k M, N[l]/ J~+k-~N[l])o = O, it follows Ext~-~ ( M, N[l]/ J~+k-~N[l])o = and Zxt~ (M, N[l]/ jl+k- ~ N [/])o = 0. Therefore: Ext~h ( M, g~+~-~N[/])o -~ Ext~ ( M, Nil])0. Now~k-~M and J~+~-~N[l] are both generated in degree k - 1. Hence, wehave an isomorphism: x ~ Fx. EXtrA (ilk-1 M, J~+~-~ Y[/])o ~- Ext~o~(f(J ~+k-~Nil]), F(fl k-~ M))o given as follows:

KoszulAlgebrasand the GorensteinCondition

139

If x E EXt~A(~k-IM,J~+~-IN[l])o is the exact, sequence: x : 0 ~ Jl+~-lN[l]

-4 E -~ ~-~M ~ O,

then Fx is defined as the exact sequence: 0 --¢ F(~/C-IM) -y F(E) ~ F(J~+I-~N[I]) -~ We have isomorphisms: Ext~ ( M, Y[/])0 ~ Ext~ (M, jI+k-1N[/])o -~ Ext~h ( ~-~ ( M), j~+k-~ Will)0 - ~ =~ Ext~

~-Zxt ~

op(F(Jt+k-~ N[l]), F(~k-I M))o

o~(~+~-~ F(N)[I]

, gk-~ f(M))o

By an argument similar to the given above, Ext~+o~(F(N)[l], Jk-~F(M))o ~Ext~+,~ ( F( Y)[/], F( M) )o. 3

GORENSTEIN

RINGS

A ring A will be called Gorenstein [1], [2] if it has finite injective dimension both as left and right module. The rings considered may not be noetherian but we will consider moduleswith minimal projective resolutions consisting of finitely generated projectives, in case A is graded; the modules, the maps and the extension groups will be graded. PROPOSITION 3.1. Let A be a Gorenstein ring, let M be a module with minimal projective resolution consisting of finitely generated modules, assume there exists an integer n such that Ext~h (M, A) = 0 for k ~t n. Then Ext~ (M, A) satisfies the following conditions: Ext~op(Ext~(M,A),h °~) = 0 for i # n. Ext~o~ (Ext~ (M, A), h°p) ~ M. In case A is graded the isomorphism is as graded modules. Proof. Assume Mhas finite projective dimension. Then pdM = n. °~) Let n = 0. Then Mis projective and M*is also projective, hence, Ext~ho~(M*, A = 0 for k different from zero and M**~ M. Assume pdM < c~ and n > 0. Let 0 ~ Pn ~ Pn-~ -~ "’" -~ Po -+ M ~ 0 be the minimal projective resolution. Dualizing with respect to the ring we obtain a complex: ,) 0 ~ P~

P{ ~ ... -~ P; -~ O. By hypothesis, it is exact except at the index n where the homology is Ext~(M,A), then the sequence: 0 --~ P~ -~ P{ -~ ... -~ P~ -~ Ext~(M,A) --~ is a minimal projective resolution of Ext~ (M, A). Dualizing again and using the fact that the complex: 0 -~ P~* -~ P~*_~--+ ... ~ P0** -~ 0 is exact except at zero, where the homologyis M, we obtain: Ext~ho,(Ext~(M,A),A °p) = 0 for i ~ n and EXt~o~(Ext~(M,A),A °~) ~- M. AssumeM has infinite

projective

dimension and n = 0.

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Let ... ~ Pj -+ Pj-1 ~ ...P1 -+ P0 -~ M -+ 0 be the minimal projective resolution of M. Dualizing we obtain an exact sequence: 0-~ M* ~P~ ~P~* ~ ""P~+I

~P~+2 -~ Y~O

Suppose idhA = k. Then for i > 0 we have isomorphisms: Ext~Aop(M*, h°n) -~ Ext~hop (12k+2(Y), h°p) ~ ~o~+i+: (v Ao~) = The exact sequence: 0 ~ M*~ P~ ~ ~k+~ (y) ~ 0 induces an exact sequence: 0 ~ (~k+~(y)).

~ p~. ~ M** ~ Ext~o~(~k+l(y),h

°~) ~ O.

Since Ext~,~ (flk+~ (Y), °p) ~Ext~](Y, A°p) = 0, thesequence 0 ~ (~k+l(y)). ~ p~. ~ M**~ 0 is exact. The sequence 0 ~ ~k+~(y) ~ p~ ~ ~(y) in duces an e xac t sequ ence:

0

P;*

O.

Since Ext~.~ (~ (Y), °~) ~Ext~ (Y , A°p) = 0, thesequence: 0 ~ (~(Y))* ~ P;* ~ (~+~(Y))* ~ 0 is We have proved the sequence: P~* ~ P~* ~ M**~ 0 is exact. It follows M~ M**and Ext~o~ (M*, A°~) = 0 for i different from zero. Assume pdM = ~ and n > 0. Let ... ~ Pn ~ Pn-~ ~ "’" ~ Po ~ M ~ 0 be the minimal projective resolution of M. Weobt~n by dualizing the complex:

which is exact except at the index n where Ext~ (M, A) ~- Kerffn+~/Ira f~. Let C = P~/Im f,~ and X -- Im f,~+l -- Ker~+2 = 1. P~/Kerff~+ Wehave an exact sequence: ,) 0 -+ Ext~(M,A) -+ C -~ X -~ Consider the exact sequence: 0 -~ X -~ P~* n-bl "-} P~+~"-} "’" "-} *P~+k~ Y "-} O. Then 12kY=’~ X. Suppose idhA = k. Then for i _> 1 we have isomorphisms: Ext,(X,

A) ~ Ext~(akY, A) -- Ext~+k(Y, A) = O.

From the exact sequence: .) we obtain the exact sequence: Ext~hop(X, A°~) ~ Ext~to~ (C, A°~) -~ Ext~o~ (Ext~ (M, A), °n) - + Ext~+o~ (X, A°v) ~ Since Extk(X,A) ~- Extk+I(X,A) = fo r i _>1, it fol lows the sequence: 0 --~ X* ---~ C* ~ Ext~(M,A)* ---~ is exa ct and EXt~ o~(C,A °p) ~Ext~o~ (Ext~ (M, i), h°p) for i _> 1. Since M*= 0 and the sequence: 0 -+ P~ -~ P~* --~ ... -~ P.* ~_~ -~ P~ * -+ C-+O is a minimal projective resolution of C the projective dimension of C is n.

KoszulAlgebrasandthe GorensteinCondition

141

From the fact that the complexes:

are isomorphic it follows HomAop (C, °p) -~ f ~n+lM, the module Extihop ( C, A°p) = °p) -~ 0 for i ~ n and Ext~o. (C, A M. °p) = 0 for i ~ 0 and i ~ n and Therefore: Exti~o~(Ext~(M,A),A °’) Ext~o~ (Ext~ (M, A), ~M. Consider the following exact diagram: 0 P’n-1

"---)" Kerffn+ 1 ---~

~d$ P’"

Ext,(M,

A)

~ ~

P:

Pr~+l

~

Im fn*+~ 0 Dualizing we obtain the diagram: 0 (Im f~*+l)*

n+l

0 ~ Ext~(M,A)*

(Kerffn+~)*

with exact rows and exact middle column. Hence; pf~+~ = 0 = st implies t = 0. By five’s lemma, t is an epimorphism. Therefore: Ext,(M, A)* = 0. PROPOSITION 3.2. Let A be any ring and let M be a finitely presented indecomposable A-module with minimal projective presentation P~ --~ Po --~ M --> 0 such that Ext,(M, A) = Ext,(M, A) = 0. Then the following statements are true: i) trM ~- f~(M)*. ii) f~2(M)is reflexive. iii)

f~(M), f~(M) are indecomposable.

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Martinez-Villa

Proof. By hypothesis we have exact sequences: 0 --~ M*--~ P~ --~ fl(M)* -> 0 and 0 -~ ~(M)*-~ P~* --> fl2(M)* Gluing the two sequences we obtain an exact sequence: *) 0--~ M* -~P~ ~ P~* -~ ~2(M)* It follows trM ~ ~(M)*. Dualizing the sequence .) we obtain a commutative exact diagram:

0 ~ fl2(M)

-~ P1 -~

0-~ fl~(M)** ~ P~** -~ Po** Hence: ~2(M) -~ ~(M)**. From the fact trM is indecomposable follows f~2(M)* is indecomposable. Assume fl~(M) = X @ Y with X ~ 0 ~ Dualizing we obtain fl~(M)* = X* ~ Y*. *Since ~22(M) C_ P1 the modules X* and Y * are not zero. Therefore: ~22(M) decomposes, which can not hapen because f~2(M)* = trM and trM is indecomposable if Mis. Wehave proved [~2(M) is indecomposable. Assume now f~(M) = X ~ Y with X ¢ 0 ~ Y. Then fiX(M) = ~(X) @ therefore either X or Y is projective. Suppose f~(M) = X @Q with Q projective. Taking push outs, the projection map q : f~(M) --~ Q induces a commutative exact diagram: -~ M -+0 0-~ ~(M) Po -~

q4

J, M

o

---~0

o

Since Ext~ (M, A) = 0 the projection map q factors through P0, there exists map t : Po ~ Q with ti = 1. Let s : Q ~ f~(M) be a map such that qs = 1. Then tsj = 1 and t is a split epimorphism, contradicting the fact fl(M) C_ radPo. We have proved ~(M) is indecomposable. COROLLARY. Assume M has a minimal projective resolution with all projectives finitely generated and ExtrA(M, A) = 0 for 1 < i < n - 1. Then fl~(M) is reflexive for all 2 < i < n - 1, if Mis indecomposable, then for 1 < i < n - 1 the module ~(M) is indecomposable. 4

GORENSTEIN

KOSZUL

ALGEBRAS

In this section we study graded Gorenstein algebras A which are in addition Koszul. Using the duality, we will describe relations between the graded A simple satisfying the Gorenstein condition and the indecomposable projectives over the Yoneda algebra F satisfying the dual property.

KoszulAlgebrasandthe GorensteinCondition PROPOSITION 4.1.

143

[10] Let A be a Koszul algebra and F = ~ Extk(Ao,Ao) its

Yoneda algebra. Then for any graded simple A-module S the conditions 1) and 2) are equivalent: 1) The simple S satisfies

the following conditions:

i) pdS = n. ii) Ext~ (S, A) = 0 for i ~ n. iii) Ext,(S, A) = S’ is a graded simple A°V-module. 2) The module F(S) = ~ Ext,(S, ~>_o Ao) is projective injective

of finite

length.

Proof. First we show 1 implies 2. Let 0 ~ Pn ~ Pn-~ ~ "’" ~ Po ~ S ~ 0 be the minimal projective resolution of S, the simple S’ has a minimal projective resolution: 0 ~ P~ ~ P~ ~ ... ~

~_, ~ P~ ~ ~0. Dua]izing wi~hrespec~ ~o ~he~e]dwe obtaina minimal injec~ive coresolution o~

D(S’): 0 ~ D(S’) ~ D(P~) ~ D(P~_,) ~...~ D(P;) LetI~ : D(P~_~) be ~he~-injec~ive in ~hecoresolu~ion. Then~hereis a chain of isomorphisms: socI~ = soc~-~D(S’) = D(D~-~D(S~)/JD~-~D(S’))=~

~ P~_~IJP~_~ ~ ~-~S/j~-~S. {~o injec~ive. ~e h~vethefollowing isomorphisms: soc}+~I/soc}I:~ D(J~D(I)/J}+’D(I))

’,A~)) -~D(E~, ~

~ ~ ~* "~D(P~ /JP~ )~ ~

Ho~ao(~-~S/J~-}S, Ao) -~ E~-~(S, Ao) =~ PJ~-}/PJ~-}+~. In particular, socI ~ PJ~. Le~T be a simplein ~hesoc]eof P. Then~hereexists~n in~ege~ 1 <} < ~ -1 suchtha~T ~ PJ~ and T is no~ contained in PJ~+~.Hence;PJ} : T ~ X. Applyin~ Koszuldualitywe haveisomorphisms: ~ E~(J}P, Fo) ~ F-~(PJ~) ~ D~F-~(P) ~ ~(S) ~ F-~(T) ~ F-’(X). ~o Set Q ~- F-I(T) and F-I(X) ~- Y.

144

Martlnez-Villa We have an exact commutative diagram: ~

Q~ Y

-~ ~-~(S)

-~0

w

-~ ~-1(S)

-~0

-~

Q

x : 0--~

Pk-1

o

o

But x 6 Extl~(f~k-*S,Q) ~- Ext~(S,Q) = 0 implies the projection p factors through Pk-1, contradicting Q c_ JP~_~. Wehave proved socP ~- J"P, hence P C_ I. Since P and I have the same composition factors, P - I, which shows 1 implies 2. AssumeF(S) = is projective inj ective of fin ite len gth and Ext~(S, A) ~ 0 f or some k > 1. Wehave an exact sequence: 0 ~, Q --~ W -~ ~qk-~(S).--~ and an ind uced commutative exact diagram:

0-~

~(S)

0~

Q

-~ P~_I -~ ~-~(S) ~

W ~

-+0

~-~(S)

If Im f is not contained in JQ, then Q is a summandof ilk(S), this is: fl~(S) Q ~9 X and we have isomorphisms: F(~%~(8)) ~- J~F(S) ~- F(Q) ~9 F(X). The module F(3) is an indecomposable projective injective of length n, this implies k = n and X = 0. If Im f C_ JQ, then there exists an integer t _> 1 such that Im f ~ J~Qand Im f is not contained in J~+iQ. Wehave a commutative exact diagram: O ~ ~k(s) 0 ~ jt-lQ

~

Pk-~

~ ~k-~(S)

~

Z

~ ~k-l(S)

~

with 0 ~ Je-~Q ~ Z ~ ~k-~(S) ~ non sp lit an d J~-~Q,Z, ~k -~(S) generated in degree k - 1. Applying F we obt~n a non split exact sequence: *) 0 ~ Ja-~F(S) ~ F(Z) ~ ~-~F(Q) Since F(S) is projective injective, the inclusion map: j : J~-~F(S) ~ F(S) extends to F(Z). This is: there exists a degree zero map ~ : F(Z) ~ F(S) with ]i = j, but Im~ ~ Jk-~F(S), contradicting the sequence .) does not split. Nowassume Homh(S, A) ~ 0, then there exists an indecomposable projective Q such that S ~ J~Q ~d S is not contained in J~+~Q. As above, J~Q ~ S ~ X. Applying F, there is an isomorphism: ~F(Q) ~ F(S) ~ F(X), with F(S) projective injective. Let 0 ~ ~F(Q) ~ P~_~ ~ fl~-~F(Q) beexact with P ~_~ t he pr ojec tive cover of ~-~F(Q). Since F(S)is injective the inclusion F(S) ~ P~_~ splits, contradicting F(S) ~ JP~-~.

KoszuiAlgebrasand the GorensteinCondition

145

Therefore: HomA(S,A) ---- 0. It remains to prove that Ext~ (S, A) -~ Mis simple. Consider the minimal projective resolution of S : 0 -~ Pn -~ Pn-1 -+ "’" ~ P0 -~ S ~ 0. DuMizing we obtain an exact sequence: 0 ~ P~ ~ P~ ~ ... ~ P~_~ ~ P~ ~ M ~ O. Since P; is generated in degree -j and P;_~ is generated in degree -j + 1 the module Mis Koszul up to shifting. Applying Koszul duality F°in the opposite ring to Mwe obtain isomorphisms: JtF°(M) ~ F°(~t(M)), hence; there is a chain of isomorphisms: JtF°(M)/J~+~F°(M)

~

= $ Ext~(~(M)’A°)/~

~

= $kE0 Ez h.~+~ (M, Ao)/

~ Ext~ ~p(~t+~(M),A°) _ ~ ~+~+~ (M, ho) _

~

~o =

~= Homho~(~(g)/~(M)J,A

ho) In the other hand we have isomorphisms: soc~+~DF(S)/soc~DF(S)

~ D(J~F(S)/J~+~F(S))

~ D(Ext~(S,

p /jp We have proved soc"-~+~ DF(S)/socn-~DF(S) ~ J~F(M)/J~+~ In particular, JnF(M) ~ socDF(S). Suppose there exists a simple T ~ JkF(M) and T is not contained in J~+~F(M). Then jkF(M) ~ T ~ and ~k(M) ~ F-~(T) ~ F- I(X), th us F-~(T) is a p ro jective summandof fl~(M), but Ext~(M,A) = 0 for all k ¢ n. It follows k = n

o F(M) J"F(M). ~omthe fact DF(S) is injective it follows F(M) is isomorphic to a submodule of DF(S), but F(M) and DF(S) have the same length, hence; F(M) ~ DF(S). Therefore: F(M) is projective, hence M is simple. ~ DEFINITION. Let A = KQ/I be a greed quiver algebra. A ~r~ed simple S generated in degree zero satisfies the Gorenstein condition if there exists a non negative integer ns, called the depth of S, such that: i) Ex~(S,A) = for k ¢ n~. ii) There exists a graded A°" simple S’ generated in degree zero and an integer l such that Ext," (S, A) ~ S’[-l], as graded A°’-modules. Wewill denote by G (A) the set of isomorphism closes of graded simple satisfying the Gorenstein condition. PROPOSITION 4.2. Le~ A = KQ/I be a Gorens$ein graded quiver algebra, this is: idhA < ~ and idAh < ~, such that all graded A and A°~ simple have projective resolutions consisting of finitely generated projectives. Then there exists a bijection ~ : ~ (A) ~ ~ °~) between th e se ts of graded A a ndA°~ simple sati sfying the Gorenstein condition.

146

Marffnez-Villa

Proof. Let S be in g (A) and define ~o(S) = S’, where S’ is a graded A°p simple such that Extnh" (S, A) ~- S’[-l]. Wemust prove that S’ We know by proposition 3.1, Ext~op(S’[-l],A °p) = 0 for k ¢ n.,,. Hence; Ext~op (S’, A°p) = 0 for k ~ ns. Wehave isomorphisms:Ext~[~ (Sxt" h" (S, A)[/], A°P)) - S[-l] ~ Ext~’o~ (S’, h°P). Defining in a similar way a map~o’ : G(A°r) -~ Q(A), it is clear that are inverse maps. [] DEFINITION.Let A = KQ/I be a graded quiver algebra, a graded indecomposable projective Q satisfies the co-Gorenstein condition if there exists a non negative integer ne, called the co-depth, such that: i) The module Ext~ (A0, Q) = 0 for k # nQ. ii) There exists a graded °r s imple S’ g enerated i n degree z ero a nd a n i nteger 1 such that Ext AnQ(A0, Q) ~- S’[-l], as graded A°r-modules. Wedenote by CG(A)the set of indecomposable projectives Gorenstein condition.

satisfying

the co-

THEOREM 4.3. Let A be a Koszul K-algebra and F its Yoneda algebra. Then the Koszul duality F(M) = $ Ext~(M,A) induces a bijection °p) ¢ : g (A) -~ C~(F given by ¢(S) = F(S). Proof. Let S be in ~(A). By hypothesis, given S there exist two integers n and such that Ext~(S,A[m])o = if k ~ n or m # I and Ext~(S ,A[l])o = S’[- l] is a A°r-simple concentrated in degree I. By theorem 2.1, for k > -m there is an isomorphism of K-vector spaces: Extk^(S, A[m])o ~ Ext-gr+o~m(f(A[m]), f(S))o, where F(A)~ p and F(S) = is an indecomposable projective. Hence; Ext~+.~m(Fgr[m],Q)o= 0 for k +m _> 0, unless k = n and rn = I. We have proved Ezt[o.(rg ~., ,.,., E Xtro, n÷l ~op ( o [/], Q)a = Ext,(S, A[/])o = S’[l]. n+l op . Therefore: d~mK Extro~ (Fo , Q) = 1 = dim S’[l]. It follows Extro~ (F0 , Q) is simple as F°rK-module, hence ~oro~ T[/], with T a F simple. It was proved above ~b(S) F(S) = The converse is proved using again theorem 2.1. Let Q be an indecomposableprojective F°P-modulesuch that there exist integers n and l with Ext~op(r~r,Q) --- 0 for k ¢ n and EXt~o,(F~P,Q) ~ Till, with T a simple. Then we have Ext~o~ (r~r[m], Q)o = 0 if Let S be a graded simple such that F(S) ~- Q and let t be a non negative integer such that Extra (S, A) ~ 0. Then there exists an integer m with t + m _> 0 such that Ext~h (S, i[m])o ¢ 0. By theorem 2.1, Ext~ (S,

~ Extro h[m])o = t+m p (F(A)[m], F(S))o

Ezt[+oF(r~P[m],Q)o# O.

KoszulAlgebrasandthe GorensteinCondition

147

It follows t + m = n and m = l. Hence; EXtrA(S, A[m])o = 0 unless k = n - l and m = I. Wehave the following vector space isomorphisms: Ext~-~(S, A) = Ext~-~(S, A[/])0 ~ Till. Therefore: dimKExt~-t(S, A) = 1 and Ext~-~(S, A) is a simple concentrated in degree I. We have proved Ext~-~(S, A) ~ S’[-l]. [] One example of the situation considered above is the standard Auslander algebra. Weknow by [6] that a simple S over an Auslander algebra A satisfies the Gorenstein condition if and only if S has projective dimension 2. If F is the Yoneda algebra of A and F is the Koszul duality, then F(S) is projective injective if and only if S has projective dimension 2. It is clear that the conditions of the proposition are satisfied. PROPOSITION 4.4. Let A = KQ/I be a Gorenstein algebra such that all graded A and A°~’ simple have finite projective resolutions and let ~(A) be the set of graded simple satisfying the Gorenstein condition. Then for any simple Sj E G(A) there ezists an indecomposable projective Q:,(j) and an integer l such that Ext] sj (Sj, Q,,(.¢) [/])o ~ and a is an inj ective fun ction ~omG(A)to th e g raded indecomposable projective A-modules. Proof. Wehave the following isomorphisms: Ext~(Sj,A)

~- ~ Ext~(S~,A)k kEZ

with Sj generated in degree I. Since dimKSJ = 1 there exist isomorphisms: Ezt~(S~, A) = Ext,(St,

m A)t ~- Sj ~- ~ Ext,(S1, Qk[/])0.

It follows there exists someinteger a(j) such that Ext~ (S1, Q~(j)[/])o ~ 0 and

Ext2(Sj,Q~[l])o= 0 for k ~ a(j). Then there are isomorphisms: Ext s~ (Sj, Q~(j))~ ~- ~ Ext~(Sj, Q~(j))m ~- Ext~(SI,Q~(j)) k=l

Wewill show nowa is injective: Assumefor S~ ~ ~(A) there exist an isomorphism Qa(j) ~ Qa(k). Then both Sxt] s~ (S~, Qa(j)) and Sxt] s~ (S~, Qa(k)) are different from zero. There exists natural isomorphisms: Sxt~S~ ( S~, Q~(j) ) ~ Ext~S~ ( Sj, A ) ~ Q~(~) ~ S~ ~ Q~(~) Zxt] s~ (Sk, Q,(~)) ~ s~ (Sk, A) ~ Q,(k) ~ S~ ~ Q,(~)

148

Martlnez-Villa

Dualizing we have: D(S~ @~Qa(j)) -~ HomA(S~,D(Qa(j))) and D(S~k ~h Qa(k)) ~- Homh(S~k,D(Q~(~))). Hence; S~ ~- socD(Qa(~)) and S~k ~- socD(Qa(k)). Since Q~(~) Q~(k), th en By proposition 3.1 Sf DEFINITION. Wesay that an (graded) algebra is weakly Gorenstein if there exists °p an integer n > 0 such that for all (graded) A-modules M and all (graded) A modules N of finite length and all integers k > n we have Ext,(M, A) = 0 Ext~,~(N,A°P). THEOREM 4.5. Let A be a Gorenstein Koszul algebra such that all graded simple satis~ the Gorenstein condition and let F be the Yoneda algebra of A. Then all graded F and F°p simple satisfy the Gorenstein condition, in particular F is weakly Gorenstein. Proof. By proposition 4.4, for each graded simple Sj there exists a unique indecomposable projective Q¢(i) and an integer l such that Ext] s~ (Sj, Q~(j)[/])o ~ 0 hence; Extrasj (Sj, Qa(i)) ~- Extro~ (F(A)[I],F(S~))o There are isomorphisms: Ext2s~ (Sj, A[/l)o ~’ Ext]sj (St, Q~(~)[/])o ~ Extro, (F(Q¢(j))[I], F(S~))o. Set Ta(j) = F(Qa(j)) and F(Sj) = Assumethere exists an indecomposable projective F°P-module P~ and integers k and m such that Ext~op(T~(j)[k],P~)o There is an isomorphism: Ext’~op (T~(j)[k], P~)0 = Ext’~-k (F-~ (P~), Q~(~)[k])0. By proposition 4.4, Ext’~-k (F-~ (P~), Q¢(1)[k])o ~ 0 and Extnas~ (Sj, Qa(j)Ill)0 ~ 0 imply ns~ = m - k, F-~ (P~) = Sj and k = Wehave proved for each F°P-simple Ta(j) there exists an integer m such that dimKExt~o~ (Tao), F°~) = 1 and Ext~o~ (Tao), F°p) = 0 for k ~ m. It follows Ext~°~(T~(~), F°~) is a F-simple. If we consider A°p instead of A we obtain in a similar way that all graded F-modules satisfy the Gorenstein condition. [] 5

TENSOR PRODUCT CONDITION

OF ALGEBRAS

AND THE GORENSTEIN

In this section we will construct examples of algebras such that all graded si~nple satisfy the Gorenstein condition, we knowthat for selfinjective and generalized Auslanderregular algebras (see [4], [10], [11] and section 6, below) all simple satisfy this condition, we will prove that given two algebras A~, A: such that all the graded simple satisfy the Gorenstein condition the tensor product A~ ® A2 has the same property, hence the tensor product of a selfinjective and a generalized Auslander

KoszulAlgebrasand the GorensteinCondition

149

regular algebra will be an example of an infinite dimensional algebra of infinite global dimension such that all graded simple satisfy the Gorenstein condition. Wewill start by recalling Kiinneth relations, refereeing to [5] for the proof. PROPOSITION 5.1. Let A be a ring, A a left A complex and C a right complex, H(A), H(C), H(A ® C), the homology of the complexes A, C, A ~h C, respectively. Then there exists an exact sequence with a a degree zero map and ~ a degree one map: 0 -~ H(A) ® g(c) ~ H(A ® C) £ Tor~(H(A), Explicitly,

H(C))

for any integer n an exact sequence:

0 -~ ~, H(A)p ® Hq(C) ~ Hn(A ® C) -~ ~ Tor~A(Hp(A), p+q=n

Hq(C))

p+q=n--1

LEMMA 5.2. Let R,T be two K-algebras over a field, M,X left R-modules and N, Y right T-modules with M, N finitely presented. Then there exists a natural isomorphism: ¢ : Homn(M, X) ® HomT(N, Y) -~ Homn®T(M® N, X K given by ¢(f ® g)(m ® n) = y(m) ®g(n). Proof. If R = Mand N = T, then ¢ is an isomorphism, since is the composition of the natural isomorphisms: HomR(R, X) ® HomT(T, ~- X ®Y ~- Homn®T(R ® T,X®Y). AssumeN -- T and let R"~ --r R’~ -~ M-4 0 be a presentation of M, it induces exact sequences: R"~ ®T ~ Rn ®T-~ M®T--r O, n ® T, X ® Y) 0 --+ HomR®T(M® T, X ® Y) -r HomR®T(R ~ Homt~®T(Rm ® T, X ® Y) 0 ~ Homrt(M, X) ~ Homl~(Rn, X) -~ Homrt(R m, X) n, X) ® HomT(T, Y) 0 --~ Homl~(M, X) ® HomT(T, Y) --r Hom1~(R -~ HomR(Rm, X) ® HomT(T, The maps: n ® T, X ® Y) ¢ : HomR(Rn, X) ® HomT(T, Y) --~ HomR®T(R ¢ : HomR(Rm,x) ® HomT(T,Y) -~ HomR®T(Rm ® T,X @ Y) are isomorphisms, since the first ¢ is a composition of the following isomorphisms: HOml~(Rn, X) ® HomT(T, Y) ~- (~ HornR®T(R, X) ® HomT(T, Y) ~~(HomR(R, X) ® HomT(T, ~- ¢9 Homl~®T(R ® T,X ® Y) ~HomR®T(@ R ® T, X ® Y) ~- HomR®T((@R) ® T, X ®

150

Martlnez-Villa

Similarly, for the second ¢. Hence they induce an isomorphism: ¢ : HomR(M, X) ® HomT(T,Y) ~ HoraR®~r(M ® T, X Assumenow N has a presentation: exact sequences:

T~ ~ Tt -~ N -~ 0, it induces the following

M ® T~ -~ M ® Tt -~ M ® N -~ O, 0 -~ HomR®T(M® N, X ® Y) ~ Hora~®T(M t, X ® Y) -~ HomR®T(M® k, X® Y) 0 --> HoraT(N, Y) ~ Hom~(T~, Y) -~ Hom~(T~, Y) 0 ~ Hom1~(M, X) ® HomT(T, Y) -~ Homn(M, X) ® ~, Y) ~ -~ Homn(M, X) ® HomT(T~, Y) The natural

isomorphisms:

¢ : Hora~(M, X) ® HomT(T~, Y) -~ Ho~®T(M® ~, X® Y) ¢ : Hom~(M,X) ® HomT(T~,Y)

-~ Hom~®T(M ®T~,X

Induce an isomorphism: ¢ : HOml~(M, X) ® HomT(N, Y) --~ HomI~®T(M® N, X as claimed.

[]

PROPOSITION 5.3. Let R, T be two algebras over a field K and M, X left Rmodules, N, Y right T-modules. Assume M, N have projective resolutions consisting of finitely generated modules. Then for all n >_ 0 there exists a natural isomorphism: Ext~®T(M ® N,X @ Y) ~- ~ Ext,(M, i+j=n

X) ® EXt~T(N,

Proof. Consider projective resolutions of Mand N as R and T-modules, respectively, and assume all projectives in the resolution are finitely generated: *) " " -~ Pk f~-~ Pk-~ --+ " " P~ [-~ Po l--~ M-+ **) "’" ~ Qt a!~ Qt-~ ~ ’"Q~ ~ Qo "~ N ~ O. Since K is semisimple, it follows by Ktinneth formulas that the following sequence is a projective resolution of M® N as R ® T-modules:

***)...

i+j=n

~ P~®Q~ ~ P~®Q~-~...P~®OoePo®Q~ i+j=n- 1

~ Po ® Qo --+ M ® N --~ O Applying the functors Homn(-, X), HOmT(-, andHornn®T(--, X ® Y) to the sequences: *), **), * * *), respectively, we obtain complexes:

151

KoszulAlgebrasandthe GorensteinCondition ~) 0 -~ Homn(Po, X) ~ Homn(P~, X) ~... oo) 0 ~ HomT(Qo,X)

~ HomT(QI,X)

Homn(P~, X)

~ ...HOmT(Qm,X)

o o o) 0 ~ HOmR@T(Po ~ Qo, X ~ Y) ~ Homn~T(P~ ~ Qo ¯ Po ~ Q~, x ~ Y) ~ ...

Wehave natural

H~(ai

isomorphisms:

nPi~Q~’X~Y)~i+j=n

~ H~n~T(Pi@Q~,X@Y)~

~ Homa(Pi, X) @ HomT(Q~, Hence; o o o) is isomorphic to the tensor product o o oo) of the complexes o) Whereo o oo) is 0 ~ Hom~(Po,X)

@ H~T(Qo,Y)

~ H~n(Po,X)

@ H~T(Q~,Y)

~Homn(Px, X) ® HomT(Qo, Y) -~ "" ~ Homn(Pi,X)®HomT(Qj,Y) iTj=n

Let A be the complex o) and C the complex oo). Then A ® C is isomorphic the complex o o o). We have isomorphisms: Hp(A) ~- Ext~n(M,X), Hq(C) ~- Ext~(N,Y) and H,~(A ® C) ~ Ext~®T(M ® N, X ® The result follows by Kiinneth relations. [] COROLLARY. Under the conditions of the proposition for each integer n _> 0 there exists a natural isomorphism of (R ® T)°P-modules: Ext~®T(M ® N,R ® T) ~- ~ Exth(M ,n) iq-j=n

® EXt~T(N,T).

Proof. It is clear that the map: ¢ : Homn(M, R) ® HomT(N, -~ HomR®T(M® N, R ®T) K

given by ¢(f ® g)(m ® n)=f(m) ® is an iso morphism of (R ® T)°P -modules. From this it follows that the isomorphismof extension groups given in the proof of the proposition is an isomorphism of (R ® T)°P-modules. PROPOSITION 5.4. Let R, T be two algebras over an algebraically closed field K of small injective dimension si(R) and si(T), respectively. Then R ® T has small injective dimension and si(R ® T) = si(R) + si(T).

152

Martinez-Villa

Proof. Since K is algebraically closed, all R ® T-simples are of the form S = X ® Y, with X an R-simple an Y a T-simple. Set n = si(R) and m = si(T). Then for any integer k > n + m: ExtkR®T(X ® Y, i+j=k

Ext~T(Y, T) = Let M be an R-module of finite length with Ext~(M,R) ~ and N a Tmodule of finite length with Extra(N, T) ~ O. Then ExtR®T(Mn+m Ext,(M, R) ® Ext~(N,T) Wehave proved is(R ® T) = n + m. THEOREM 5.5. Let R and T be two graded quiver algebras over an algebraically closed field K, let Si and Sj be R and T graded simple, respectively, satisfying the Gorenstein condition. This is: there exists non negative integers nl and nj such that: Ext~(Si, R) = 0 for all k ~ ni and Ext~T(Sj, T) = 0 for all l ~ nj Extn~’(Si,R) = S~[l] is a graded R°T-module and Ext~(Sj,T) = Sj[m] is a graded T°P-simple. Then Si ® Sj is a R ® T-simple satisfying the Gorenstein condition. Proof. It follows from the isomorphisms: Ext~®T(S~

®Sj,R®T)

~- ~ Ext~(Si,R)

®Ext~(Sj,T)

sTt=k

ifs#niort#nj

and

12~X$R® T I,

Oi ® Sj,

R ®

T) ~-- Ext~~ (Si, R) ® Ezt r (S~, T)

COROLLARY. Let R, T be two graded quiver algebras over an algebraically closed field K such that all graded simple satisfy the Gorenstein condition. Then all graded R ® T-simple satisfy the Gorenstein condition. Weend this section with an example: Let A = Tn(K) be the triangular n x n matrix ring. Then all non projective simple satisfy the Gorenstein condition and all have depth 1, the unique projective simple has depth zero, but since its dual with respect to the ring is not simple, it does not satisfy the Gorenstein condition. Wedo not known examples of algebras such that all simple modules satisfy the Gorenstein condition and some of them have different depth. 6

APPENDIX:

SELFINJECTIVE

KOSZUL

ALGEBRAS

In [10], selfinjective Koszul algebras and their Yonedaalgebras were studied, in particular the following generalization of a theorem by Bondal-Politshchuk and P. S. Smith was proved. (see also proposition 4.1). THEOREM 6.1. Let A = KQ/I be an indecomposable following two conditions are equivalent:

Koszul algebra

then the

KoszulAlgebrasand the GorensteinCondition

153

Yoneda algebra of A is a selfinjective 0 and rn+l = O, with n >_ 2. 2) The algebra A satisfies

algebra with radical r such that

the following conditions:

i) There is an integer n >_ 2 such that all graded simples have projective dimension n. ii) Given a graded simple S for all k ~ n, we have: Ext,(S, A) = 0. iii) There exists a bijection Ext~ (-, A). The algebras satisfying Auslander regular.

between the graded A and A°P-simple given by

the conditions of the theorem were called generalized

DEFINITION.A graded quiver algebra A has small global sup (pdMIMis of finite length}.

dimension n if n =

The aim of this section is to prove the following: THEOREM 6.2. Let A be a noetherian generalized Auslander regular Koszul algebra of global dimension n. Let 0--~ A-~ Eo --~ El -~ E2 -4 ... -~ En --~ 0 be the minimal injective coresolution of A. Then En ~- D(A)[n]. (See [1] for related results). Wewill use freely the results and definitions from [9], [10]. PROPOSITION 6.3. Let A be a generalized Auslander regular Koszul algebra of small global dimension n. If M is a graded torsion ~ree module of finite projective dimension such that all projectives in the minimal projective resolution are finitely generated, then pdM < n. Proof. Weknow by [9], there exists a Koszul submodule N of Msuch that L = M/Nis of finite length. Since N is a Koszul torsion free module, it follows from [9], that pdN < n. Hence; pdL = n and pdM<_ n. Let 0 -~ Pa -~ ... ~ Pn-1 ~ P0 -~ M-~ 0 be the minimal projective resolution of M. The exact sequence 0 ~ N -~ L ~ M-+ 0 induces an exact sequence: Ext,(L,

A) ~ Ext,(M,

A) ~ Ext~(N,A).

Since Ext~(N,A) = an d Ext~(L,A) is of fin ite len gth as A°~-module, Ext~ (M, A) has finite length as A°P-module. Therefore: pdExt~ (M, A) = n. Let 0 -~ Q~ -+ Q~ -~ ... -~ Q~ -~ Ext~(M,A) -+ bea min imal pro jective res olution of Ext~ (M, A). Since A°p is generalized Auslander regular and Ext~ (M, A) is of finite length, Ext~o~(Exth (M, A), °~) =0 for i ~ n. Wehave a complex: 0 -~ P~ --~ P~* --+ ... --~ P~ -+ 0 with n - th homology group Ext,(M, A), hence a presentation: P~_I-~ P~ --* Ext,(M, A) -~ 0.

154

Martfnez-Villa Wehave the following commutative exact diagram: 0

0

O~

P

-~

P --~0

0-~

H

-> P,~

Ext~(M,A)

~

J, O~ f~(Ext~(M,A))

Ext~(M,A)

~

$. 0 where P is a projective Set

0

0

module.

M= Ex~(M, A).

There exists a decomposition of H, as H = P $ f~(M). The projective cover H is P $ Q~-I and there exist an epimorphism: P,~-I -~ H -> 0. It follows that P~-I has a decomposition P~-I -~ P $ P’ $ Q~,-1 and there exists a commutative square:

P ~9 P’ ¯ Qn-~

~ P ~ Q~

Dualizing, we obtain a commutative square:

(lo) ~ P*$P’*~Q~*__~ A

Since Im fn-~ C_ rP, it follows P = 0 and P,~ is the projective cover of M. Wehave an induced map of complexes:

with hi an isomorphism.

KoszulAlgebrasandthe GorensteinCondition

155

Dualizing, we get a commutative exact diagram:

The module Mis torsion free and Ext~o~ (M,A°p) of finite

length, therefore:

It follows, there exists a maps : Q~*-~ P~* with f~*s = h*o*. The equalities: f~*(h~* - s.gl) = f~*h~* - h~*gl = imply th e existence of a map: sl : Q~* ~ P~* such that f~*sl = h~* - s.gi. By induction, there exist homotopies s~ : Q~* ~ ~** i+~ with h~* = s~_~.g~ + J~isi, in particular, h~Ll = s~-~.gn_i + f~* sn-~. It follows h~:ig~ = f~* s~_ign = y~*h~*. The map f~* is a monomorphism,hence sn-~gn = h~* and gn splits. A contradiction. ~ COROLLARY. Let A be a noetherian generalized Auslander regular Koszul algebra of global dimension n. Then any torsion free module Mhas pdM < n. Proof. Let M be torsion

free. Then M= lim Ma with Ma finitely generated and torsion free. By [9], Ext~(lim M~, A) -~ lim Extn~(M~,A). --+ ~By the proposition, Ext~(Ma,A) = fo r ea ch a. Therefore Ext~(M,A) = O It follows, pdimM< n. []

Wecan prove now theorem 6.1. Proof. Let A be a noetherian generalized Auslander regular Koszul algebra and let 0 -~ A -+ Eo -~ E~ -~ E2 -~ ... -~ En -¢ 0 be the minimal injective coresolution of A. Weproved in [9], that En ~- D(A)[n] (9 E~ with socE~ = O. Consider the exact sequence: 0 -~ ~-n+l(A) --~ En_l --~ En -+ 0 and assume

E:~# 0.

"~ ’ = ExtA(E~,fl-~+I(A)) By the corollary, Ext A ( E~, ) i ~ This implies that the top row in the pull back:

0-~ f~-~+~(A) 0-~

[2-~+~(A)

-~ W -~ E~ -~0 -~

E~_~

-*

E~

splits. Hence; E’~ is a summandof En, contradicting the minimality of the coresolution. We have proved E~ = 0. [2

156

Martinez-Villa

REFERENCES [1] K. Ajitabh, S.P. Smith, J.J. Zhang, Auslander Gorenstein rings and their injective resolutions, preprint, (1999). [2] M. Auslander and M. Bridger, Stable Module Theory, Mem. of AMS94, Providence 1969. [3] A. Beilinson, V. Ginsburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. [4] A.I. Bondal, E. Politshchuk, Homologicalproperties of associative algebras: the method of helices, Russian. Acad. Sci. Izv. Math. 42, no. 2 (1994), 219-260. [5] E. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956. [6] E. L Green, R. Martinez Villa, Koszul and Yoneda algebras I, Rep. Theory of Algebras, CMSConference Proceedings, Vol. 18 (I996), 247-306. [7] E. L Green, R. Mart/nez Villa, Koszul and Yoneda algebras II, in "Algebras and Modules II" CMSConference Proceedings, Amer. Math. Soc. Providence, Vol. 24 (1998), 227-244. [8] P. Jorgensen, J. Zhang, Gourmet guide to Gorensteinness, preprint (1999). [9] R. Mart/nez-Villa, Serre Duality for Generalized Auslander Regular Algebras, Contemporary Math. Vol. 229 (1998), 237-263. [10] R. Mart/nez-Villa, Graded, Selfinjective (1999), 34-72.

and Koszul Algebras, J. Algebra. 215

[11] S. P. Smith, Some finite dimensional algebras related to elliptic curves, Rep. Theory of Algebras and Related Topics, CMSConference Proceedings, Vol. 19 (1996), 315-348. [12] J. Zhang, Connected Graded Gorenstein Algebras with Enough Normal Elements, J. Algebra, 189 (1997), 390-405.

Some remarks about the "double extension" bra of a finite poset

alge-

TERESITANORIEGADpto Ecuaciones Diferenciales, Fac Mat-Comp, Universidad de la Habana, San Lazaro y L Habana 4, Cuba, emaih [email protected]

ABSTRACT In [4],^Bautista and Martinez introduced what we will call the "double extension" algebra A of a finite poset S. Wewill denote by A the incidence algebra of S. The tilted algebras were introduced by Happel and Ringel in [5] and since then, have been proven to be a powerful tool in the study of different classes of algebras. The main result of this paper is a theorem that relates the property of/~ being tilted to the same condition in A. Somecorollaries give more precise results in the cases: Ais hereditary, A is tilted, A is tilted and of finite representation type. 1

INTRODUCTION

Throughoutthis paper, k will denote a fixed algebraically closed field. Wewill consider finitely generated right modulesover a finite dimensional (associative with unit ) k-algebra A (or finitely generated left modulesover °p).If Mis an A-m odule we denote by EndA (M) the ring of endomorphisms of M, by RadMits radical, by SocMits socle and by pdAM(idAM) its projective dimension (injective dimension). S will be considered a finite and connected poset. The incidence algebra of S is the quotient of the path algebra corresponding to the Hasse diagram of S modulo the ideal generated by all commutativity relations. The "double extension" algebra ~ of S is the quiver algebra k~/I where ~ is the Hasse diagram of ~ = St~ {rn, f} with m ~ s and s ~ f Vs E S and I is the ideal generated by all differences of .paths sharing the same initial and end points. /~ has a unique projective-injective A-moduleP,~ = If and this we will denote by /~. Werecall somebasic definitions. DEFINITION 1. [1] Let A be a finite dimensional k-algebra, let TA be a finitely generated A-module, we say that TA is a tilting module i] and only if it satisfies: 1. pdTA <_ 1 157

158

Noriega

2. EXtrA (T, T) = 3. There exists a short exact sequence 0 ---~ AA ~ T~A ~ T~ ---~ 0 where and T~ are direct sums of direct summandsof TA. The third condition is equivalent to the following: "The number of indecomposable non-isomorphic summands of TA is equal to the number of non-isomorphic simple modules in A." There is also the dual notion of co-tilting moduleif and only if it satisfies:

module. Wesay that TA is a cotilting

1. idTa <_ 1 2. Ext~ (T, T) = 3. There exists a short exact sequence 0 --~ T~ ---} T~ ---} D(A) ---> where T,~ and T~ are direct sums of direct summandsof TA. DEFINITION 2. [1] An algebra A is called a tilted algebra of type ~ (where ~ is a finite connected quiver without oriented cycles) if there exists a tilting moduleTB over the path algebra B = k~ such that A = End (TB) or equivalently if there exists a tilting A°P-module AU such that End (AU) = k~. REMARK 1. [1] If PA is an indecomposable projective-injective is a direct summandof any tilting module TA.

A-module then it

DEFINITION 3. [1][5] A class ~ of non-isomorphic indecomposables A-modules is called a complete slice in modA if it satisfies: 1. U = (~Me8 M is a sincere module (That is, HomA(P, U) ~ 0 for any zero projective A-module P). 2. If Mo ~ M1 ----} ... ---4 M,~ is a sequence of non-zero non isomorphisms in rood A, with Mo, MnE S, then Mi ~ S for all 0 -~ i -~ m. 3. If 0 ---} L ~ M ---} N ---~ 0 is an almost split sequence, then at most one of L and N lies in S.Furthermore, if an indecomposable summandof M lies in S, then either L or N lies in If S is a complete slice in A then U = I~)Me8 Mis a tilting A-moduleand it is called the slice moduleof ,~. A complete slice and its existence characterize tilted algebras according to the following theorem: THEOREM 1. [6] If B is hereditary and TB is a tilting module with A = EndTB, then the class of all indecomposable A-modules of the form Hom~(T, I) with indecomposable injective is a complete slice in mod A. Conversely, if ~q is a complete slice in rood A, then UA = (~MeS M, is a tilting module with B = End UA hereditary and thus ~q is isomorphic to a complete slice of the previous form. In A it will be important to consider the indecomposable sincere module H with k at each vertex and all mapsthe identity.

"DoubleExtension"Algebraof a Finite Poset 2

MAIN

159

THEOREM

Our main result is the following: THEOREM 2. /~ is tilted if and only if A is tilted and there is a tilting T such that H is a direct summandof T and Endh (T) is hereditary.

A-module

Proof. 1.-Supposethat ,~ is tilted. Since ~i, is tilted its Auslander-Reiten quiver contains a complete slice ~ as a connected full sub-quiver. Let :~ be the slice module for ;~. As P is a projective-injective/~-module it is a direct summandof every tilting -~-module and so lies on ~. The only almost split sequence of/~-modules which contains/5 is 0 -~ rad/5 -4/5 ~9 rad/5/soc/5 -~/5/soc/5 -40. Thus ~ contains either radP or P/soc/5. Let I be an injective/~-module. Then top/--- top/5 is not a composition factor of radP and so there is no non-zero map from I to radP. Dually, there is no nonzero map from P/soc/5 to a projective/~-module. It follows that, without loss of generality, we may assume that ~b = p $ radP ¢ radP/socP ~9 T’

(1)

for some h-module T’. Write T = r^adP/s^oc/5 $ T~. Then 5b =/5 ~ rad/5 $ T is a ~lting module if and only if ~ =/5$P/socP$T is a tilting module. Wedenote by S’ the slice corresponding to Weshall show that T is a tilting A-module with End^(T) hereditary. T is a A-module. We have to prove that T can’t have TopP or Soc/5 as composition factors,

it

is enough to prove that Hornh (X,/5)

if X is a summandof T, then Hornh (/5, X) = 0 and = 0. Suppose that there is a summand X of T such that

Hornh (/5, X) ~ O. Let o be a non-zero map from /5 to X, then o factorizes through P/SocP and hence/5/Soc/5 will belong to the slice ~, but this is imposible as RadP = V (/5/Soc/5) belongs to ~. Suppose that Hornh (X,P) ~ with X a di rect su mmand ofT. From (1) we had that ~ is tilting if and only if ~ is tilting. Consider now~ and let y be a non-zero map from Z to/5.

Then ~ factors

through RadP = r (~/Soc/5)

which is impossible as P/SocP belongs to ~. REMARK 2. Notice that RadP/SocP is isomorphic

to H as a A-module.

160

Noriega

pdAT <_ 1. As Pdh~ = Sup(pdhP,pdhRad#,pdhT } and pdh~ <_ 1, we have pdhT :<_ 1. So we have a projective resolution:

0 ---~h P1 --~h Po---~h T --~ 0

(2)

of T as h-module. Using Proposition 1.1 of [2], we get AT ~-- Hornh (-fi,

T)

where P = (~eAo P~ and thus pdAT <_ 1. Ext^ (T, T) = It suffices to prove that ExtlA (Ti,T¢) = 0 Vi,j, where the T~’s are the direct summands of T. Since pdAT~<_ 1, we have Extra (Ti, Tj) ~_ D HomA(T~, rATi). It will be enough to prove that there is no path from Tj to ~’aTi. Suppose the contrary, that is, there is a path from T1 to vhTi. Let’s consider O ---~ TATi ~-~ F --~ Ti ---~ 0

(3)

the almost split sequence in modA.

o ---~ r x T~~--~E A-~T~--~ o

(4)

the almost split sequence in mod/~. The sequence (3) considered as a sequence in mod/~is exact non split, have the following conmutative diagram: 0

~

0 ---~

~’ATi rAT~

~ l--~

F

-~

E -~

Ti T,

~ ----~

so we will

0 0

Then we can find a path

Since Ti and T~. belong to the slice ~, it follows that rhTi should belong also, which contradicts the fact that slices are sectional. The number i of indecomposable non-iso summandsof T is equal to the number s of simple non-iso A-modules Weknow [5] that the number of indecomposable non-iso summandsof ~b is equal

"DoubleExtension"Algebraof a Finite Poset

161

to the number of simple non-iso /~-modules. Let us call both of them n. Since ~’ -- ~ (~ Rad~ ~]~T, we have i = n-2 and by construction of A, we have s = n-2 too, so i = s. Endi (T) is hereditary Let ~’ = ~ ~]~ PadS’ (~ T = T" (~ T. Then:

i?t=

Endh (~)

)( Endh Horn(h T" (T", T) End O) h (T)

The bound quiver of EndA (T) = Endh (T) is a full convex bound subquiver of [/. Hence, EndA (T) is hereditary. 2.-Suppose that h is tilted and that there is a tilting is a direct summandof T and EndATis hereditary.

A-moduleT such that H

Wewill first establish two lemmas. LEMMA 1. Let A be an algebra and T a tilting A-module such that End~ (T) hereditary. Let To be a summandof T. I] ~ = A[To] is the one-point extension [7] of A by To and P is the indecomposable projective T-module which is not a A-module, then: 1. T (~ P is a tilting

~-module.

2. End~. (T (~ P) is hereditary. Note. Results similar to 1. are known,see [6]. Proof. pd~ (T (~ P) 1. pd~ (T (~ P) = Sup {pd~T, pd~P} = pd~T = pdhT <_ Ext~ (T (~ P,T (~ P)

Exth (T (~ P, T (~ = Ext h (T, T (~ P)~ Exth (P, T ~ As P is ~-projective,

we have Exth (P, T (~ P) = sowe have to prove tha t

which is the same as Ext~ (T, T) = and Ext~ (T , P) = O Let us consider the sequence: O--~T-~E--~T--~O

(5)

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Noriega

an exact sequence of E-modules. A is a full convex subcategory of ~, so modA is closed under extensions and (5) will be an exact sequence of A-modules. As Ext~h (T, T) = O, we have Ext~ (T, T) = Wewill consider now the exact sequence in mod E: O--+ To---~ P---~ S----> O (notice that To ~- RadP). Applying to it Hom~(T,-),

we get:

¯ .. --~ Exth (T, To) --+ Exth (T, P) --~ Exth (T, S) Ext~ (T, S) = 0, because S is E-injective, Exth (T, To) = 0 because To is a direct sumand of T and Ext~ (T, T) = 0, hence Ext~ (T, P) = The number ~ of indecomposable non-iso summandsof T (~) P is equal to the number ~ of non-iso simples in As the number n of indecomposable non-iso summandsof T is equal to the number of non-iso simples in A, then by definition of one-point extension we get: ~=n+l=~. Endr, (T (~ P) is hereditary. We have: End~.(T(~P)=

( Homr, End~.(T) (P, T) Hom~.(T,P) Endr~ (P)

)

Hom~.(P, T) = 0, because if there is a non-zero morphism ] : P ~ T, then T will have topP as composition factor and that is impossible. So End~. (T ~ P) is the one point extension of End~ (T) by Hom~. (T, P). End~. (T) = End^ (T) and is hereditary, we need only to show that Horn~. (T, P) is Endr. (T)-projective. We claim that Hom~(T, P) Homr~ (T , To Let f : T --+ P be a E-linear map. Then ] is not epi, because if f is epi, as P is Z-projective, f splits and P will be a direct summandof T, which is impossible. P is indecomposable projective with a unique maximal submodule RadP, so Ira] C_ RadP ~_ To, and then: Hom~. (T, P) = Horn~ (T, To) = gomh (T, To) But HomA(T, To) is EndA (T’)-projective,

since T = To (~ T’

Endh (T)= HOmA(T, To) (~ Homh (T,T’). Therefore Hom~.(T, P) is End~ (T)-projective ditary.

and hence End~. (T ~ P) is here[]

Wewill now give the following lemma, dual of Lemma1, without proof.

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LEMMA 2. If F is tilted with U a cotilting module and Endr (U) hereditary, let be a direct summandof U, then the one-point coextension [7] U = [U0] F of F by Uo is tilted with cotilting moduleU (]~ r, where I ~ i s t he i ndecompasable nnjective i Fr-module which is not a F-module and Endr (U ~]~ ~) i s h ereditary. Wewill nowprove that/~ is tilted. First we consider A’ = h [H] the one-point extension of A by H. Let us call P~ the new projective. By Lemma1, A~ is tilted with tilting A~-module T~P~ and End~,, (T (~ P’) hereditary. Nowby Lemma2 we can consider the one-point coextension [Pt]A’ of At by P~. The indecomposable injective [~P’]A’-module which is not a A~-module, will be a cotilting moduleover [P~]A~ _~ A, by abuse of language, we will call it/5. But: pdh (T ~ P’ (~ /5) = h (T(~ P’) = pd^ (T ~ P’) So T (~ Vr~]~/5 is a tilting hence/~ is tilted. The following corollaries

/i,-module

with E.dh (T(~ P’(~/5) hereditary

and []

can be obtained from the theorem.

COROLLARY 1. If A is hereditary then ~ is tilted

¢=~ H is not regular.

Proo]. By [5], H not regular ~ H is postprojeetive (in [5] they are called preprojective) or preinjective. As H is an indecomposable and sincere A-module, there exists a complete slice in rood h on which H lies. Then by the theorem,/~ is tilted. [] COROLLARY 2. If A is tilted,

then ~ is tilted

¢==~H is directing.

Proof. If H is directing, since it is an indecomposable and sincere A-module, it belongs to a complete slice and applying the theorem A is tilted. If h is tilted, then H belongs (by the theorem) to a complete slice, then also to connecting component and then by [7], H is directing. [] COROLLARY 3. If A is tilted and of finite representation type, then fi, is tilted. Proof. This follows from Corollary 2, since if A is of finite representation type, all indecomposable A-modules are directing. [] ACKNOWLEDGEMENTS The author would like to thank Dr. Sheila Brenner and Dr Ibrahim Assem for manyusefull and enlightning discusions and sugestions.

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REFERENCES [1] I. Assem. Tilting Theory, an introduction. part 1, 1990, pp 127-179.

Banach Center Bulletin.

Vol 26,

[2] I. Assem and P. Brown. Strongly simply connected Auslander algebras. versit~ de Sherbrooke. preprint No 160, June 1995.

Uni-

[3] M. Auslander, I. Reiten and S. Smalo. Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, Vol 36. Cambridge University Press. 1994. [4] R. Bautista and R. Martlnez. Representation of posets and 1-Gorenstein Artin algebras. Proceedings of the 1978 Antwerp Conference. Marcel Dekker Inc., 1979, pp 385-433. [5] D. Happel and C.M. Ringel. Tilted Algebras. Transactions Mathematical Society. Vol 274, No 2, 1982. pp 399-443.

of the American

[6] C.M. Ringel. Tamealgebras and integral cuadratic forms. Lect. Notes in Math. 1099, Springer Verlag. 1984. [7] C.M. Ringel. Representation theory of finite Math. Soc. Lect. Notes 116, 1985. pp 7-81.

dimensional algebras.

London

Coil algebras which are derived-tame JOSE ANTONIODE LA PEI~A Instituto de Matem~ticas~ UNAM,Ciudad Universitaria, Mdxico 04510, D. F., Mdxico, E-mail: jap~penelope~matem.unam.mx

BERTHATOMI~Depto. de Matem~ticas, Facultad de Ciencias, UNAM,Ciudad Universitaria, Mdxico 04510,D~ F., Mdxico, E-mail: [email protected]

ABSTRACT Let k be an algebraically closed field and A be a finite dimensional k-algebra~ A classification of the coil algebras A having n0mnegative Euler form XAis presented in order to prove our main result: the repetitive algebra ~ of a coil algebra A is tame, if and only if the Euler form XAof A is non-negative.

Let A be a finite-dimensional algebra over an algebraically dosed field k. Following [14],.we say that A is derived-tame if g/dim A < o~ and the repetitive category ~ of A is tame. It has been shownthat the class of derived~tame algebras is closed under derived equivalence~ and includes the hereditary tame algebras, the. domestic tubular and tubular algebras and. the pg-critical algebras (see [2,10,11,14]).. Ifgl. dim A
166

de la Pefia and Tom~

The ¯results of this work were ¯presented by the second author during the Conference in Representation Theory of Algebras held in Sao Paulo in July 1999. Both ¯ authors acknowledge support from CONACyT,M~xico. 1 ~PB, ELIMINAP,~IES 1.1: Let A be a basic, finite-dimensional algebra over an algebraically closed field k. It is well knownthat A may be written as A = kQ/I, where Q is a finite quiver and/is an admissible ideal :of the path algebra kQ. Weassume moreover thai; Q is connected and has no oriented cycle. Weconsider A.as a k-category whose objects are the vertices Q0 of Q and in which the morphism space A(x, y) from x to y e~Ae~, where ez denotes the primitive idempotent associated to .ghe vertex x. See [9]. The repetitive category ~ is the k-category with objects Q0 x Z (which are denoted by 8[i] for s E Qo and i E Z) and in which the only possibly non-zero morphism spaces are ~(r[i],s[i])= A(r,8) {i} an d ~(r[i],s[i + 1]) = DA(s,r) × {i }, where D = Hom~(-, k) denotes the usual, duality. See [10]. 1.2.. Let A = kQ/I be as above. Werecall that theone-point extension AIM] of the algebra A by a module Mis the k-category with objects Qo O {8} and m0rphism spaces A[M](s,x) = M(x) forx ~ s, A[M](8,s)= k, and such that A is a full convexsubcategory.’ See .[16]. Let j be asink in Q. The reflection SfA of A at j is the quotient of the onepoint extension A[Ij] by the..two-sided ideal generated by ej, where Ij denotes the injective envelope of the simple S~ at j. The reflection S~-A of A at a source i of Q is defined dually. It was shownin [18] that S~A and.A are tilting-cotilting equivalent, that is, there is a sequence of algebras A= Ao,AI,... ,Am = SfA and asequence of modules a~Ti with 0 _< i < m such that Ai+t = Enda~Ti and Ti is either a tilting or a cotilting module. In particular, this implies that SfA and A are derived equivalent, that is, they have triangle equivalent derived categories. ’Let B be ~ny finite-dimensional k-algebra. Then by [12], the repetitive categories ~ and B are :isomorphic if and only if A and B are reflection-equivalent, that is, *~here ~.s a seq~er_ce of a/gebras A = A0,At,... ,Am = B where Ai+~ = S~(1)Ai or S+r.(i)Ai for.some source a(i) or somesink ~’(i) of Ai (0 <_ i < m). 1.3. An algebra A is tame if for every d fi ~l there is a family M~,... ,M~(d~ of A - k[t]-bimodu!es which are finitely generated and free as right k[t]-modules and such that almost every indecomp0sable A-module X of dimension d is of the form Mi ®~[t] S~ for some 1 < i < ~(d) and A ~ k, where S~ k[t]/(t - A)is a s it nple k[t]rm0dule. An:infinite category, such as the repetitive category ~ of an algebraA, is said tO be tame if. every full finite subcategory is tame. Clearly, if A is derived:tame, then A is tame. 1.4. Werecall that a 2-tubular extension is a one-point extension D[M]of a critical ¯ algebra D of type II~m by an indecomposable regular module Mof regular length 2

Coil Algebrasthat Are Derived-Tame

167

and period m- 2. Forbasic definitions see [16]. Therefore, we call.an iterated one-point extension DIM1,..., Mr] of a critical algebra D of type ]~,~ by indecomposable regular modules of regular length 2 and period m- 2 such that Mi ~ roMi+l for 1 < i < r-1, and Mr ~voM1, an iterated 2-tubular extension. 1.5. For an indecomposable module Min an inserted tube of a domestic tubular algebra, the notion of level was defined in [17], where it was used indistinctively for domestic tubular or cotubuiar algebras. Here we will distinguish the two cases. Let D be a domestic branch extension of a critical algebra C, and. let 7" be an inserted tube in r~9..The ray modules in 7, will be called of level 1, and a module ME 7" will be called of level n > 1 if there is a sectional path in T of length n - 1 M =Mn ~ Mn-1 ~ "’"

-~ M2 -+ M1 =E

with E a ray module. The notion of colevel is defined dually. Let D and T be as above, and let ME T. By [16], if B is any critical algebra of tubular type the extension type of D over C, ¯then there exist a tilting B-module T = T1 $T2, with T1 preprojective and T~ regular, such that D = End/~ (T), and indecomp0sable regular B-m0dule X such that .M = EX, where E = HomB(T,-). It was shownin [17] that the level of Misn if and only if the regular length £(X) of X is n. The following result is also taken from [17]. LEMMA. Let D and M be as above. If XoIMI is non-negative then level (M) < 2. Moreover, if level (M)= 2, ~hen C is of type ~m, and the extension type olD over C is (n - 2,2,2), where n> ra >_ 4. 1.6. Let B be an algebra, ¢ be a standard component of FB and X be an indecomposable ¯module in C. In [3], three admissible operations (ad 1), (ad 2) and (ad were defined ¯ depending on the shape of the support of HomB(X,-)lc in order ~. obtain a new algebra B (ad i) If the support of HomB(X,--)It is of the ¯ X = Xo ~ X~ -~.X~ ~ ... weset B’ = (B x D)[X~ Y,], where D is the full t x t lower triangular algebra and Y~ is the indecomposable projective-injective D-module. (ad 2)If the support of HomB(X,-)le is of the Yt +- "-- ~- Y~ ~- X = Xo ~ X1 -+ X~ -+ ... with t _> 1, sO that X is injective, we set B~ = B[X]. (ad 3) If the support of Homn(X,-)lc isof the

X=

Xo

~ X~

~ ...

~ X~_a

matrix

de la Pefia and Tom~

168 with t _> 2, so that X~-xis injective, we set B! = B[X].

In each case, the moduleX and the integer t are called, respectively, the pivot and the parameter of the admissible operation. Moreover, the componentCI of rB, containing Xis standard under certain conditions satisfied in this work. The dual operations are denotedby (ad 1"), (ad 2*) and (ad Following[5], an algebra A is a coil enlargementof the critical algebra C if there is a sequence of algebras C = Ao,A1,... ,Am = A such that for 0 _< i < m, Ai+x " is obtained from Ai by an admissible operation with pivot in a stable tube of Fc or in a Component(coil) of FA, obtained, from a stable tube of Fc by means of the admissible operations done so far. WhenA is tame,;.we call A a coil algebra.. If A is a coil enlargement of a critical algebra C, thenthere is a maximalbranch coextension A- of:C inside A which is full and convex in A, and such that A is obtained from A- by a sequence of admissible operations of types (ad 1), (ad : and (ad 3)~ Dually, there is a maximal branch extension A+of C inside A which Is full and convex in A, and such that A is obtained from A+ by. a sequence of admissible operations of types (ad 1’)~ (ad 2*) and (ad 1.7. For a coil enlargement A of a critical algebra C, we define the type t(A) of A as follows: Let T = (7~)~6pi(k ) be the separating tubular family of rood C. For each )~ E P~(k), let n~ be the rank of 7~, r~ (respectively, c~) be the numberof (respectively, corays) inserted in 7~ by the sequence of admissible operations that leads from C to A, and t~ =.n~ + rx+ c~. Finally~ let t(A) ---- (t~)~e_o~,,~i, where write downonly those $~ ~ 1. 1.8. A coil enlargement A of a critical algebra C is called a branched-critical algebra if A is obtained from C by a sequence of admissible operations of types (ad 1) and (ad 1") such that the pivot of each operation is both a ray and a coray module (see

[8]).

¯ Weobserve that Ringel’s branch extensions or coextensions of critical algebras are branched-critical algebras A for which A- or .A÷ is trivial. Moreover, if A is branched-critical and A-and A+ are non-trivial, then A can be written as A = A~ [Mi, Ki]~=~, where the Mi are both ray and coray A--modules and.the Ki are branches. 2

COIL

ALGEBRAS

WITH

NON-NEGATIVE

EULER

FORM

2.1. Wedevote this section to the characterization of the coil enlargements A of a Critical algebra C, not of type ~k~, having non-negative Euler form. The following proposition together with [5] and [15] shows that these algebras are tame, that is, coil algebras. PROPOSITION. Let A be a coil enlargement of a critical algebra ~, not of tgpe ouch that both A- and A+ are non-trivial, If)ca is non-negative, then A- and are domestic. Proof. + Since Xa is non-negative, so are X~- and XA+. Thus, by [15], A- and .A

Coil Algebrasthat AreDerived-Tame

169

are either domestic or tubular. Assumethat A- is a tubular algebra which is a branch extension of Co and a branch coextension of Coo = C. Then indA-=P0YToV

V ~VTooVZoo. .7eQ+

Let.z0 be the minimal .positive generator ofrad 2:c0, and let E.be a simple regular Co-module of period 1. Without loss of generality, we may assume that A is obtained from A~ by a single admissible operation. Let X E ind A= be the pivot of the admissible operation and let Pwbe the indecomposable projective Amodule Such that X is a direct summandof rad Pw. Since E E To, X ~ Too, Tois separating, and pdimAE = pdimA-E = 1, (dim

Pw,zO)A

= (dim E, dlmPw) = dim~HomA(E, Pw) "dim~Exth.(Ei = dim~HomA-(E,X) >0.

Then xa(dlrn P~ - 2zo) =1 - 2(dlm Pw,zo) < 0, a contradiction. is domestic:¯ : Dually, one shows that A+ is domestic.

Pw)

Therefore A[]

2.2. Since the coil algebra A can.be obtained from A- by a sequence of admissible ¯ operations of types (ad 1), (ad 2) and (ad 3), and XAis. non-negative, (2.1) (1.5) show that the pivots of such operations which, belong to rood A, must of colevel at most 2. Assumefirst that Ais branched-critical, that is, A = A-[MI,K~]i=I " ~ , where the M~areboth ray and coray A--modules and the.K~ are branches. Let A, = EndB(T), where B is a critical algebra of tubular type the coextension type of A= over C, and T = T1 ~B T2 is a cotilting B-modulewith T1 regular and T2 preinjective. For 1 ~ t, let M~= EX~, .where Xi is a simple regular B-module and ~. = DHomB (-,T). Finally, let B’ B[X~, K~].~=~. . LEMMA. With. the notation introduced above, there is a cotiltin 9 B’-module T’ such that A = End,, (T’). Proof. By induction on t. Whent = 1, A.= A-[M~,K~] can be obtained from Aby a sequence of one-point extensions and coextensions.. The proof of this case then follows from [17]. [] 2.3. For .branched-critiCal algebras, we achieve our goal with the following result. PROPOSITION~ If A is a branched-critical only if t(A) is Dynkin or Euclidean.

algebra, then X~ is non-negative if and

Proof. Wemay assume that both A- and A+ are non-trivial. Under any of the two hypothesis, A- is domestic, :and hence A is cotilting equivalent to a branch extension B’ = B[XI, K./]~=~, where B is a critical algebra whose tubular type is the same as the coextension type of A-~over C. Therefore t(A) is. theextension type of B’ over B. The assertion then follows from [15]. []

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de la Pefia and Tom~

¯ 2.4. Assumenowthat A is not branched-critical, that is, the sequence of admissible operations that leads from A- to A contains an operation of one of the following kinds: an operation (ad 1) whose pivot is not a coray module, an operation (ad an operation (ad 3). Weconsider the first operation of this kind that appears the sequence, and we denote its pivot by N. By [5], we may assume, that all the operations of type (ad 1) whose pivot is a coray module and which can be carried out before the operation with pivot N precede it in the sequence. Let A~ be ~the algebra obtained from A- by the operations mentioned above, that ~ = A-[Mi, N coil in FA-[M,~:,]~ffi has no exceptional mesh(see [3]). Note Hence we can is, Acontaining Ki]~=I [N], where aA-[MI,K~]~= that the 1 is branched-critical. keep the notions of level and colevel as defined in (1.5). As in ¯(2.2), A-[Mi, Ki]i=~ = Ends,(T’), where B’ B[XI, Ki ]i=l is a b ra nch extension of the critical, algebra B whose tubular type is the coextension type of A- over C, and T~ is a cotilting B~-module. If N is an A-.module, ¯then N = EZ, where Z is an indecomposable regular B-module with regular, length l(Z) = colevel (N) > 1. If N is not an A--module, then .N " ~ (R, HomA-(Mi,R), 1) for some indecomposable A--module R and some 1 < i. < r. As above, R = EZ, where Z is an indecomposable regular Bmodule with l(Z) = colevel (R) = colevel (N) > 1. Then N= ~ = E~, where (Z~ H0ms(Xi, Z), 1) is an indecomposable B~-modulewith eolevel (~) £(Z) > Hence, in both cases, we obtain that N = EY for some indecomposableB~-module Y with level (Y) _> colevel (Y) = colevel (N) ¯ . Let. B" = B’[Y],. then there is a cotilting B"-module T" such that A~ = EndB,,(T"). Since XA, is non-negative, so is X~,. The next result, whose proof is similar.to that in (2.1), showsthat ~ i s d omestic. LEMMA. LetB’ be a branch extension of.a critical algebra B and Y be an indecomposable B’-module in the separatingtubular family of rood B’ obtained from the tubular family of rood B by ray insertions. If Xs,t~, ~ is non-negative, then B’ is domestic. 2.5. The next two lemmasshowthat the colevel of N, as defined in (2.4), is LEMMA. Let A be obtained from a branched-critical eration of one of the following kinds:

algebra by an admissible op-

i) (ad 1) with parameter t = 0 and pivot a module of colevel greater than ii) (ad ~), iii) (ad 3).

.

Let N: be th e pivot of such operation. If XA is non-negative, then the colevel of N is 2. Proof. As in (2.4), A is cotilting equivalent to B" = B’[Y], where B’ is a branch extension of a critical algebra B, and Y is an indecomposable B~-modulelying in an inserted tube of F/~,. and satisfying level (Y) > colevel (Y) = colevel (N) Since X~ is non-negative, B~ is domestic, and therefore tilting equivalent to a critical algebra D whose tubular type is the extension type of B~ over B. Moreover,

Coil Algebrasthat AreDerived-Tame

171

Y = I2U, where U isan indecomposable regular D-module with g(U) = level (Y). Then B" = B’[Y] is tilting equivalent to D[U], and so XDIvl is non-negative. By [13], g(U) = 2. Therefore c01evel (N) = 2. 2.6. LEMMA. Let A be obtained from a branched-critiCal algebra bit an operation (ad 1.) whose pivot. N is not a corait module. If XA is non-negative, then¯ the parameter t of the operation is O. o r Proof. Let A B[M~,¯ Ki]~=x denote the branched-critical algebra from which A is obtained, If t.E !, then A is obtained from A’ = A-[MI, K~][=I[N] by a sequence oft one-p0int coextensions.. Without loss of generality, we mayassume that t = 1. Let 0 denote the extension vertex of A~, Then A = [I~]A’, where I~ is the simple injective At-modulecorresponding to the vertex 0. As in (2.5), A’ is cotilting-tilting equivalent to D1 = D[U], which is a 2-tubular extension. Then A is co~ilting-tilting equivalent to D2 = [Io]Dx, where Io i s the simple injective Dl-modulecorresponding to the vertex 0. Since XA is non-negative, so i s Xo~. Since D1 is a 2-tubular extension, tad XD:has 2 generators: the minimal positive generator z of rad Xo and dim W+ eo, where Wis an indecomposable preinjective D-module such that .dim~H0mD(U,W) = 2 (see [17]). Let w be the coextension vertex of D~ = [Io]D~. Then Xo~ (e~ + 2(dim W + co))< 0

(e~, dim W+ Co) = (dim W+ Co, e~) = (Co, ew) = -1. Therefore t. = O.

~

2.7. Note that. the admissible operations in the sequence that leads from A~ = ¯ A-[MI, Ki]~=~[N]toA fall into two classes: a) those operations whose pivot arises from the one-point extension by N, and hence cannot.be performed before this operation, b) those operations which can be performed before the .one-point extension by N. A case by case inspection shows that there is no operation of the first kind. LEMMA.In the sequence of admissible operations that leads .~om A’ = A~-[Mi, Ki]~=i[N] to A there is no operation whose pivot arises from the one-point extension by N. Proof. Weanalyze only the casein which N is an (ad 2)~pivot. The other cases are treated, similarly. Since the colevel of N is 2~ the parametert of the operation (ad 2) is 1,. and there is an arrow from N to a simple module S .lying on the mouth of the coil ~ that contains N in rA,. Any operation whose pivot arises from the one-point extension by N .is either of type (ad 1) with .pivot S, or has pivot a module of the form R = (R, HomA-[M,,g,] (N, R), 1)i where R is an indecomposable A- [Mi, Ki]i=~module lying on the faystarting at N. First, we mayassume that A = A’[S]. Then, keeping the notation introduced in (2.5) and (2.6), A is cotilting-tilting equivalent to Dz = Dx IV], where V is inverse translate in modD of the regular socle of U. Since X~ is non-negative,

172

de la Pefia andTom~

¯.so is Xo2- Let w be the extension vertex of D2 and dim W+ e0 be one of the generators of rod Xol. Since pdimolV = pdimDV= 1, then

(e~,dlm W+ e0)m = -(dim V, dlm W) + (e~,eo) = -dim~Homo(V, W) + dim~Ext~(V,

W) =

and therefore Xoa (e~ + 2(dim IV + e0)) Next, we may assume that A -- A’[R]. Then A is cotilting-tilting equivalent to Da = D~ [’~, where ]7 = (~ Homo(U, V), I) and V is an indecomposable D-module lying on the.ray starting at U. Thus Xo~ is non, negative. Let w and dim W+ eo be asabove. Since pdimD~ <2 and ~o~ is a regular D-module, (ew, dim W + e0)Da = "(dim ~,dim W).+ (e~,e0)

=

.dim~Homo, (~, W)dim~Ext~ (~ , W) - d im~Ext~l (V,W) - 1=--

¯ = -dim~Ext~(~7, and therefore:

W).- 1 <

Xo~(e~, + 2(d|m W+ e0))

[]

2.81 Thus¯ far we have ¯shown that the sequence of admissible operations that leads from ~,he bra~_ched-cr~_~,icalalgebra A-IMp,K~][=~~o A co_-_s!s~,s of operations of ~,ype (ad 1) with parameter 0 and Operations of types (ad 2)and (ad 3), all of which as pivot¯ an A-[Mi, K~] [=~-module of colevel 2 (and hence the operations commute between them). Weare nowable to prove the following¯ result which completes our classification. PROPOSITION. Let A be acoil enlargement of a critical algebra. C not of type ~n. IrA is not branched-critical, then Xa is non-negative if and only if i) C is of type ~,~, m > 4, ¯ ii) t(A) = (n - 2, 2,2) with n > m, and iii) A is cotilting-tilting,

equivalent to an iterated 2-tubular extension.

Proof. :Let A’ = A-[Mi, Ki]~=i[N]be as in the discussions above. Ifxa is non. negative, we knowthat A’ is cotilting equivalent to B" = B[Xi, Ki]~=x[Y], where B is a critical algebra with tubular type the coextension type of A- over G. In turn, B" is ~ilting equivalent to the 2-tubulax extension D~ = D[U], where D is a ¯ critical algebra with tubular type the extension type of B~ = B[Xi,K~]I=a over B. ¯ Hence, the extension type of B~ over B (which is the type of the branched-critical algebra A- [Mi, Ki]~=t) is (P- 2, 2, 2)for some > 4. Then i) fol lows from thefact that C is not of type A~, ii) follows from the considerations above the proposition, and iii) from the same considerations and repeated application of [17]. Conversely, we. mayassume that A is cotilting-ti!ting equivalent to one of the following iterated 2-tubular extensions whose Euler forms are easily seen to be non-negative.

173

Coil Algebras that Are Derived-Tame

XD(x)

1.

1

1

+~(2~, - ~s)2 +... + ~Cx.-, - ~)2+ ~(~... 2~.+1 _.~,.:)2 1 1 +~(~.+~"x~+2- ~2)2 + ~(~.+2- ~.+a)2 +". 2 + 1~(X.-2+~(Zn-a -- Zn-2) +~(Zn-I

zn-1 - Xw~)2

_x._x.+~_x~ +[(x._x )2 ~.+~.2

2.9. Combinin~ (2.3)~d (2.8)

we ob~n:

THEOI~M.Let A be a coil enlargement Of a critical algebra. C not of type ~. Then XA is non-negative if and only if A is branched-criticaland t(A) .is Dynkin or Euclidean, or A is cotilting-tilting equivalent to an iterated 2-tubular extension and t(A) = (n- 2, 2, 2) for some n >_ 4. 3

ALGEBRAS

DEI~IVED

EQUIVALENT

TO COIL

ALGEBRAS

3.1. This section contains the main result of our work. Westart by proving that the coil algebras that appear in the Classification given in (2.9) are derived-tame. PROPOSITION. For a branched-critical algebra A, the following are equivalent: i) X~ is non-negative. ii) t(A) is Dynkin or Euclidean. iii) .A is derived-tame.

174

de la Pefia andTom~

ProofTheequivalence between i) andii)wasestablished in (2.3). Therewe showed thatift(A)is Dynkinor Euclidean, thenA is Cotilting equivalent to a branch . t ~ extension B’ = B[X,,KI]i=I of a critical algebra B, and the extension type of B ¯ over B is t(A). Therefore B~ is either a domestic tubular or a tubular algebra and hence A, which.is derived equivalent to B’, is derived-tame. Thus we have shown that i) implies,iii). .Finally, assume that t(A) is neither Dynkinnor Euclidean. Weshall prove that A is not derived, tame. Wemay assume that both A- and A+ are non-trivial and, as in (1.8)~ we may write A A-[MI, Ki ]~=l., where th e M~are bothray a nd c oray A--modules and the K~.are branches. Moreover, we may assume that A- is tame, for otherwise A is not derived-tame. Therefore A- is either a domestic tubular or a tubular algebra. If A- is a tubular algebra, then indA’=7)0VToV

V ~vT"ooV~oo

and the extension modules Mi belong, to the.coinserted tubular family 7"00. The ¯ algebra [M~]A-is a full subcategory of the repetitive category -~. By [13] (see also [1]), [Mi]A-" is of wild type and hence A isnot derived-tame. ¯ If A- is domestic, then as in (2.2), A is tilting equivalent Co a branch extension B’ = .B[Xi, K~]i=t of a critical algebra B, andthe type t(B’) = t(A). By [!5], B’ is of wild type and.consequently neither B nor A are derived-tame. [] 3.2. LEMMA. IrA is an iterated 2-tubular extension then A is derived-tame. . Proof As in (2.8), we may:assumethat A has the following shape (the other case ¯ being similar)

A:

The algebra S~,~S[_i...S~S~S~A obtained by reflections equivalent to the algebra BI below

from A is tilting

with commutative squares. The algebra S~ S~_l... S~+~S~S~i Bx .is. alent to the algebra Bl below

tilting

equiv-

175

Coil Algebrasthat Are Derived-Tame

with commutative squares. Repeating the procedure, we get that A is derived equivalent to an algebra D of the shape ¯

_..o ,,----.-~’

o--

¯ "

-,,,.,,,,,,/ \___,...___,,z ,,,.___....___..,,

with all squares .commutative. Clearly,. every convex subcategory of ]~ has again the above shape and:hence/~ is tame as observed in [14]. 3.3~ THEOREM. Let A be a coil enlargement of a critical algebra C not o] type ~,,. Then A is derived-tame if and only if XA is non-negative. Proof: Assumefirst that XAis non-negative. Because of the classification established in section 2, either A is branched-critical with t(A) Dynkinor Euclidean or A is tilting, cotilting equivalent to an iterated 2-tubular extension. By (3.1) and (3.2), ,4 is derived-tame. " Assumenow that,x~ is non-negative. IfA is branched-critical, then A is not derived-tame by. (3..1). Otherwise, we may assume that A = A~[N], ~ where A is branched-critical and derived-tame, and N is either an (ad 2) or (ad 3)-pivot or an (ad 1)-pivot which is not a coray module. Since XA, is non-negative, A~ is cotilting equivalent to a tame branch extension B~ of a critical algebra whosetubular type is the coextension type of A- over C. Moreover, A is. cotilting equivalent to B = B~[Y], where Y is an indecomposable module belonging to the separating tubular family of rood B~ that contains projectives. IfB~ is tubular, then B is wild by [1,13], and so neither B nor A is derived-tame. If B~ is domestic, then A is tilting-cotilting equivalent to a one-point extension D[U] of a critical algebra D by an indecomposable regular module U, and XD~v~is not non’negative. By [13], D[U] is wild and A is not derived-tame. [] REFERENCES [1] I. Assemand A. Skowrofiski. Quadratic forms and iterated.tilted Algebra 128 (1990) 55:85. I. Assem and A. Skowrofiski. On tame repetitive [2]’ (1993) 59-84.

algebras.

algebras. Fund. Math. 142

[3] I. Assemand A. Skowrofiski. Indecomposable modules over multicoil algebras. Math. Scand. 71 (1992) 31-61. [4] I. Assem and A. Skowrofiski. Multicoil algebras. Proc. ICRAVI Canadian Math. Soc. Conference Proc. 14 (1993) 29-67. [51 I. Assem, A. Skowrofiski and B. Tomd. Coil enlargements of algebras. Tsukuba J. Math. 19 No. 2 (1995) 453-479.

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[6]M. Barot.

Representation-finite derived tubular algebras. Arch. Math. 74 (2000)

89-94.

[7]M. Barot

and J.A. de la Pefia. Derived tubular strongly simply connected algebras. To appear in the Proceedings of the Euroconference on ComputerAlgebra. in Representations of groups and algebras:. F, Coelho, J.A. de la Pefia and B. Tom~. Algebras whose Tits form weakly controls the module, category. J. of Algebra 191 (1997) 89-108.

P. Gabriel and A. V. Roiter. Representations of finite-dimensional Algebra VIII, Encyclopaedia of Math. Sc. Vol 73. Springer (1992).

algebras.

[10] D: Happel. Triangulated categories in the Representation Theory of finite mensional algebras. London Math. Soc. LN 119 (1988).

di-

[11] D. Happel and C; M...Pdngel. The derived category of a tubular algebra. In Representation Theory I. Springer LNM1177 (1984) 156-180. [12] D. Hughesand Ji Waschbiisch. Trivial extensions of. tilted algebras. Proc. London Math. Soc. 46 (3) (i983) 347-364. [13] J. A. de la Pefia. On the representation type of one-point extensions of tame concealed algebras. Manuscr. Math. 61 (1988) 183-194. [14] J.A. de la Pefia. Algebras whose derived category is tame. ContemporaryMathematics VoL 229 (1998) 117-127. [15] J.A. de la Pefia and B; Tom~.Iterated tubular algebras. J: Pure Appl. Alg. 64 (1990) 303-314. [16] C.M.Ringei. Tame algebras. Representation Theory I. Proc. ICP~AII (Ottawa, 1979), Springer LNM831 (1980) 137-287. [17] B. Tom~. One-point extensions of algebras with complete preprojective components having non-negatire Tits forms. Communications inAlgebra, 22 (5) (1994) 1531-1549. [.18] T: Wakamatsu.Stable equivalence between universal covers of trivial sions. Tsukuba J. Math. 9 (1985) 299-316.

exten-

One-point extensions of quasitilted ules on stable tubes

algebras by mod-

JOSE ANTONIODE LA PElqIA Instituto de Matem~ticas, UNAM., Ciudad Universitaria, M~xico 04510, D. F, M~xico, E-mail: [email protected]

SONIA E. TREPODEDpto de Matem£ticas. FCEyN, Universidad de Mar del Plata, 7600 Mar del Plata. Pcia. Bs. As., Argentina, E-mail: [email protected]

ABSTRACT In this note we study some necessary and sufficient conditions for the one-point extension of a quasitilted algebra by an indecomposable module to be again quasitilted. In particular we showthat if A is a quasitilted algebra with a convex standard stable tube 7-, then the one point extension by a module of the mouth of 7" is again quasitilted. Wecharacterize the one-point extension of quasitilted algebras by simple modules which are quasitilted. In particular we show that in case that a tilted algebra A has a strong sink, then the one-point extension of a tilted algebra by the simple projective associated with this strong sink is always quasitilted. Finally we study the double extension of a tame hereditary algebra by indecomposable modules.

Quasitilted algebras were introduced in [4] as a natural generalization of two classes of algebras extensively studied in the last years: the tilted algebras and the canonical algebras. Weare going to consider only finite dimensional algebras over an algebraically closed field k. Let 74 be a locally finite hereditary k-category, that is, for every couple of objects X and Y in 7/the spaces Homn(X,Y) and Ext,(X, are finite dimensional vector spaces and Ext2 vanishes on 74. An object T in 74 is a tilting object if Ext~(T,T) = 0 and if Homn(T,Z) = 0 Ext~(T,X)implies that X = 0. For a tilting object T the k-algebra A = Endn(T)°p is said to be a quasitilted algebra. In case 74=modHfor a finite dimensional hereditary k-algebra H, then A is said to be a tilted algebra. In case 74 is derived equivalent to CohX the category of coherent sheaves on a weighted projective line :K, then A is said 177

178

de la Pefia andTrepode

to be quasitilted of canonical type. It has been conjectured that every quasitilted algebra is either tilted or quasitilted of canonical type. The conjecture has received a positive answerin several special cases, see [5], [10], [12], [14]. Werecall that by definition the one-point extension of an algebra A by a Amodule M is the algebra AIM] = ( AO Mk ) with the usual matrix operations. It is knownthat any quasitilted k-algebra is a one-point extension of a quasitilted algebra. Manyauthors have studied the one-point extensions AIM], where A is an indecomposable quasitilted algebra and Mis a decomposable A-module([1], [2]) and also when A is a decomposable quasitilted algebra ([7], [8]). Along this work A will be an indecomposable k-algebra. In [10] we showed that the conjecture can be stated in the following way: CONJECTURE: For every quasitilted algebra A there exists module M such that AIM] is quasitilted.

an indecomposable A-

In this note we study, in certain cases, conditions for the one-point extension of a quasitilted algebra by an indecomposable module to be again quasitilted, using these results and [10] we obtain our main result: THEOREM: Let A be a quasitilted algebra with a convex standard stable tube in ~ A ~l~’~A. Theneither A is a tilted algebra or a quasitilted algebra of canonical type. The note is organized in the following way: In section 1 we discuss some basic facts about quasitilted algebras. In section 2 we showour main theorem. In section 3, as an application we caracterize the one-point extension of quasitilted algebras by simple moduleswhich are again quasitilted. In particular we showthat in case that a tilted algebra A has a strong sink, then the one-point extension of a tilted algebra by the simple projective associated with this strong sink is always quasitilted. Finally we study some cases concerning double extensions. In particular, we discuss the double extension of a tame hereditary algebra by a simple regular module and by a postprojective module, in those cases such extensions are tilted or quasitilted of canonical type. We thank the referee for several suggestions which clarified authors acknowledge finantial support from CONACyT,M~xico. 1

the work. The

PRELIMINARIES

In the following, A will be a finite dimensional indecomposableh-algebra. It has been proven in [4] that A is a quasitilted algebra if and only if A satisfies the following homological properties: gldimA _< 2 and for every indecomposable A-moduleeither idAX _< 1 or pdAX_< 1. Werecall from [4] the definition of two important classes of modules in modA. Wesay that Y is a predecessor of X if there exists a path Y --+ X1 ---~ ¯ .. --~ Xn --~ X of non-zero morphisms which are non-isomorphism between indecomposable modules. Dually we define a successor of a module. Then we have ~:A = { X E indA: for all Y predecessor of X we have that pdAY <_ 1}.

One-PointExtensionsof QuasitiltedAlgebras

179

TIA = { X E indA: for all Y successor of X we have that idAY <_ 1} Let A be/a quasitilted algebra. According to [4], the following holds: i)

T~AU ~ A .~-

indA.

ii) 7~A contains all indecomposable injective indecomposable projective modules. iii) HomA (~A

-’:

/~A, T~A CI/~A) = 0 =

A-modules, and £A contains all

HOmA(7~A f’l £A, £A -

~’~A)

= HomA(T~A,£A -iv) If/~A contains an injective A-module,then A is a tilted algebra. Werecall that a map f : Mr ~ X is said to be decomposable is there exist a number 0 < s _< r and maps fl : Ms ---+ X1 and f2 : Mr-s ~ X2 such that either s < r or X2 ~ 0 and isomorphisms ¢1 and ¢I,~ making the following diagram commutative: r M

-~

Ms r~ sM

X

-~

X1 ¯ X~

REMARK 1. a) f : r ~X is an ind ecomposable map if a nd only if th e A [M]module Z = (k r, X, f) is an indecomposable module. b) Let Z = (k ~, X, f) be an indecomposable A[M]-module with f ~ O. If Z T~A[M], then X ~ add~A. c) Let Z = (kr,X,f) be an indecomposable A[M]-module. If ~ ~. A[M], th en X ~ addl:A. Proof. a): is clear. Wegive the proof of b) and c). b): Suppose X ~ add~A. Then there exists X~ an indecomposable direct summandof X such that X~ ~ ~A- Then there exists Y successor of X~ such that idAY = 2. Weare going to construct a successor Z~ of Z with idA[M]Zt = 2 which is a contradiction with Z ~ addT~A.Since Y is a successor of X~ there exists a path of non-zero, non-isomorphism maps between indecomposable modules of the form:

If ¢I’n’"¢~f ~ 0 then any indecomposable summand Z’ = (ks,Y,f~) of (k r, Y, Cn"" ¢~f) is a successor of Z with idA[M]Z~ = 2. Otherwise the module ~ = 2. Z~ = (0, Y, 0) is a successor of Z with idA[M]Z C): Suppose X q~ addEA then there exists X1 an indecomposable direct summand of X such that X~ ¢ £A. So there exists L a predecessor of X~ such that pdAL = 2. Hence the A[M]-moduleL is also a predecessor of Z, and Z ~ I~A[M]. [] LEMMA 1. Let M be an indecomposable A-module such that AIM] is quasitilted. If there exists a non-zero indecomposable morphismf : Mr ---+ X such that f is a monomorphism and X ¢ addT~A or ExtrA(M, X) ~ O, then pdacokerf < 1.

180

de la Pefia andTrepode

Proof. Observe that the A[M]-moduleZ = (k r, X, f) is indecomposable. Using the same commutative diagrams in [4], III.2.1 and the fact that f is a monomorphism, we get that pdA[M]Z = 2. Since X ¢ add~A or Ext~A(M,X) ~, then Z ¢ ~’~A[M]. So we obtain that AIM] is not quasitilted, a contradictior~ a~gainst our assumption. :, [] 2

MAIN

RESULTS

Werecall the following lemmafrom [4]. Here a(Y) denotes the number of indecomposable direct summandsin the middle term of the almost split sequence ending at Y. LEMMA 2. Let A be an ar~in algebra and M = NI -~ N2...

Nn_I ~=~ Nn = N

be a sequence of irreducible monomorphismsbetween indecomposable non-injective modules and assume a(TrDM) = 1 and a(TrDNi) = 2 for 2 <_ i < n. f : M ~ X be a map where X is indecomposable and not isomorphic to Ni for 1 <_ i < n. Then there is some h : N ~ X such thatht = f where t = un-1 .... ul. Werecall that a component C of the Auslander-Reiten quiver of A is said to be convex if for every pair of non-zero maps between indecomposable modules f :X ~ Yandg:Y--~ ZwithX, ZECwehavethat Y is also in C. Ina similar way than in [4] we get the following lemma: LEMMA 3. Let A be a quasitilted algebra with a convex standard stable tube 7" in £A N T~A. Let M be an indecomposable module in the mouth of T. Suppose that there is a non-zero indecomposable map f : Mr ~ X such that EXtrA(M, X) ~ Then this morphism is of the form f : M --~ X with X an indecomposable module in T. Moreover, pdAcokerf <_ 1. Proof. Let f : Mr ~ X be a non-zero indecomposable map such that Ext~A(M,X) ~ O. Then there exists X1 an indecomposable direct summand of X such that Ext~A(M, XI) ~ O. Hence there exist non-zero maps g : X~ --~ wM and f~ : M~ X~. Since T is convex it follows that X1 E T. Write X = X~ ~ L where we can choose X1 in T such that the length, of X~ is maximal among the indecomposable direct summands Y of X such t:hat ExtrA(M, Y) ~ Writer = (a b) Mr_ 1 --+ X~ ~ L, where we can ~ssume t]hat c d : M $ a : M ~ X1 is not zero. Then a must be a composition

of irreducible monomorphismsbetween indecomposables modules in 7", since 7" is standard. Weknow that ~(TrDM) = 1 and a(TrDM~) = 2, for 2 _< i < n. Since no summandof L is isomorphic to any Mi for i < n, we get from Lemma1 that c : M---+ L factors through a : M---~ X~, i.e there exists e : X1 ---+ L such that

181

One-PointExtensionsof QuasitiltedAlgebras

c= ea. Further b: Mr-1 ---~ X1 can be written asahwhere h : Mr-1 ~ M, since dim~ HomA(M,X1) = 1. Then we get the following commutative diagram: M @ Mr-1 MI~L

r~ 1M ~ M ---4

X1 ~gL

Since f is an indecomposable map, then r = 1 and L = 0. The rest of the proof is easy. [] PROPOSITION 1. Let A be a quasitilted algebra with a convex standard stable tube 7- in ~:ACq~A.Let M be an A-module in the mouth ofT-. Then AIM] is quasitilted. Proof. It is clear that’ gldimA[M] _< 2. Any indecomposable A[M]-modules of the form (0, X, 0) does not have both projective and injective dimension equal by [4], III.2.5. So, let Z = (kr,X,f) be an indecomposable A[M]-module with f # 0. Since M E T~A and f # 0 is an indecomposable map, we have that X ~ addT~A, i.e, idAX < 1. If ExtlA(M,X) = O, then by [4] III.2.2 it follows that idA[M]Z _< 1. However, if EXt~A(M,X) ~ O, by Lemma3 we obtain that r = 1, X is an indecomposable module in T, and pdAcokerf _< 1. Note that f is a monomorphism.Then by [4], III.2.1, it follows that pdA[M]Z_< 1 and so AIM] is quasitilted. [] Werecall a theorem in [10] (see also [7]): THEOREM 1. Let M be an A-module such that AIM] is quasitilted. A is a tilted algebra or a quasitilted algebra of canonical type.

Then either

Using this result we get: THEOREM 2. Let A be a quasitilted with a convex standard stable tube T in £.A ~q T¢A. Then either A is a tilted algebra or a quasitilted algebra of canonical type. Proof. Take M in the mouth of the tube T. By Proposition 1 we have that AIM] is quasitilted. Theorem1 implies that A is a tilted algebra or a quasitilted algebra of canonical type. [] REMARK 2. In the situation above note that if T is sincere the result follows from the main theorem in [13]. 3

APPLICATIONS

Weare going to show nowsome further aplications of the results in section 1. PROPOSITION 2. Let A be a quasitilted algebra, and S be a simple module in CA. Then A[S] is quasitilted if and only if for any non-zero indecomposable map f : S~ ~ X with X ¢ addTiA or ExtlA(S,X) # O, we have pdAcokerf <_

182

de la Pefia andTrepode

Proof. Suppose that A[S] quasitilted. Let f : Sr ~ X be a non-zero indecomposable map with X ~ addTCAor EXtrA(S, X) ~ O. Since f is an indecomposable map and Sr is semisimple then f is a monomorphism.Under the hypothesis, it follows from Lemma1 that pdAcokerf <_ 1. For the converse observe that gldimA[S] _< 2. Any indecomposable A[S]-module of the form (0, X, 0) does not have both projective and injective dimension equal by [4], III.2.5. So, let Z = (k r, X, f) be an indecomposableA[S]-modulewith f ~ 0. Note that f is a monomorphism.In case X ~ addT~A or EXtrA(S, X) 0, then by our assumptions and [4], III.2.1., it follows that pdA[s]Z _< 1. Otherwise idAZ ~_ 1 by [4], III. 2.2. Hence A[S] is quasitilted. [] REMARK 3. a) Let A be a tilted algebra of Dynkin or Euclidean type and S be simple A-modulesuch that A[S] is quasitilted. Then A[S] is tilted or quasitilted of canonical type. b) Let A be a quasitilted not tilted algebra and S be a simple A-modulesuch that A[S] is quasitilted. Then A[S] is quasitilted of canonical type. Proof. a): It follows from our results in [11], prop 2.1, prop 2.2 (see also Proposition 6 below) and prop 2.4, that if A is tilted of Dynkinor Euclidean type then A[S] is tilted or quasitilted of canonical type. b): If A is quasitilted but not tilted, then it follows from [7] Th.13 that A[S] i~ quasitilted of canonical type. [] Let a be a sink of QA, we say that a is a strong sink if does not exist a path of non-isomorphisms between indecomposable modules I--~ X~ --’~ X2 ~ "" "--~ Xn ----~ I,~ with I an injective module and Ia the injective envelope of the simple module S~ associated with the vertex a. Note that our definition of strong sink is not the standard definition, since the injective I could coincide with I~. LEMMA 4. Let A be a quasitilted A is a tilted algebra.

algebra such that there exists a strong sink. Then

Proof. Weare going to show that if a is a strong sink then Ia E/:A, which implies that A is tilted. Suppose that I~ ~ £A, then there exists X a predecessor of I~ such t:hat pdAX -- 2. Hence there exists an indecomposable injective module I such that HomA(I, 7X) ~ and wehave a p at h: I---~rX

.-~ E--~ X --~ ...--~L,,

which contradicts the fact that a is a strong sink. REMARK 4. In case A is a tilted

[]

algebra there is not always a strong sink in Q A.

Proof. Let A = [N]C with C tame concealed and N simple regular such that A is tilted. Wecall Iw the injective associated with the new vertex w, then I~ belongs to a coray tube and there exists a path from Iw to Iw, then w is not a strong sink. []

One-PointExtensionsof Quasitilted Algebras PROPOSITION 3. Let A be a tilted A[S~]is quasitilted.

183

algebra with a strong sink a in QA. Then

Proof. It is clear that gldimA[Sa] _< 2. Moreover the indecomposable A[Sa]modules of the form Z = (0, X, 0) have projective or injective dimension at most Let Z = (k r, X, f) be an indecomposable A[S~]-module, with f ¢ 0. Since f ¢ is an indecomposable map and S~ is semisimple, f is a monomorphismand. kerr is projective. Weshow first that if f : 5~ ----r X is an indecomposable map then pdAX< 1. Suppose pdAX = 2 then there exists XI an indecomposable direct summandof X such that pdAXx = 2. Then HomA(I, TXX) ~ for so me in decomposable in jective module I. Since f is an indecomposable map, it follows that HOmA(Sa,XI) ~ and HoraA(X~, Is) ~ O. Then we have the following path: I---~ rX1 ---+ E----~ XI ---~ I~, contradicting the fact that a is a strong sink. Hence pdAX< 1. Nowwe are showing that pdAcokerf _< 1. Since f is monomorphism then we obtain the following inequality: pdAcokerf < max{pdAX, pdS~ + 1} = 1.

Nowwe are going to point out some results about double extensions of quasitilted algebras by an indecomposable module. PROPOSITION 4. Let A be a quasitilted algebra and M an indecomposable directing module in £.A f~ TiA. Then A[M][M]is tilted. Proof. By [9] it is enough to show that AIM] is a quasitilted algebra and Mis a directing module in £A[M]NT~A[MI. Observe that, in fact, we are showing that AIM]is tilted (using again [9]). By [9], A is tilted. Since Mis a directing modulein I:A f~ 7~A, by [4], III.2.6, it follows that AIM] is quasitilted. Nowwe see that Mis a directing module in £AtMIN7~AIMI. Let P be the new indecomposable projective A[M]-module. Since AIM] is quasitilted we have that P E £A[M]. Since Mis a predecessor of P, we get that M E £.A[M]. Suppose now that Mis not a directing A[M]-module. So, M belongs to a cycle in modAIM]. Then there exists a proper A[M]-moduleN in that cycle (otherwise Mwould not be a directing module in mod A). Hence, there is non-zero morphism from P to N, and therefore P is not a directing A[M]-module. Then [6] implies that Mis not a directing A-module. Finally we show that M ~ ~’~A[M], Since Mis an indecomposable A-module, it is easy to verify that all successor X of Min rood A satisfy ExtrA(M, X) = 0. Hence, since ME 7~A, it follows that each indecomposable A[M]-moduleZ = (k r, X, f) with f ~ 0, or Z = (0, Y,0) with Y be successor of M in mod A, is such that [] idA[M]Z_< 1. In conclusion M ~ T~A[M], and this finishes the proof. As an application of Proposition 1 we prove the following result. PROPOSITION 5. Let A be a tame hereditary algebra, N be a simple regular Amodule and M be a postprojective A-module. Let C = A[N]. Assume the following conditions hold:

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de la Pefia andTrepode

a) There exist non-zero indecomposable maps 9 : Mr ~ X where X is an indecomposable C-module and idcX = 2, and for any such map we: have that ker9 is projective. b) For any non-zero indecomposable map f : Mr --~ X with idcX = 2 we have pdccokerf <_ 1. Then A[N][M]is quasitiited. Proof. First observe that C = A[N] is quasitilted, by Proposition 1. Moreover, it is easy to verify that Malso is a directing C-modulein Lc. So is clear that gl dim C[M] _< 2, and that each indecomposable C[M]-module of the form (0, X, 0) has projective or injective dimension at most 1. Let Z = (k r, X, f) be an indecomposable C[M]-module with f ¢ 0. Since f ~ 0 is an indecomposable map and M is an indecomposable directing C-module, then Ext~(M,X) = O. If idcX _< 1, then by [4]. III.2.2, we have that idc[M]Z _< 1. But, if idcX = 2, then there exists an indecomposable direct summandX’ of X such that idcX ~ = 2. Hence, taking the canonical projection r : X ---+ X’, we have that g = 7rf : Mr ~ X’ is a non-zero indecomposable C-morphism, and from a) it follows that kerg is a projective Cmodule. Then kerr as a submodule of kerg is also projective, l~rom b) it follows that pdccokerf <_ 1, and therefore in that case, by [4], III.2.1, pdc[M]Z_< 1. So A[N][M] is quasitilted. [] Werecall the following result from [11]: PROPOSITION 6. Let A be a representation infinite tilted algebra of euclidean type and let M an indecomposable A-module such that AIM] is quasitilted. The following holds: Q If Mis postprojective or preinjective,

then AIM]is tilted.

ii} If Mbelongs to a semiregular tube, then A[M]is quasitilted of canonical type. REMARK 5. Let A, M and N be as in Proposition 5. (a) If there is no non-zero indecomposable map g : r - ---r X with X in decomposable and idcX = 2 then A[N][M] is quasitilted. (b) Under the assumption of Proposition 5 and Remark 5.1 , A[N][M] is ~ilted or quasitilted of canonical type. Proof. a): The proof is contained in the one of Proposition 5, since any non-zero indecomposable C-morphism f : Mr --~ X we have that idcX < 1. b): If A[N] is quasitilted but not tilted, then by [7] Th.13 we get that A[N][M] is quasitilted of canonical type. If A[N]is tilted, then it is tilted of euclidean type. Since Mis a postprojective A[N]-moduleit follows from Proposition 6 above, that A[N][M] is tilted. [] NOTEADDEDIN PROOF. After the submission of this paper we learned Happel has proved the Conjecture recalled in the Introduction.

that

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185

REFERENCES [1] F. Coelho, I. Martins and J. A. de la Pefia. Quasitilted one-point extensions L Proc. of AMS,to appear. [2] F. Coelho, I. Martins and J. A. de la Pefia. Quasitilted one-point extensions II. Journal of Algebra 227 (2000) 582-594. [3] D. Happel. Quasitilted Algebras. Canadian Math. Soc. Conf. Proc. 23 (1998), 55-82.

[4]D. Happel, I.

Reiten, and S. Smaloe. Tilting in abelian categories and quasitilted algebras. Memoirs AMS,575 (1996).

[5]_D. Happel and I. Reiten. Hereditary categories with tilting 232 no. 3, (1999), 559-588. [6] D. Happel and C. Ringel. Directing projectives (1993), 237-246.

objects..

Math.Z.

modules. Arch. Math., Vol. 60

[7] D. Happel and H. Slungard. On quasitilted algebras which are one-point extensions of hereditary algebras. Coll. Math., vol 81, no. 1, (1999) 141-152. [8] D. Happel and H. Slungard. One-point extensions of hereditary algebras. Canadian Math. Soc. Proc. 24 (1998), 285-291. [9] J. A. de la Pefia and I. Reiten. Trisection of modulescategories. In preparation. [10] J. A. de la Pefia and S. Trepode. Algebras with quasitilted one-point extensions. Preprint (1999). [11] J. A. de la Pefia and S. Trepode. Quasitilted one-point extensions of tilted algebras. Preprint (1999). [12] H. Lenzing. Hereditary noetherian categories with tilting 125(1997), 1893-1901.

complex. Proc. AMS

[13] I. Reiten and A. Skowrofiski. Sincere stable tubes. To appear in J. Algebra. [14] A. Skowrofiski. Tamequasitilted algebras. J. Algebra 203 (1998), 470-490.

Combinatorial Partial Brauer Star Algebras

Tilting

Complexes for the

MARYSCHAPSDepartment of Mathematics and Computer Science, versity, 52900, Ramat-Gan,Israel. 1 Department of Mathematics, EVELYNE ZAKAY-ILLOUZ Jordan Valley, Israel

Bar-Ilan

Uni-

Jordan Valley College,

ABSTRACT In modular representation theory, the Brauer star algebras play a very special role, being the "local" blocks corresponding under the Brauer correspondence to the "global" blocks of a group ring of cyclic defect group. Every block of cyclic defect group can be obtained by tilting from a Brauer star algebra. Among the various tilting complexesfor the Brauer star algeras, we distinguish a subclass, which we will call two-restricted tilting complexes, of a special combinatorial character. In this paper we determine necessary conditions for a two-restricted complex to be a partial tilting complexfor the Brauer star algebra. 1

INTRODUCTION

This paper is intended to be the first of three based on the doctoral dissertation of the second author. Since we cannot refer most readers to the original dissertation, which is in Hebrew,we intend to give a fairly detailed presentation in the published version. The general subject is tilting complexes and deformations for blocks of cyclic defect group. The work was motivated by two problems: First, to find a DonaldFlanigan deformation of blocks of cyclic defect which would be more natural than that given in [$2]. Second, to try to understand the combinatorics of the Brauer tree in terms of the tilting theory. Rickard had already shown that a connection existed in [R2], where he constructed a tilting complex from a block of the "global" Brauer tree to the Brauer star. Wechose to work in the opposite direction; to find 1Workdonefor a doctoral dissertation at Bar-Ilan University,and partially supportedby the Bar-Ilan ResearchAuthority 187

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SchapsandZakay-Illouz

tilting complexes for the Brauer star algebra from which one can "read off" the global Brauer tree directly. Since not every tree is the Brauer tree of a block of a group ring, our results are not restricted to blocks of a group ring, and in fact, we will work with the larger class of algebras called Brauer tree algebras, over fields of arbitrary characteristic. However, the main application is to modular representation theory of finite groups. This paper corresponds to Chapters III and IV of the thesis, classifying two-restricted partial tilting complexes. The second paper, corresponding to Chapter V, will demonstrate a one-to-one correspondence between Brauer trees with an additional structure, called a pointing, and equivalence classes of two-restricted tilting complexes modulo cyclic rotation of the Brauer star. The third paper, Chapter VI of the thesis, will give a "natural" deformation based on tilting theory. Let e, m be natural numbers, with e > 1. Let K be any field containing a primitive e-th root of unity ~. Let ~-= em ÷ 1. Then the cyclic group Ce of order e acts on the truncated polynomial ring A = K[x] /x ~ by letting the generator d of Ce send x to ~x. Wenow construct the skew group ring b = A[Ce], where the algebra generation according to the relation

x of A and the group generator

d of Ce commute

xd = ~dx or

d-lxd = ~x. The most important case of this construction occurs when K is a field of characteristic p, e is a divisor ofp - 1, and m = (pC _ 1) /e. In this case b -% g[Cpo >~C~]. The generator d of Ce induces an action on Cpc = (a) given by r, d-lad =: a where r is an integer which is a primitive root of unity modulo e. The isomorphism between this case and the previous, more general case, is established by setting 1 ~-~ X-~-

e

Z r-JarJ" j=o

Then d-~xd = rx, as desired. This algebra b has been intensively studied in the context of modular representation theory. For a discussion of the quiver of b in a more general context, see [MS] or [SSS]. Alternatively, one can see b as a special case of the Brauer tree algebra, which we will discuss below. Returning to the situation where K has an arbitrary characteristic, we set A = K[x]/x ~ as before, and define idempotents of b = A[C~] as e--1

fi = ~ ~-iJd j, i = 1, ..., j=O

e.

Partial Tilting Complexes

189

This is a complete set of primitive orthogonal idempotents of b, and thus the indecomposable projectives are 1~i =bfi, i = 1,...,e. Wewill denote the identity map on Pi by ida. LEMMA 1.1. The set { fiX$ I s=O,...,f~l; i= l,.. ,e}. is a basis o] eigenvectors, with eigenvalues ~s, for the action of d. It satisfies xS fis= fk x

, where

k =_ i - s

(mode).

Proof: Since { 1,x, ...,xn-1} ’is a basis for A, and {1, ..., de-1} are the elements of Ce, a basis for the skew group algebra A[Ce] is given by {xSdi [ s = 0... ,fi - 1; i = 1,..., e}. Wecan make a change of basis for each s to {x’fi[ s O, , h - 1, i = 1;...,e}. Now,

=$kX s,

where

k=-i-s

(mode).

DEFINITION.The basis {xSfi} will be called the normalized basis. The natural grading of K[x]/(x n) induces a grading of A[Ce] in which the space of elements of degree s is spanned by xsfl, ...,xSf~. The unique b-homomorphismfrom Pk to Pi sending fk to Xsfi is said to be normal homogeneousof degree s. COROLLARY.If k = i, then there are m + 1 normal homogeneous maps q : Pk --~ Pi, fors = O,e, 2e,...,me. Ilk 7~ i, and {i-k}e is the residue mode, then there are m normal homogeneousmaps, for s = {i - k}e + ee,g = O, ...,m - 1. 2

BRAUER

TREES

AND TILTING

MODULES

DEFINITION.Let T be any tree, together with a cyclic orientation on the edges adjacent to each vertex. Assumethat T has e edges, and that there is one designated vertex, called the exceptional vertex which is assigned a multiplicity m _> 1. T is called a Brauer tree of type (e, m). To any field k and any Brauer tree T, we can associate a k-algebra AT called the Brauer tree algebra. Wedescribe the algebra AT in terms of composition series. For each edge frl and each vertex v of the edge which is not a non-exceptional terminal vertex, we define the standard circuit at v denoted by S(v, frl) to be

190

Schapsand Zakay-Illouz

the sequence of edges in the cyclic ordering around v either one time or, if v is exceptional, m times. Wedescribe the composition series of the projective modules Pi, i := 1,..., e. The top and socle of Pi are the same simple module Si. Rad(Pi) / Soc(Pi) is direct sum of either one or two uniserial components, whose composition factors are the simples corresponding to the edges encountered in a standard circuit of the vertices at either end of the edge. If one of those vertices is a non-exceptional terminal vertex, then the corresponding factor is empty. A detailed description in the case of blocks of a group ring is given in [A]. In [R2], Rickard proved that any two blocks of a group ring with the same Brauer tree are derived equivalent. By the various reductions in [R1], this means that there is a complex Q of projective A-modules such that B = EndA(Q), where Q" is a tilting complex according to the following definition. The notation Q’[n] denotes shifting the complexn places to the left. DEFINITION.Let R be a Noetherian ring. A bounded complex of finitely-generated projective R-modulesis called a tilting complex if (1) HomD~(R)(Q’ , Q’[n]) = 0 whenever n ¢ 0. (2) For any indecomposable projective P, define the stalk complex to be the complex P’ : 0-~ P-~ 0. Then every such P’ is in the triangulated category generated by the direct summandsof direct sums of Q’. Q" satisfying (1) and (2) is called ti lting co mplex for R. If it sat isfies (1) , it called a partial tilting complex. Rickard proved his result [R2] on derived equivalence of blocks of a group ring by showing that every block of a group ring which is a Brauer tree algebra can be tilted to an algebra Morita equivalent to the Brauer tree algebra of a star, i.e., to an algebra like the b defined above. Membrillo proved a similar result with. regard to the generalized Brauer tree algebras, in which each vertex is allowed arbitrary multiplicity [M]. Our interest in the subject of Brauer tree algebras came from deformation theory. Wewanted to transfer the very easily described deformation of b given in [$2] to an arbitrary Brauer star algebra B, using the Rickard result [R3] on deformations of tilting complexes. Where Rickard and Membrillo were passing from B to b, we need to go from b to B. Furthermore, we wanted to get deformations which would be "homogeneous" with regard to the deformation parameters in a sense which will be made explicit in the sequel [SZ2]. Thus we restricted ourselves to tilting complexes of a particularly combinatorial form. DEFINITION. A complex of projectives over b = A[C~] will be called two-restricted if it is a direct sumof shifts of complexesinvolving no more than two indecomposable projectives of the form Sr

:

Trt :

0-+Pr

~0

O ~ P~ -~ Pt --> O r # t.

Partial Tilting Complexes

191

where the map h is homogeneous of minimal degree. The complexes Sr[n], Train] will be called elementary. NOTE.One can show that any indecomposable partial tilting complex containing at most two terms is elementary, and Theorem2 below will establish that the elementary complexes are all indecomposable. Werecall somegeneralities about tilting complexesfrom [R.1]. If A has e simple modules, and Q’ is a tilting complex over A, then Q" is homotopic to a complex which is the direct sum of indecomposable complexes from exactly e distinct isomorphismclasses. A partial tilting complex has no more than e distinct elementary complexes. If we take a tilting complex Q" over A, and replace ~t by a complex Q with the same distinct indecomposables but a different number of copies of each, we get an algebra B~ = EndA(Q") which is Morita equivalent to B = EndA(Q’). Wewill generally work with tilting complexes in which there is a single copy of each indecomposable, so that the tilted algebra B will be basic. An indecomposable complex can be recognized because its endomorphism ring is local. It will be a corollary of the results in this paper that the complexesSr and Tr~ are indecomposable. Thus if we can find a sum of e complexes Sr and T~ which form a partial tilting complex, it will be a tilting complex. There are manyother indecomposable complexes from which tilting complexes could be constructed; for example Pr ~ P~ ~ P~ -~ ... ~ P~, where the map s takes the top of P~ to the socle of P~. Weconcentrate on the complexes S~ and Tr~ because they are particularly "combinatorial" in the following sense: they provide a minimal encoding of the information contained in an edge of the Brauer graph. NOTATION. The set of all two-restricted partial tilting complexes will be denoted by PTC2, and the subset of two-restricted tilting complexes by TC2. In this first paper, we will classify the elements of PTC~, and determine the homomorphismsbetween their indecomposable components. In the second paper, we will show that there is a one-to-one correspondence between elements of TC~ modulo rotation of b and Brauer trees with an additional structure we will call a pointing. This will show that TC2 is adequate to produce all Brauer trees, but also will show that there are generally manydifferent elements of TC~ which will produce a given Brauer tree. Finally, in the third paper, we will derive the existence of manydifferent representations of AT as a graded algebra, all equally "natural", and all giving the same deformation.

3

STATEMENT

OF RESULTS

In order to state the results in full, we need to introduce one more concept, that of a short sequence of indices. DEFINITION.Let b be the Brauer star algebra K[x]/(x~’~+~)[Ce], as above. We

192

Schapsand Zakay-lllouz

let {xSf~}~__m0 be a basis for P~, and define the following maps ~: P~ -+ P~

~i(f~) -- xef~

~z~j : P~-~ P~

~j(fi)

= xk f~,

k ~ j - (mod e) 0 ~ k <

For i ~ j, we will denote ~ by h~, and for i = j by ida. The homomorphism e~ : P~ ~ P~ will be cMled the socle map, for the obvious reason that it maps the top of P~ into its socle (xemf~). DEFINITION. Consider a sequence {r~}~=~ of elements of {1, ..., h ~~ ~ -o,..

o ~r~

e}. Set

~ ~r~hrlrt. ~-

Then the sequence is short if ~ = 0 and long if ~ > O. Wegenerally represent the sequencein the form rl -~ r2 --~ ... -~ rt. EXAMPLES. r -~ r -~ 1 -~ 1 -~

s r 2 3

-+ r -~ s -~ 3 -~ 2

is is is is

long short short long.

Wegather together the following facts about short and long sequences. (1) Cyclic permutationproperty: If the sequence rl --~ r2 --~ ... --~ rt is short (or long), then the cyclic permutation rl -~ r~+l -~ ... -~ rt --~ r~ ~ ... -+ ri-~ is short (or long), respectively. (2) Restriction property: If rl -+ ... -+ re is short, so is every subsequence. (3) Mixing property: If all the ri are distinct, then up to cyclic permutation there is a unique permutation a of indices such that r~(1) -~ ra(2) -~ ... -~ r~(e) short. (4) Refinement property: If rl ~ ... -~ rt is short, and c~1, ..., such that for k = 1, ..., £ - 1

at-~ are sequences

rk --~ ~k _~ rk+l

is short, then rl

2 --~ ... -+ re --~ O/1 --~ r2 "-~ O~

is short. Proof. The proofs of (1), (2_) and (4) are simply repeated applications of the ciability law for the mapshr~r~+~. The proof of (3) is by induction, using the fact that if r --+ s --~ t is short, then r --~ t ~ s is long. []

Partial Tilting Complexes

193

The first theorem will describe which pairs of indecomposable elements of PTC~ can occur together in a partial tilting complex. THEOREM 1. If Q’ is in PTC~, then: (a) All elementary components of Q’ of the form Sr[n] must have their non-zero term in the same degree, which we will assume to be zero. (b) If two elementary components of Q" have a commonterm Pr, it must occur with the same degree in each. (c) If Trs[n] and Sk occur together in Q" then r -~ k -~ s is long. (d) /f Trs[n] and Tt~[n’] occur in Q’, with r,s,t,u following sequences must be short.

all distinct,

then one of the

(1) r -~ s -+ t --~ (2) r -~ t-~ u--~ (3) r --+ s--~ u--~ REMARK. Wedivide the circle into e segments with end points numbered counterclockwise from 1 to e, and let ~-5 and ~’~ be directed counterclockwise segments. Then (1) represents the case where the ones ~’~ and ~-~ are disjoint with compatible orientations, (2) the case where Iu is included in V~, and (3) the case r-~ included in tu. The three excluded cases represent partial intersections

r --+ u -~ s-~ t r -~ u-~ t ~ 8. This approach in terms of segments of a circle originated with [KZ1]. The actual problem treated by KSnig and Zimmermannis different, and the two term sequences represent projective resolution in a hereditary order. Also, in their case all the tilting complexesare two-restricted, which is not the case for Brauer tree algebras. However, the combinatorics which were developed independently, turned out virtually equivalent. The case of Green orders treated by Zimmermannin [KZ2] presumably provides a link, since it uses methods similar to [KZ1] and reduces, modulo the prime, to the case of Brauer tree algebras. The proof of Theorem1 will depend on Propositions 1 - 5 in the next section. In these propositions, we will determine all homogeneouschain maps between indecomposable complexes which are not homotopic to zero. This information about the non-trivial chain maps will also be important in the sequel for constructing the endomorphism ring of a tilting complex Q in TC2, so we will summarize in Theorem 2. However, since the statement of Theorem 2 will also include information about the degree of these chain maps, we first pause to show that this is a well-defined concept for chain maps in TC2.

194

Schaps and Zakay-Illouz

DEFINITION. A chain map ~" between C’ and D’ is called each vertical map is normal homogeneous.

normal homogeneous if-

LEMMA3.1. Any chain map ~" between irreducible combination of normal homogeneous chain maps. Proof: If there is only one non-zero vertical only consider the case

elements

of PTC2 is a linear

arrow, this is obvious.

Thus we need

Set \ i=0 g2 u =

bi e \ i=O

Since ~" is a chain map, the diagram is commutative, (*)

and thus

~2hrs=htu~l.

Define a and f~ by htu

We compute the two compositions

~t = eu

in (*):

htu~l

becauset # u. m--1

Similarly, ~2 hrs = E bi e~+~fir~. Set a_~ = b_~ = O. Thenfor 0 _< k _< j=O m -- 1 + 5ru, we get ak-a = bk-z. Dividing into cases according to cq and B~, we get the following decompositions

of

Partial Tilting Complexes

195

~. = (~, ~) : a=0,~=0: fdh ~ hsu) 5,tam(e~,O) 58ub~(0,e~) i=0

~=0,

~=1:

= ~=1,

+

(0, 7 %.)

~=0: m--1

j=0 ~ = 1, D = 1 : In this case, if r = u, then am-1 = b~_~. Thus, If r C u,

" ~h "’h ’

(ep-~h,,,O) bi n-1 (0, em-lh

If r = u, ~--I

This lemma means that if there is any non-zero homomorphismbetween indecomposable complexes, then there is a normal homogeneous non-zero homomorphism, so it suffices to study the normal homogeneouschain maps, of which there are a finite number. Since we have reduced ourselves to the study of normal homogeneous chain maps, it will also be useful to knowthat if such a mapis homotopic to zero, then homotopy can also be chosen with the same property. LEMMA 3.2. If a normal homogeneous chain map between two indecomposable elements of PTC2 is homotopic to zero, then we may choose the homotopy to be normal homogeneous. Proof: Let C" and D’ be indecomposable elements of PTC2. If there is a homogeneous non-zero chain map g" from C to D which is homotopic to zero with homotopy map T’, we have

0

~

D1

--¢

D2

---¢

D3

~ 0

where T" is a non-zero homotopy between g" and the zero map. Wemay assume that we have removed from T" any homogeneous components whose composition with the relevant horizontal maps is zero. Since each of C’ and D" is of width no greater than two, we have four cases with non-zero homotopy: a square, a triangle with the base at the top or at the

196

SchapsandZakay-ll~ouz

bottom and a parallelogram. In the first three cases it is not hard to show that the mild assumption made on T’, that it not contain any irrelevant homogeneous factors, insures that it is normal homogeneous. Thus we will give the proof only for the more difficult case of a parallelogram. Wehave

40

0

rl

~

Pr

-~

Ps

Pt

~

Pu

~

0

9’2

h,u

9.0

where h,u T~ + Tuhrs = ~. Wehave assumed that g’ is normal homogeneous, so g~=¢u~o~r~ We have

,

O_<~o_<m-l+~,

rn--1 i=O

Choose a, fl such that htu~tr t

_-- a ~

Note that ~r, = 1 =~ a = 0 and ~su = 1 ~ ~ = 0 = e u hru :

ai -u

¯

k i=O

k j=0

Thus, for every k ~ io, 0 < k < m - 1 + 6ru, we have a~-a + b~_~ = 0. Thus we may substitute for T’ a new homotopy T" in which a~_~ = b~_~ = 0 for k ~ io. For k = io, we get aia-a

+ bio-~

= 1.

io - a and io - ~ cannot both be -1. Thus if we se~ one of a~0_a or b~o_~ equal to one and the other equal to zero, we ge~ the desired normal homogeneoushomotopy ~’.

DEFINITION.If ~" : C’ -+ D" is a normal homogeneous chain map with one nonzero map, then the degree of ~" is the degree of its non-zero component. If ~ has two non-zero maps, one of which is a map between two identical projectives, the degree of ~’ is the degree of the mapbetween the identical projectives. REMARK. It will be a consequence of the main theorems of this paper that every normal homogeneous chain map between two indecomposable components of an element of PTC~fulfills one of these two conditions and thus has a well-defined degree.

Partial Tilting Complexes

197

THEOREM 2. Between two elementary complexes in PTCo., there are chain maps not homotopic to zero in the following cases:

(i) From Si

to Sj there are m + 5ij normal homogeneousmaps. I] we let (j represent the residue of j - i moduloe, then the maps have degree {j - i}e + ke, From T~j to itself the socle map.

there are two normal homogeneousmaps, the identity

and

Between two elementary components with one commonindex, there is a normal homogeneousmap of degree zero in one direction and of degree em in the opposite direction. 4

HOMOMORPHISMS COMPLEXES

BETWEEN

INDECOMPOSABLE

Our eventual aim, summarized in Theorem 1 given in the previous section, is to describe all the partial tilting complexesQ" E PTC~,as direct sums of shifts of the complexes S~ and T~’k of projective b modules. The condition for Q’ to be in PTCo. is that HomD~(b)(Q’,O’[n])=O for n~O. By the standard properties of homomorphisms of direct sums, this is equivalent to showing that for any pair R and/T of elementary indirect summands of Q’, we have HOmD~(b)(R,R’[n D = 0 for n ~ 0. Since our elementary complexesare all of a length less than or equal to two, there are at most three possible relative shifts for which a non-zero projective of R is positioned over a non-zero projective of R. Where a non-zero chain map exists we must decide whether or not it is homotopic to zero. Since, for those partial tilting complexeswhich are actually tilting complexes, we will be interested in calculating the endomorphismring, we want to determine all maps in HOmb(R,R~)[n]) which are not homotopic to zero. This information, which will be summarized in Theorem2, will be determined from the same case-bycase study of homomorphisms between elementary components. Weconsider the following five cases,which will be treated in the subsequent five propositions. Case 1:

S with S.

Case 2:

S with T, and no commonindices.

Case 3:

S with T, and a commonindex.

Case ~:

T with T, and no commonindices.

Case 5:

T with T, and commonindices.

Once we have determined all non-trivial maps, we can combine the results into Theorem1 by using the principle that if there are two or more relative positions which produce non-trivial maps, then the two elementary complexes cannot appear

198

SchapsandZakay-Illlouz

together at all in partial tilting complex. If there is exactly one, then that must be the relative position of the two elementary complexes in any element of PTC~. Finally, if there are no non-trivial maps between the two elementary complexes, then they can appear together without any restrictions on their relative position. Since in classifying all the maps we also show that they have a well-defined degree and determine what it is, we then get Theorem2 more or less for free. We will also establish that the elementary complexesare in fact indecomposable, since their endomorphismrings will be local. PROPOSITION 1. Two complexes, Sr[n] and St[n’], can appear together in a partial tilting complex in PTC2if and only if n = n~. The normal homogeneousmaps from Sr[n] to St[hi are the/ollowing Ckt ~rt , O < k < m- l ÷ ~rt. If { t - r } e is the residue moduloe, the degrees are {t-r}e+ke

,

O
Proof: There are no non-zero chain maps from Sr[n] to St[n’] unless n = n’. If n ¢ n’, then there is a mapafter a shift of n’ - n, so Sr[n] and Sr[n’] cannot occur together. Whenn = n’, the normal homogeneous maps and their degree were calculated in Lemma2.1. [] Henceforward we will assume that Q" has been shifted so that if there i:s any indecomposable St, the non-zero projective module appears in degree zero. PROPOSITION 2. If Sr and Tst[n] have no common index, then they can appear together in a partial tilting complex in PTCzif and only if r -~ s --r t is short. Proof. (¢=) Assumethat r --~ s -+ t is short. Wewill show that every non-zero chain map in either direction is homotopic to zero. Case I. ~" : Sr -~ Tst. If ~1 = eishrs is non-zero, so is hst o ~1 = e~hrt, and thus the following diagram does not commute. S~:

0 ---+

Tst:

0

---4

P,~ Ps

---->

0

-~

Pt

Case 2. g" : S~ ~ Tst[1], If el = e~h~t is non-zero, then T~ = ¢i~hrs is a homotopy

Partial Tilting Complexes

199

Case 3. ~" : Ts~ ~ St. As in Case 2, there is a non-trivial homotopy. Case 4. ~ : Ts~[1] -+ St. As in Case 1, there are no well-defined chain maps. (=~) Nowsuppose that r -~ s -~ t is long, which implies, by the cyclic permutation property, that s ~ t -~ r is long. Weobtain that these are two different non-trivial mapsfrom Sr to Ts~, with different shifts, so Sr and Tst cannot appear together in any relative position.

Case(I). ~’ : S~ ~ T~.

This mapis well defined because hs~ o (esm-lhrs) = 0, and no homotopyis possible. Case(2). ~’ : S~ -~ Ts~[1].

0 0

-~

~

P~ --~

P~

~

0

Pt

~

0

This mapis clearly well defined, and no homotopyis possible because h~ o (~.ishrs) t ,~rt, and this cannot equal hrt. PROPOSITION 3. If Sr and Ts~ or Tst[1] have a common index, $hen they can appear together in a partial tilting complex in PTCaif and only if the commonindex appears in the same degree in each. The normal homogeneous homomorphisms are then the following: (i)

ffr=s

:£’:Sr~Trt

has £~ = e~ o] degree em.

~" : Tr~ ~ Sr has ~ = idr of degree O. (ii) ff r = t : Sr ~ T~[1] has £2 = id~ of degree O. ~" : Ts~[1] ~ S~ has ~2 = e~ of degree era. Proof. Wewill first show that the four maps given in the proposition are welldefined and not homotopic to zero. This will establish that the commonprojectives must be in the same degree in any partial tilting module, since otherwise we would get a mapto a shift. (i) If £ : S~ ~ Trt is normal homogeneous,then £1 must be e~. In order for the diagram to commute,we must have hr~ e~ = 0, which will only be true if i = m. By the shape of the diagram, there is no possible homotopy,so £x = e~ is a non-trivial homomorphism of degree em. ~ : T~ ~ S~ is well-defined for any ~ = e~, but if i > 0, then we have a homotopy T~ = e~ ~-~h~. Wheni = 0, there is no normal homogeneous T such that T o hr~ = idr.

200

SchapsandZakay-II~louz

(ii) is dual to (i). It remains to showthat if the corresponding indices are not positioned one over the other, then there is no non-trivial chain map. (i’) r = s : Consider ~’ : Sr -+ Try[l], with ~1 : P~ -+ P~ equal to eJh~. This has a homotopy T = e~ for any j. Nowtake the opposite direction ~’ : T~[1] ~ S~. If ~ ¢ 0, then fl = e~h~, with j < m. Then, ~ cannot be well-defined because e~h~r o hr~ = ~+1 ~ 0.

.~ 0

N~ --~

~ ~1---- ~Jrhtr P~o

(ii’) r = t : Dual to (i’).

The following is the most difficult of the five propositions: PROPOSITION 4. Iy Trs[n] and T~u[n’] appear together in a partial tilting complex in PTC2,with all four indices distinct, then at least one .of the following three sequences of indices must be short: (1) r --~ s -+ t -~ u ( 2 ) r -~ t -~ u -~ (3)

r~s~u~t

This implies, be short.

by the m~ing prope~y in Lemma3.1, that none of the following can

(4)

r~t~s~u

(5)

r~u~s~t

(6) r ~ u ~ $ ~ Proof. Wewill first show that in C~es (1), (2) and (3), there are no non-trivial homomorphisms,and thus these elementary complexes can appear together in some Q" ~ PTC~with any desired shift. Since interchanging r, s with t, u produces the same three c~es, it suffices to show that there are no non-trivial homomorphisms from Trs to T~u[n], for n = -1,0, 1. Case (1). r -+ s -~ t ~ is short. n = -1 There is no well-defined map, as in case (i ~) of Proposition 3. The first square does not commute.

0

~

P~

--+

P~

Partial Tilting Complexes

201

n=0

0

---+

Pt

~

P~

~

This is Well defined if i - j, but then T = ¢~hst is a homotopy. n=l

0

~

-- ¢t hrt,

T2 = ¢~h~ is a homotopy.

Case (2). r ~ t -~ u --+ is short. n = -1 This is not well defined, as in Case 1. n=0

This is well defined only if i= j + 1, but in that case n = 1 There is a homotopy, as in n = 1 for Case 1. Case (3). r ~ s ~ n ~ is short. If we rewrite thi s as t ~ r ~ s -+u, it is cle arl y dual to Case 2, where r is interchanged with t, and s with n. Wenow show that the Cases (4), (5), and (6) cannot occur Q’ e PTC2, by exhibiting, in each case, two different shifts which produce non-trivial homomorphisms. Once again the cases are symmetric under the interchange of T~8 with Tt,,. Wewill in general leave to the reader the verification that the mapsare not homomorphic to zero.

Pu

~

0

202

Schapsand Zakay-Illouz

The second map is well defined because hs~, o hr, = hr,~ = ht~, o h,t. There is no homotopy because hs, o hrs ~ Crhrt. Case (5).

t -+ r -+ u --~ s is short.

0

~

0 --~

Pr --~

P.

Pt

Pu

0

-~

~

The second homomorphism is well defined because hs~, o hrs = ¢~,hr~, = h,~, o hrt. r ~ u -~ t --+ s is short. 0---~

P,-~ P~ --~ 0

0---~

P, ~ P~ ~

The final proposition is parallel to Proposition 3, dealing with the case of commonindices. PROPOSITION 5. Tr8 can occur in a two-restricted partial tilting complex, but not together with any shift of itself or of Tsr. If Trs and Tt~[n] have unique cornmon index, then they can occur together in a two-restricted partial tilting complex if and only if the commonindex occurs in the same degree and 1. If n = -1, so that s = t, then r ~ s ~ u is short. There is one normal homogeneousmap in each direction

Pr --~

P8

Partial Tilting Complexes

203

2. If n = 0, and r = t, then s and u are symmetrical, and we may assume r -+ s -+ u is short. There is one normal homogeneousmapin each direction

P,. ~ P, P,. ~ If n = O ands=u, then r, short.

t are symmetric so we may assume r --+ t -~ u is

3. I] n = 1, we get the case dual to n = -1, and must have t -~ r ~ s short.

There are two normal homogeneousmaps from Trs to itself, the socle maps.

the identity

and

Proof: Since we always have id : Trs -~ Trs, in order to showthat Tr, can occur in Q’ E PTC2, it suffices to show that there are no non-trivial homomorphismswith a shift

The left hand map is never well defined for any i, and the right hand map always has the indicated homotopy. Trs cannot occur together with any shift of T~r because the following two homomorphismsare non-trivial and thus whatever relative position is chosen, there is always a diagonal map

Weare nowprepared to consider a pair of elementary complexes, T~s and Ttu[n], for n -- -1, 0, 1, with one commonindex. The case n = 1 is dual to the case n = -1, and for n = 0, the case r = t is dual to s = u, so we will give the details only for n -- -1 and n -- 0 with s = u. Wewill first show that for the cases given in the proposition there are no non-trivial homomorphisms for other shifts. In addition, in the case n = -1, we must show that if r -~ s -~ u is long, there are two possible shifts which give non-trivial maps.

204 n = -1 We first suppose that r -~ s -+ u is short, indicated in the proposition are non-trivial

SchapsandZakay-lllouz and show that

the maps

The map from top to bottom is well defined because e~hrs = 0 = eumhsu, and no homotopy is possible. The map from top to bottom is automatically well defined, and there is no homotopybecause id~ does not factor through hr8 or hgu. Any homomorphism

has the indicated homotopy and so does not give a non-trivial

map

This is e~ homotopywith i = j if r --~ s --~ u is short. If r --~ s --~ u is long, it is a homotopywhen i = j - 1 for j > 0, but the case j - 0 gives a non-trivial homomorphism. Thus, ifr -~ s -~ u is long, Tr8 and T~[-1] cannot appear together because there is a non-trivial homomorphism with a different shift, and if r -+ s -+ u is short, they can. n = 0 We do the case s = u, and assume rts is short.

In the left hand map there is no homotopybecause id~ does not factor through h~. In the right hand case, if T= ~r’m-~hsr, then T o hes ~ 0. It remains to show that there are no well-defined shifted maps

The left hand map is not well defined because the indicated diagonal is non-zero, and the right hand map has the indicated homotopy since we did not need the condition r ~ t -> s is short. The same is true for maps from T~s to Trs. []

Partial Tilting Complexes 5

PROOF

OF

205

THEOREMS

Nowthat we have Propositions 1 - 5, the proofs of Theorems1 and 2 are simply matter of organization. Proof of Theorem 1 (a) By Proposition 1, all Si[n] have their non-zero term in the same degree. Shifting Q’ if necessary we may assume that this is degree zero. (b) By Proposition 3, if Sr and Ttu[m] have a commonprojective, it occurs in the same degree in both. By Proposition 5, this is also true for Trs[n’] and Ttu[n]. (c) By Proposition 4, if Sr and Ts~[n] occur in Q, then r -+ s -~ t is short. (d) This is Proposition 4. Proof of Theorem Z (i) FromProposition (ii) FromProposition (iii) From Proposition 3 and Proposition In conclusion, having shownthat in a two-restricted partial tilting complex all occurrences of Pt axe in the same degree, we want to summarize the information given by Propositions 1, 3, and 5 in a way that considers all Sr together, and all irreducibles with a commonPr together. As before, we let {t - s}e be the residue modulo e of t - s between 0 and e - 1. THEOREM 3. (Local order theorem) 1. If { Sil , . . . , &. } are a set of& in Q’ E PTC~,such that il ~ i2 -~ ... -+ it is short, then we have homomorphismshit~,+ 1 : Sit ~ Sit+, such that hi~_lit o...

ohit+lit+2 ohiti~+~ --cit

fork = 1,...,r.

}j=l are a set of irreducibles in Q~, we can arrange the indices so that is short. All of the maps

will be identity on Pr, whereas Tst~[1] -~ Trt~ will be em on Pt. If St is in Q’, then S~ can be inserted between Ttt~ and T~v.[1], with identity maps in P~. Proof. (1) This is direct from Proposition 1 and the definition of a short sequence. (2) That each of the indicated maps is the identity on Pr is a consequence Proposition 5, as is the fact that the mapfrom Tttt to Ts~t[1] is e~. The possibility of inserting & comes from Proposition 1. []

206 6

Schapsand Zakay-Illouz APPLICATIONS

Our aim in studying these two-restricted partial tilting complexes was to give a classification to all TC2in combinatorial terms, which will be done in the sequel, [SZ1]. It was, furthermore, important to show that the homomorphismsbetween the irreducible complexes could be taken to be homogeneous,for this allows a study of a class of homogeneousdeformations of the Brauer tree algebras [SZ2] which is muchmore natural than that given in [$2], in the sense that it is local rather than global, and also that it is derived from a tilting complex. Although the "two-restricted" condition means that we have considered only a subclass of all possible tilting complexes,it is a sufficiently large subclass to allow us to reach every possible Brauer tree algebra. In fact, it is too large for it is possible to reach each Brauer tree algebra in manypossible ways. In the next paper we consider all the different possible "foldings" of the tilting complex.It had already been shown by Rouquier [R] that each tree can be reached by a "completely folded" complex with only two non-zero terms. In [RS] we showthat Rickard’s combinatorial tree-tostar complex and the corresponding star-to-tree two-restricted complex give inverse equivalences, and reduce to the Rouquier two-term complex in the completely folded case. As Zimmermannpointed out in a very helpful conversation, if we go over to two-sided tilting complexes[Z] we can study the different possible tilting comple×es which give the same Brauer tree algebra in terms of the group of two-sided selfequivalences of b. This group might be quite large; for the very simple case of the Brauer star algebra of type (2,1) Rouquier and Zimmermann[RoZ] calculated that, modulo shifts, it was the modular group, which is the free group on two generators of order 2 a~d order 3. The entire subject merits further investigation. Wealso hope that it might be possible to use this approach for some ge~teralization of the Brauer star algebra with abelian but not cyclic defect group. This might lead to some generalization of the Brauer tree. One such generalization has already been made by Benson [B], for dihedral defect groups. REFERENCES [A] Alperin J.Local Representation Theory, Cambridge Studies in Advanced Mathematics, vol. 11, 1986 [B] Benson D. Representations and Cohomology Cambridge Studies in Advanced Mathematics, vol. 30, 1991 [H] Happel D. On the derived category of a finite Math. Helvetici, vol 62, 339-389, 1987 [KZ1] KSnig S. and ZimmermannA. Tilting vol 24, 1996, 1893-1913

dimensional algebra Comment.

hereditary

orders Comm.in Algebra

[KZ2] KSnig S. and ZimmermannA. Derived equivsalence vol. 1685, 1998

for group rings LNM

[MS] Mejer C. and Schaps M. Separable deformations of blocks with abelian defect group and of derived equivalent global blocks Canadian Math. Soc. Conf. Proc. vol 18, 1996, 505-518

Partial Tilting Complexes

207

[M] Membrillo F.H. Homological Properties of Finite Dimensional Algebras Ph.D. Thesis, Oxford University, 6-13, 78, 1993 [R1] Rickard J. Morita theory for derived categories J. LondonMath. Soc., vol 39, 2, 436-456, 1989 [R2] Rickard 3. Derived categories and stable equivalence J. of Pure and Applied Algebra, vol 61, 1989, 303-317 [R3] Riclmrd J.Lifting theorems for tilting 1991

complexes J. of Algebra, vol 142,383-393,

[RS] Rickard J. and Schaps M. Folded titlting preprint

complexes for Brauer tree algebras

[P~o] P~ouquier P~. and Zimmermann A. Picard groups for derived modular categories preprint [RoZ] Rouquier R. Fromstable equivalence to Rickard equivalences for blocks with cyclic defect group Proc. of Groups 93, Galway, St. Andrews, London Math. Soc. Lecture Notes Series, vol 212, 512-523 [S1] Schaps M. Deformations of finite Trans. AMS,1988, 843-856

dimensional algebras and their idempotents

[$2] Schaps M. A modular version of Maschke’s theorem for groups with cyclic p-Sylow subgroup J. of Algebra, vol 163, 1994, 623-635 [SSS] Schaps M., Shapira D. and Shlomo D. Quivers of blocks with normal defect group preprint [SZ1] M. Schaps and Zakay-Illouz E. Pointed Brauer trees preprint [SZ2] M. Schaps and Zakay-Illouz algebras preprint

E. Homogeneous deformations of Brauer tree

[ZI1] Zakay-Illouz E. Basis-graphs and deformations for non-abelian groups of order 2j ¯ 3i _< 24, i,j >_ 0 with abelian p-Sylow subgroups Master’s thesis, Bar-Ilan Univesity, 1993 [ZI2] Zakay-Illouz E. The Green Correspondence between Separable Deformations Ph.D. dissertation, Bar-Ilan University, 1999 [Z] Zimmermann,A. A two-sided tilting complex for Green orders and Brauer tree algebras J. of Alg., vol 187, 1997, no. 2,446-473

Almost split sequences in categories tions of Quivers II

of Representa-

SVERREO. SMALO Institutt for matematiske fag, NTNU,7491 Trondheim, Norway, email: [email protected]

ABSTRACT Let k be a field, Q a connected quiver and fd(Q, k) be the category of finite dimensional representations of Q over k. In this note it is proved that for a quiver Q the subcategory fd0(Q, k) of fd(Q, k) consisting of the representations having composition factors from the discrete simples, has almost split sequences if and only if Q is either path finite (without oriented cycles if Q is finite) or consists entirely of a single oriented cycle or is a subquiver of A~with linear orientation. 1 INTRODUCTION In this note a quiver Q is an oriented graph where only a finite number of arrows are adjacent to each vertex. A quiver is called path finite if there is no oriented path of infinite length. Let k be a field and let fd(Q, k) denote the category of finite dimensional representations of Q over k. For each vertex p in Q, the representation given by a one-dimensional k-space at p, all other spaces being zero and all maps associated with the arrows being zero, is a simple representation. It will be denoted by Sp and called the discrete simple attached to the vertex p. Finally, let fd0(k, Q) denote the full subcategory of fd(k, Q) consisting of representations having composition factors only amongthe discrete simples attached to vertices of Q. The aim of this .note is to prove the following result. THEOREM 1 Let Q be a connected quiver and k be a field. (a) In case Q is finite, fdo(Q, k) has almost split sequences if and only if one of the following two properties are satisfied. (i) Q contains no oriented cycles 209

210

Smal~ (ii) Q is -~n for some n with cyclic orientation.

(b} In case Q is infinite, fd0(Q, k) has almost split sequences if and ~,nly if one of the following two properties are satisfied. (i) Q is path finite (iO Q is either A~ with linear orientation or Ao~ with linear orientation. Let C be an abelian Krull-Schmidt-category, i.e. the indecomposable objects in C have local endomorphismrings and each object in C can be decomposed as a finite direct sum of indecomposables, and then in a unique way up to isomorphism by the Krull-Schmidt theorem. The category C is said to have right almost split morphisms if for each indecomposable object X in C, there is a morphism f : Y --r X in ~ such that the c0kernel, Coker(,f), of the natural morphism Homc(,f) as a functor takes the value 0 for all indecomposable objects Y not isomorphic to X and the value Homc(X’,X)/rad(X’,X) for the indecomposable objects X’ isomorphic to X. Here rad(X’, X) is the set of nonisomorphisms from X’ to X which is a subI, X) since the endomorphism ring of X is local. The morphisms group of Homc(X are sent to their residues. A morphismf : Y --+ X satisfying the above property is called a right almost split morphismin d. Dually, the category C is said to have left almost split morphismsif for each indecomposable object X in C, there is a morphism f : X ~ Y in C such that the cokernel, Coker(f, ), of the natural morphism Homc(f, ) satisfies the condition as above, i.e. Coker(f, ) applied to an indecomposable object ~ i n C is Homc(X, X’)/rad(X, X’) where rad(X, I) i s t he s ubgroup o f n onisomorphisms from X to X*. A morphism f : X -~ Y satisfying the above property is called a left almost split morphismin C. The category ~7 is said to have almost split morphismsif it has both right and left almost split morphisms.It is said to have almost split sequences if in addition, for each indecomposable nonprojective module X in C there exists an exact sequence 0 -~ Z --+ Y --r X ~ 0 with Z indecomposable and where Z --> Y is left ahnost split in C and where Y -> X is right almost split in C; and for each indecomposable noninjective module Z in C there exists an exact sequence 0 ~ Z ~ Y --r X -~ 0 with X indecomposable and where Z -r Y is left almost split in ~ and where Y -+ X is right almost split in C. For background on the representation theory of artin algebras including finite dimensional algebras, and on the theory of representations of quivers, the reader is referred to the book [ARS]. A celebrated result of Auslander and Reiten is the following theorem (see JARS]). THEOREM 2 If A is an artin algebra, then the category of finitely generated left A-modules has both left and right almost split morphisms as well as almost split sequences. THE PROOF

OF THEOREM

1

Nowto the proof of the result of this note.

AlmostSplit Sequencesin Categoriesof Representations of Quivers

211

Let us start by giving the arguments that if (i) or (ii) in (a) is satisfied, fdo(Q, k) has almost split sequences. If (a) (i) is satisfied, then fdo(Q, k) is fd(Q, k), which again is equivalent to the category of finite dimensional modulesover the path algebra kQ. Since the quiver is finite and has no oriented cycles, the path algebra kQ is finite dimensional and therefore the category of finite dimensional modules and the category of finitely generated modules coincide, and hence by the result of Auslander and Reiten quoted above, the category fd0(Q, k) has almost split sequences. If (a) (ii) is satisfied, then Q = ~n for some n with cyclic orientation, fdo(Q, k) is a category with n + 1 simple objects and where all indecomposable objects are uniserial. The indecomposable objects are then given by their socle and their length, so numberingthe vertices of Q by O, 1, ..., n one obtains a natural indexing of the indecomposableobjects by {0, 1, ..., n} × N. With this indexing the indecomposable objects fit together into exact sequences 0 --~ (j, m) --~ (j, m+ 11 ((j - 1) , m - 1)-- ~ ((j - 1), where (s) denotes the residue modulon + 1, and the module(t, O) is zero. It is to see that these sequences are almost split (see [S] for more details on this). This showsthat if either (i) or (ii) is satisfied, then fd0(Q, k) has right and left split morphismsas well as almost split sequences. To prove the converse in (a) we prove the following somewhatstronger result. PROPOSITION 3 If Q is a finite connected quiver and contains a subquiver ~il with cyclic orientation as a proper subquiver, then fd0(Q, k) has neither left nor right almost split morphisms. Proof: Assumethat Q contains an oriented cycle and let ,~n be a fixed minimal subquiver of Q with cyclic orientation. Since Q is connected there is at least one additional arrow a starting or ~ending at a vertex q of ~n. Let ~ be the subquiver of Q consisting of the arrows in An and a together with their initial and end vertices. By duality, it is enoughto consider the case whenthe arrow a ends in a vertex q of.~. Let p be the start of a. Assumethat X -~ Sp is a right almost split morphism in fd0(Q, k). Now,take the subcategory of fd0_(Q, k) consisting of all objects where the maps corresponding to the arrows not in Q are zero. This subcategory is closed with respect to sub-objects and quotient-objects in fdo (Q, k) and it is equivalent the category fdo((~, k). The trace of this subcategory, ~-X, in X will induce a right almost split morphismrX -~ Sp in fd0(~, k). So in order to prove that there is right almost split morphismX -~ Sp in fdo(Q, k), it is enough to prove that there is no right almost split morphism Y ~ Sp in fdo(~, k). Therefore we can without loss of generality, from now on assume that Q = (~. Observethat fdo (Q, k) is the full subcategory of fd(Q, k) consisting of the objects which, when viewed as modules over the path algebra, are annihilated by some power of the ideal generated by the arrows. So for the right almost split morphismX -~ S~, there is some power, say t > 0, of the ideal I generated by the arrows in kQ that annihilates X. Then one knows by the Auslander-Reiten formulas that the minimal right almost split morphism X --4 Sp fits into an almost split sequence 0 ~ DTr~:QH., Sp ~ X -~ S~ --> 0

212

Smal~

for all m_> t. The projective resolution of S~ over kQ/Im looks like

where P8 -" 0 if p is not in An and, by the minimality of ~n in Q, s = q is the successor of p in .~,~ if p is in .2,n. Nowthe representation corresponding to the module TrkQ/i,,. Sp has a projective presentation

and its dimension is a function of m, which will tend to oo as rn grows. This gives a contradiction to the existence of a right almost split morphismX --~ To prove that there do not exist left almost split morphisms in fd0(Q, k), will use the simple representation Sq and show that there is no left almost split morphismstarting in Sq. Wecan reduce to the situation where Q = ~ by using that if Sq ~ X is a left almost split morphismin fd0(Q, k), then Sq .-~ X/(rejfdo(~,k ) X) is a left almost split morphismin fdo((~, k), where rej¢ t X, the reject of A in X, is the intersection of all kernels of all morphismsfrom the representation X to any representation in the subcategory ,4. Assumethat Sq ~ X is a left almost split morphism in fdo (Q, k). Then ~.here is a power, say t > 0 of the ideal I generated by the arrows in kQ that annihilates X, and we have by the Auslander-Reiten formulas an almost split sequence 0 ~ Sq -+ X --~ Tr DkQ/Im ,~q -~, 0 for all m _> t. Nowthe projective presentation of DSqover (kQ/I’~) °p is P~ LI P~ -~ P~ -e DSv ~0 where s is the immediate predecessor of q in ~n. Therefore the projective presentation of Tr DSq is

Pq-~ P~,I_I P8~ Tr DkQ/,~ Sq -~ 0 which shows that the dimension of Tr D~Q.H,, Sq tends to oo as m grows. This gives the desired contradiction and completes the proof of the proposition as well as the proof of part (a) of the theorem. The proof of part (b) of the theorem can be completed by using the following facts: (i) if Q is path finite, then fdo(Q, k) has enough projective and injective objects, and the construction of the dual of the transpose works locally to produce almost split sequences as in the situation with a finite quiver without oriented cycles. (ii) If Q is either Aoowith linear orientation or A~with linear orientation, then all indecomposable modules are uniserial and fit into almost split sequences. Finally, if Q contains a subquiver of the form Amand in addition there is a vertex p in this subquiver with two arrows starting or two arrows ending in p, then there is either an indecomposable representation (V, f) where no left almost split morphismstarts or an indecomposable representation (V, f). where no right almost split morphism ends. Note that in this last situation one may end up with existence of left almost split morphismsstarting in all indecomposablerepresentations or right almost split morphisms ending in all indecomposable representation, but not both.

AlmostSplit Sequencesin Categoriesof Representations of Quivers

213

REFERENCES JARS] Auslander, M., Reiten, I. and Smal0, S. O. Representation Theory of Artin Algebras, CambridgeUniversity Press, 1995. [S] Small, S. O. Almost split sequences in Categories of Representations of Quivers, To appear in Proc. Amer. Math. soc..

Cotilting

objects and dualities

ROBERT WISBAUER Mathematical Institute Germany

of the University,

40225 Diisseldorf,

ABSTRACT Tilting modules generalize projective generators and may be characterized either by weakenedgenerating and projectivity conditions or else by equivalences they define between certain subcategories. Dually cotilting modules generalize injective cogenerators and there are again principally two ways to describe them: first by weakened cogenerating and injectivity conditions, and second by dualities they induce between suitable subcategories. In this p~per we begin with several characterizations related to the first point of view, and it turns out that for properties of the second type certain finiteness conditions are needed - similar to the situation for Morita dualities for rings. INTRODUCTION Dualizing tilting conditions

modules, cotilting

modules Q in R-Modare defined in [7] by the

(1) inj dim (~Q) <_ (2) Ext~(Qh, Q) = 0, for any set A, (3) for all

N e R-Mod, Hom~(N,Q)= 0 = Ext~(N,Q) implies

In Section 1 various injectivity and cogener~ting conditions are introduced for objects (in Grothendieck categories), which result from dualizing notions of interest in the study of (self-) tilting objects. Self-tilting modules Mare those which are tilting in the c~tegory aiM], whose objects are submodules of M-generated modules, and they are precisely the ,-modules (introduced by Menini-Orsatti, see [16]). For the characterization of (self-) cotilting moduleswe introduce the category ~r[M], whose objects are factor modules of M-cogenerated modules. In Section 3 cotilting modules in riM] and R-Modare introduced by injectivity and cogenerating conditions and it is shown in 3.5 that these modules coincide with those mentioned above. 215

216

Wisbauer

Over artinian rings the two categories aiM] and riM] coincide and this is one of the reasons why in representation theory of finite-dimensional algebras tilting and cotilting modules are so closely connected by formal duality. The interest in tilting modulesarose from the fact that they provide equivalences between certain categories. So one expects certain dualities for the dual notion, the cotilting modules. Howeverit is well knownthat there are no dualities between full module categories and one has to restrict to finitely closed subcategories. Wewill see in 4.8 and 4.10 that under some finiteness conditions cotilting modules yield such dualities. Our techniques and results subsume and generalize previous work on the subject by Colby [4], Wang-Xu [13], Angeleri-Hiigel, Colpi, Fuller, Tonolo, Trlifaj [8, 3, 5, 6], and others. For the special case of a faithfully balanced bimodule over an artinian algebra related results are obtained in Zhaoyong[18]. 1

PRELIMINARIES

Throughout the paper C wiI| denote a locally finite Grothendieck category, i.e., a cocomplete abelian category with exact direct limits and a generating set of finitely generated objects (e.g., [11, Chapter V]). MoreoverR will be an associative ring with unit, R-Modthe category of unital left R-modules and R-rood the full subcategory of finitely generated R-modules. 1.1. Trace and reject. X in N is defined by

For any family 2’ of objects in C and N ¯ ¢, the trace of

Tr(X, N) = Z {ImC/)lf

¯ Homc(X, N), X ¯ X}

and the reject of X in N is given by Re(N,X) = N {ge(f)lf

¯ Homc(N,X),

X ¯ X} C

For A" = {X} we simply write Tr(X, N) and Re(N, X), respectively. 1.2. P-generated and P-presented objects. Let P ¯ C. An object N ¯ C is (finitely) P-generated if there is an epimorphismp(h) _~ N (with A finite), and is P-presented if there exists an exact sequence P(h’) -+ P(^) -~ N -+ 0, A’, A some sets. Wewrite Gen(P) and Pres(P) for the full generated, resp., P-presented modules.

subcategories

of C consisting

of

1.3. The category aiM]. For any object M ¯ C, aiM] denotes the full subcategory of C whose objects are subobjects of M-generated objects, aiM] is again a locally finite Grothendieck category and the trace functor TM : C -r a[M], N ~ TM(N) := Tr(a[M],

N),

is right adjoint to the inclusion functor aiM] ~ C, i.e., aiM] is a coreflective subcategory of C (e.g., [14, 45.11]). Notice that for any injective N ¯ C, TM(N) Tr(M, N) is an injective object in

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For P E C, Add (P) (resp. add (P)) stands for the class of modules which direct summandsof (finite) direct sums of copies of P, and obviously add (P) C Add (P) C Pres(P) C Gen(P) al P] C C, where all these inclusions may be proper. 1.4. Tilting objects. An object P G C is called tilting in C if P is Gen(P)projective, Gen(P) = Pres(P), and P is a subgenerator in C. P is said to sel ftilting if it is tilting in the category alP1. Self-tilting modulesdefine an equivalence between Pres(P) and a suitable subcategory of End(P)-Mod(see [16]). Nowwe consider notions which are dual to those presented above. 1.5. Q-cogenerated and Q-copresented objects. Let N,Q ~ ~. Then N is (finitely) Q-cogenerated if there exists an embedding N ~ QA(with A finite), and N is (finitely) Q-copresentedif there exists an exact sequence 0 --~ N -+ QA__~ QA’, A,A’ some sets (A finite). Wewrite Cog(Q) and Cop(Q) for .the full subcategories of C consisting of cogenerated, resp., Q-copresented modules, and cop(Q) for the modules which are finitely copresented by Q (notice that for this - in the defining sequence - we do not require A’ to be finite). Prod (Q) stands for the direct summmads of arbitrary products of Q in C. Notice that the notions Cog(Q), Cop(Q) and Prod (Q) depend on the category C. If it is necessary we will stress this by writing Cogc(Q),Copc (Q), and Prodc (Q) for clarity. 1.6. The category 7riM]. For any object M~ C, we denote by ~r[M] the full subcategory of C whose objects are factor objects of M-cogenerated objects. By definition we have add (M) C Prod(M) C Cop(M) C Cog(M) C ~r[M] It is easy to see that a[M] C ~r[M] and ~r[M] is also closed under direct sums, factor objects and subobjects and hence is a (locally finite) Grothendieck subcategory of C. In fact: For any generator G in C and A := Homc(G, M), we have

~r[Ml= aiM^] = Gen(G/Pue(G, M) Proof. By the canonical

monomorphismG/Re(G, M) -~ ~, we o bviously h ave

Gen(G/ae(G, M)) C ~] C ~r[ M]. For any N 6 Cog(M), there exists an epimorphism (a) - ~ Nwhich cl early yi elds an epimorphism G/Re(G, M)(~) --~ N. This implies N e Gen(G/Re(G, M)) and [] Gen(G /Re( G, M) ) ~r[M]. By the above equalities we know that ~r[M] is a coreflective subcategory of C (see 1.3). For special categories C it is also a reflective subcategory: Assume that in C products of epimorphisms are epimorphisms. Then

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Wisbauer

(1) triM] is closed under products in C. (2) The functor C -~ ~r[M], N ~ N/Re(N, 7riM]), is left adjoint to the inclusion functor I : 7riM] -~ C. (3) For any projective object (generator} P E ~, P/Re(P, 7riM]) is a projective object (generator) in 7riM]. Proof. (1) Consider any family {Nx}Aof objects in 7r[M]. Then each Nx is image of some Ux C M~x. By assumption, 1-IA Nx is a factor module of 1-[~ U~ and hence belongs to 7riM]. (2) By (1) the functor is well defined and obviously Homc(N,I(K)) ~_ Homc(N/Re(N, Tr[M]),K),

for any N E C, K ~ 7riM].

(3) This is easily verified. 1.7. AB4*categories. Abelian categories with products in which products of epimorphisms are epimorphisms are called AB4* categories. It is well known that for any associative ring R, R-Modhas this property (e.g., [14, 9.3]). In view of the relationship between the product in R-Modand its coreflective subcategories (see 1.3) it is straightforward to prove: Assume that for M E R-Mod the trace functor 7 TM i8 exact. Then in aiM] products of epimorphisms are epimorphisms. For characterizations of TMbeing exact we refer to [15, 4.6]. Recall that M E C is a subgenerator in C, provided aiM] = C. We call M a weak subgenerator in C, provided ~r[M] = C. 1.8. Weak subgenerators.

For M ~ C the following conditions

are equivalent:

(a) M is a weak subgenerator in (b) for some A, M^ is a subgenerator in C; (c) every object in C is a subfactor of some object in Cog(M); (d) every injective

object in C is a factor of some object in Cog(M);

(e) Cog(M)contains a generator (a generating set of objects) of (f) Cog(M)contains a subgenerator of C. Proof. This can be easily shown by standard arguments.

[]

Clearly any cogenerator of C is a weak subgenerator but not necessarily a subgenerator of C. For example, ~/z~ is a cogenerator but is not a subgenerator in ~-Mod (a[~/z~] are just the torsion ~-modules). The following observations are obvious consequences. 1.9. Corollary. (1) If M is a weak subgenerator in ~, then Cog(M)contains all projectives of C.

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219

(2) AssumeC has a (sub-) generating set of finitely cogenerated projective objects. Then Q is a weak subgenerator if and only if Q is a subgenerator in C. For any R-moduleM, ~r[M] has a particularly

nice form:

1.10. ~r[M] in R-Mod. (1) For any M ¯

R-Mod,~[M] = R/An(M)-Mod.

(2) An R-module Q is a weak subgenerator in R-Modif and only if Q is a faithful R-module. (3) If RR is finitely in R-Mod.

cogenerated then every weak subgenerator is a subgenerator

(4) R is left artinian if and only if, for any R-moduleM, aiM] = ~r[M]. In studying dualities the following finitely closed subgategory turned out to be of importance (e.g., [14, 47.12]). Here it will also help to relate cotilting modules to dualities. 1.11. The category af[M]. For any object M ¯ C, let af[M] denote the full subcategory of C whose objects are subobjects of finitely M-generated objects. This is a finitely closed abelian subcategory of aiM] (see [14, 47.2]). For Q ¯ C, we denote by Cog~,(Q) (resp., cog(Q)) the class of (finitely) cogenerated objects in ay[Q]. Clearly Cog/(Q) = Cog(Q)Nai[Q], where Cog(Q) is defined in ¢. cog(Q) is the same whenformed in ai[Q] or C and this also applies to cop(Q), the class of finitely Q-copresented modules (see 1.5). 1.12. Ext-functor in ~. By Ext~ and Ext~ we denote the first and second Extfunctor in C. For Q ¯ ~ and any exact sequence 0 -~ K -> L -~ N -~ 0 in C, we have the long exact sequence 0 -+ HomR(N, Q) -~ HomR(L, Q) -~ HomR(K, Ext~(g, Q) --~ Ext,(L, Q) -~ Ext~(g, Q) -~ Ext,(N, Q) and we denote the kernel of Ext~ (-, Q) ±CQ := {Y ¯ C[ Ext~(N,Q) =0}. Applied to the Grothendieck category ~r[M], for any M¯ C, we have the functors 1 2 Ext,[M] and Ext,{M] and, for Q ¯ r[M], we write ±MQ:= {N ¯ v[M][ Ext~[M](N, Q) = 0}. In particular,

for Q ¯ C we use the notation

ltQ := {N ¯ ay[Q]l Ext~I [Q] (N, Q) = 0}. For ~ = R-Modand M = R we apply the usual notation

Ext~ and Ext,.

220 2

Wisbauer INJECTIVITY

CONDITIONS

Dualizing projectivity conditions which came up in the study of tilting modules we recall and introduce injectivity properties which are of interest for oar further investigations¯ 2.1. Definitions. Consider exact sequences (*) 0 -> K -+ L --~ N --~ 0 in An object Q E C is called if Home(-, Q) is exact on (*) C-injective for all sequences (,) in self-Ext-injective provided N E Cog(Q); Cog(Q)-injective provided L, N ~ Cog(Q); cog(Q)-injective provided L ~ cog(Q), N e Cog(Q); w-H-quasi-injective w-H I-quasi-injective self-pseudo-injective

provided L = Q^, N ~ Cog(Q); provided L = Qk, k E ~W, N 6 Cog(Q);

provided K = Be(L, Q).. The notation w-H-quasi-injective should indicate that this notion is dual to wT-quasi-projective. Notice that for the definition of cog(Q)-injective the object in the sequence (,) is not required to be in cog(Q) but only in Cog(Q). We the obvious implications C-injective self-Ext-injective

=~

Cog(Q)-injective

=~ w-II-quasi-injective

self-pseudo-injective cog(Q)-injective =~ w-Hl-quasi-injective Notice that in general these implications cannot be reversed. For example, any left hereditary Artin algebra A is self-Ext-injective but not necessarily injective in A-Mod;for a non-semisimple ring R, any cogenerator Q in R-Modis trivially selfpseudo-injective but need not be self-Ext-injective; for such rings every semisimple R-module is cog(Q)-injective (w-Hf-quasi-injective) but not necessarily Cog(Q)injective (w-II-quasi-injective). Wewill see in 2.3 that for a moduleQ, w-H-quasiinjectivity is equivalent to Cog(Q)-injectivity, provided Cog(Q) = Cop(Q). Next we give some characterizations and properties resulting from this notions. 2.2.

Self-pseudo-iajectlve

modules in C.

(1) For Q ~ ~ the following are equivalent: (a) (b) (c) (d)

Q is self-pseudo-injective; for any N ~ ¢, Homc(Re(N, Q), Q) Cog(Q) is closed under extensions in g; any diagram with exact row in g,

V

where N ~ Cog(Q), can be non-trivially some a : Q --+ Q, ~3 : L -~ Q.

commutatively extended by

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221

(2) If Q is self-pseudo-injective

then Cog(Q)is closed under kernels.

Proof. (1) These properties are knownor easily verified ([12, 2.2], [15, 6.5]). (2) This is proved in [9, Proposition 4.4], (d)=~(e). Dual to the properties of w-Z-quasi-projective-modules given in [16, 3.2] we have the following observations on w-II-quasi-injective objects. 2.3. w-II-quasi-injective

objects.

(1) For any Q E C the following are equivalent: (a) Q is w-II-quasi-injective; (b) Homc(-,Q) respects exact sequences 0 ~ K -~ L --+ N -~ 0, where N e Cog(Q) and L e Cop(Q). (2) If Q is w-II-quasi-injective

then Cop(Q)is closed under kernels.

(3) /] Cog(Q) = Cop(Q) the following are equivalent: (a) Q is w-II-quasi-injective; (b) Q is Cog(Q)-injective. Proof. (1) (b)::~(a) is trivial. (a)~(b) For any morphism h : K -~ Q, we have a commutative diagram exact row and columns, 0 0

-~

K

Q

~

L

0 ~

N

-~

0

<~ ......QA ~> Q~

where A, A’, and 12 are suitable sets. By (a), there exists a : QA_~ Qa with pa = gq. It is easy to verify that (up to isomorphism) Kfp = Ke k n Ke a and hence QA/Kfp e Cog(Q). Again referring to (a) we obtain a morphism 7 : Q -~ with h = fp~ thus proving our assertion. (2) This can be seen from the above proof. Notice that this is also proved in [9, Proposition 4.4], (c)=~(e). (3) follows immediately from (1).

[]

In the next proposition we dualize the properties of Gen(Q)-projective modules (e.g., [16, 3.3]). 2.4. Cog(Q)-injective objects. are equivalent: (a) Q is Cog(Q)-injective;

If Q is a weak subgenerator in C, the following

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Wisbauer

(b) Q is self-Ext-injective; (c) for each g e Cog(Q), Ext~(K,Q) = 0 (/.e.,

Cog(Q) ±cQ).

IfC has a generator G e Cog(Q)with Ext,(G, Q) = 0 (e.g., then (a)-(c) are equivalent to:

a projective 9enerator),

(d) (i) Ext~(QA, Q) = O, for any set 3,; (ii) Ext,(N, Q) = O, for each Y E C. Proof. (a)=V(b) Let 0 -~ K ~ L -> N -~ 0 be an exact sequence with N e Cog(Q) and consider any morphism f : K -~ Q. By assumption, every object in C is a factor of a subobject of some QA, and so there is an epimorphism cz : X ~ L with X ~ Cog(Q). Weuse this to construct the commutative diagram with exact rows, 0 --~

K’

-+

X

-~ N --’,

0

0

K

-~

L

-~

0

~

N --+

Q

,

where the upper exact sequence is in Cog(Q) and the left hand square is a pushout. By hypothesis we can extend the diagram commutatively by some morphism X -+ Q, and the pushout property yields the desired morphism L -~ Q, thus proving our assertion. (b)=~(e 0 and (b)¢~(c) axe obvious. (c)*(d) Clearly Ext~(QA, Q) = 0, for any set A. For any N ~ C, there exists an exact sequence 0 ~ K -+ P -~ N -+ 0, where P ~ Cog(Q) with Ext,(P, Q) = 0. From this we obtain the exact sequence 0 = Ext~(g,Q) --> Ext~(N,Q) -+ Ext,(P,

Q)

proving Ext,(N, Q) = (d)=~(c) By the connecting morphismsof the Ext-functor (see 1.12), (ii) that ±eQis closed under submodules. Hence (i) implies Cog(Q) c ±c Q. It is easy to see that the above conditions on Q imply that Q is self-pseudoinjective in C and hence we have by 2.2: 2.5. Corollary. Let Q be a Cog( Q )-injective weak subgenerator in ~. Then Cog(Q) is closed under extensions in ~, and for any N ~ ~, Homc(Re(N,Q), Q) Most of the injectivity conditions defined in 2.1 also apply for C = af[Q] (although this is not a Grothendieck category). For example, for a self-pseudo-injective module Q in af[Q], Cog~.(Q) is closed under extensions in al[Q]. With the proof of 2.3 we obtain: 2.6. w-II~,-quasi-injective

objects.

(1) For any Q e ~ the following are equivalent:

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223

(a) Q is w-II f-quasi-injective; (b) Homc(-, Q) respects exact sequences 0 ~ K -~ L --+ N -+ O, where N E Cogf(Q) and L e cop(Q). (2) /f cog(Q) = cop(Q) the following are equivalent: (a) Q w-II I- is quasi-injective; Homc(-, Q) respects exact sequences 0 -~ K -~ L -~ N -+ 0, (b) where N e Cog/(Q) and L ~ cog(Q). Notice that every module in ai[Q] is a subfactor of some Qk. Hence a slight modification of the proof of 2.4 yields: 2.7. cog(Q)-injective

objects.

For Q ~ C the following are equivalent:

(a) Q is cog(Q)-injective; (b) Homc(-, Q) , respects exact sequences 0 -~ K -~ L -~ N --~ 0 in ai[Q] where N ~ CogI(Q); (c) for each K ~ Cog/(Q), Ext~[Q](K,Q) = 0 (/.e.,

Cogf(Q)

2.8. Corollary. Let Q ~ C be cog(Q)-injective. Then Cogl(Q) is closed under extensions in aI[Q], and for any g ~ ai[Q], Homc(Re(N,Q), Q) 3

COTILTING

OBJECTS

Dualizing the definitions of tilting and self-tilting following notions. 3.1. Definitions. Wecall Q 6 C a cotilting

modulesgiven in [16] leads to the

object in C if

(i) Q is Cog(Q)-injective, (ii)

every Q-cogenerated module in C is Q-presented (i.e.,

(iii)

Q is a weaksubgenerator of ~ (i.e.,

Cog(Q) = Cop(Q)),

~r[Q]

Q is called self-cotilting if it is cotilting in ~r[Q], i.e., if (i) and (ii) hold in category u[Q]. For a cotilting moduleQ in C we have C = ~r[Q] (by (iii)) and so essentially suffices to investigate self-cotilting objects in detail. It is obvious that every injective cogenerator is cotilting in C and (dual to the situation for tilting modules) we have a close connection between cotilting objects and injective cogenerators (see 3.4). 3.2. Q cotilting in a[Q]. For any Q e C we may ask when Q is cotilting in a[Q]. This is clearly the case when Q is an injective cogenerator in a[Q]. Notice that in general this does not imply that Q is cotilting in ~r[Q] (i.e., self-cotilting) even when Q is a weak subgenerator in C. For this consider the Priifer group ~p~, for any prime number p. This is an injective cogenerator in a[~p~] (abelian p-groups) and hence is coti!ting a[z~p~]. Moreover, as a faithful z~-module, ~p:¢ is a weak subgenerator (i.e., ~rI2~p~] = ~-Mod). However, ~p~ is not cotilting in ~-Mod by 3.4, since it is injective but not a cogenerator in ~-Mod.

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Wisbauer

Dualizing the characterizations of self-tilting 4.2]) we obtain characterizations of self-cotilting 3.3. Self-cotilting

modulesand their proofs (see [16, objects.

objects. For any Q E C the following are equivalent:

(a) Q is self-cotilting; (b) Cog(Q) = Cop(Q) and Q is w-II-quasi-injective (c) Cog(Q) = Cop(Q) and Q is self-Ext-injective

; in ~r[Q];

(d) Cog(Q) = (e) Q is Cog(Q)-injective,

±QQis closed under submodules,

(i) for N e r[Q], Homc(Y,Q) = 0 = Ext~[Q](N,Q) implies g = O, (ii) for any injective object (some injective cogenerator) W ~r[Q], th ere exists an exact sequence 0 -~ Q’ ~ Q" -+ W -~ O, where Q’, Q" e Prod (Q). Proof. (a)V~(b) and (b)~(c) follow from 2.3 and 2.4, respectively. (c)=~(d) and (c)=~(e)(ii): Consider an exact sequence 0 -~ K -~ L where L is a submodule of some Q^. With a Q-corepresentation of K (first column, X ~ Cog(Q)) and a pushout construction we obtain the commutative exact diagram 0

0

0

-4

K

-4

L -~

N -~

0

0

~

QA

-~

P

N

0

~

X

X 0

~

~

0

Assume N e ±oQ. By 2.5, Cog(Q) is closed under extensions in ~r[Q] and 1 (N, QA) = 0 the central sequence splits and hence P E Cog(Q). Since Ext.[Q] N ~ Cog(Q). This proves (d). Nowassume that N is injective in ~r[Q]. Then, for any Y e Cog(Q), we have the exact sequence t (Y,Q^) -~ Ext~[Q](Y,P) ~ Ext~[Q](Y,N) 0 = Ext.[Q] and hence Ext~[Q](Y, P) = 0. By (c), there exists a copresentation of 0 ~ P -~ Q~ ~ Y -~ 0, where Y e Cog(Q). Since Ext~[Q](Y, P) = 0 this sequence splits and hence P e Prod (Q), thus proving (e)(ii). (e): (ii)~(i) Let Wbe any injective yields the exact sequence

cogenerator in ~r[Q]. The given sequence

Homc(N, Q") ~ Homc(N, W) -~ Ext~iQ)(N, Q)

CotiltingObjectsandDualities

225

and Homc(N, Q) = 0 implies Homc(N, Q") = 0 and hence Homc(N, W) = 0, means N = 0. (d)~(c) Let N E Cog(Q) and Homc(N,Q). With the c anonical seque nce on the top and any extension with X E ~r[Q] on the bottom we have the diagram 0

-~

0

-+

N _~

QA -+

Q

X

-~

L -~ ~L-~

0 0,

which can be extended by some a : Q^ --r X (since Ext,[Q](QA, Q) = 0) and some f~ : N -+ Q commutatively. Since Q is injective with respect to the upper sequence we conclude (by the Homotopy Lemma)that the bottom sequence splits. Hence Ext~[Q](L, Q) = 0 implying L ~ XQQ= Cog(Q) and therefore N ~ Cop(Q). (e)(i)=~(d) N ~ ±QQ.Then Re(N, Q) ~ ±QQ and Homc(Re(N,Q),Q) = 0 by 2.5. Now(i) implies Re(N, Q) = 0 which means N E Cog(Q). 3.4. Corollary. For any self-cotilting

module Q ~ C, the following are equivalent:

(a) Q is a cogenerator in (b) Qis injective in 7r[Q]. Proof. (a)=C,(b) is obvious. (b)=~(a) follows from 3.3(d),since for Q injective in 7r[Q] clearly ±~Q= 7r[Q].

To make a self-cotilting object Q ~ C cotilting in C somecondition is needed to turn Q into a weak subgenerator in 3.5. Cotilting objects.

For any Q ~ C the following are equivalent:

(a) Q is cotilting in (b) Q is self-cotilting

and a weak subgenerator in

If C has a projective (sub-) generator then (a)-(b) are equivalent (c) Cog(Q) (d) (i) Ext~(Qh, Q) = o, .for any set A; (ii) Ext~(g, Q) = .fo r eac h N (iii)

for g e ~r[Q], Homc(N,Q)--- 0 = Ext~(N,Q) implies N = O;

(e) (i) and (ii) as in (e) and (iv) for some injective

cogenerator W~ ~, there exists an exact sequence

0 --~ Q’ --~ Q" -~ W--~ O, where Q’, Q" ~ Prod (Q).

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Wisbauer

Proof. The assertions follow by 3.3, 2.4, and the observation that any projective object of C belongs to ±cQ.The latter implies that in (c),(d) and (e), Q is subgenerator in C. [] For any R-module Q, ~r[Q] = R/An(Q)-Mod and hence it has a projective generator. So we obtain from 3.3 and 3.5: 3.6. Self-cotilting modules. For any left the following are equivalent:

R-module Q, put-~ = R/An(Q). Then

(a) Q is self-cotilting; (b) Cog(Q) = Cop(Q) and Q is w-H-quasi-injective (c) Q is cotilting

;

in R-Mod;

(d) Cog(Q) = .L-~Q; (e) (i) Ext-~(QA, Q) = O, Ior any set A; (ii)

nxt~-(N, Q) = O, for each N E ~-Mod;

(iii)

for N ~ ~-Mod, HomR(N,Q) = 0 = Ext-~(N, Q) implies N = O;

(f) (i), (ii) as in (e) and (iv) for some injective quence

cogenerator W ~ R-Mod, there exists an exact se-

0 --~ Q’ -~ Q" ~ w -~ o, where Q’, Q" ~ Prod (Q). 3.7. Cotilting in R-Mod.By definition, Q is cotilting in R-Modif and only if it is self-cotilting and faithful. Hence 3.6 yields characterizations of these modulesby replacing R by R in (c),(d),(e) and Notice that for this case (d) ~=~(e) was proved in Colpi-D’Este-Tonolo[7, Proposition 1.7] and (d) implies Cog(Q) = Cop(Q) is shownin [7, Proposition 1.8]. over (d)=~(f) corresponds to Angeleri-Hiigel-Tonolo-Trlifaj [3, Proposition 2.3]. From 3.5 we obtain the 3.8. Corollary. For Q cotilting

in R-Mod,the following are equivalent:

(a) Q is a cogenerator in R-Mod; (b) Q is injective

in R-Mod.

To investigate dualities we introduce a finite version of cotilting objects. 3.9. Definition. Wecall Q ~ C an f-cotilting

object if

(i) Q is cog(Q)-injective, (ii) every finitely Q-cogenerated object in C is finitely Q-copresented (i.e., cog(Q) = cop(Q)).

CotiltingObjectsandDualities

227

For example, every semisimple R-moduleis f-cotilting. Remark. In Angeleri-Hiigel-Valenta [2] finitely cotilting modules Q are defined as "cotilting" modules(with a slightly different definition) which are finitely generated R-modules such that HomR(X,Q) is a finitely generated EndR(Q)-module, for any finitely generated R-moduleX. It is easy to see that over noetherian rings such modulesare f-cotilting in the sense defined above (compare[2, Corollary 5.2]). Recall that an object Q is injective in af[Q] if and only if it is injective in a[Q] (i.e., Q-injective) and Q is an injective cogenerator in af[Q] if and only if it is an injective cogenerator in a[Q]. Similar to 3.4 we have now: 3.10. Proposition. For any f-cotilting

object Q E C, the following are equivalent:

(a) Q is a cogenerator in af[Q] (in a[Q]); (b) Q is injective in al[Q] (in a[Q]). Proof. (a)~(b) is obvious. (b)~(a) (compare proof of Fora ny s ubob ject K C Q weobta in the commutative exact diagram, where the first column is a Q-copresentation of K (k E SV, X ~ Cog(Q)), by a pushout construction 0 0

-+

K

k0

~ Q X 0

0 -~

Q ~

-~

P -~

~

X

N -~ N -+

0 0

0

Since P is in a[Q] and Q is Q-injective, the central columnsplits implying that P ~ Cog(Q). For the same reasons the central row splits and hence N ~ Cog(Q). From this we conclude that Q is a cogenerator in a[Q] (e.g., [14, 16.5]). The condition cog(Q) = cop(Q) holds trivially provided Q is a cogenerator af[Q] but it need not follow from the condition Cog(Q) = Cop(Q). In fact it related to some finiteness properties. 3.11. Proposition. Let Q ~ ~ be w-HI-quasi-injective. Then for every K ~ cop(Q), Home(K,Q) is a finitely generated right Endc( Q )-module. Proof. This is obvious.

[]

Adapting the proofs of 3.3 we obtain characterizations of f-cotilting 3.12. f-cotilting equivalent:

objects.

(a) Q is f-cotilting;

objects.

For any Q ~ ~ and S = Endc(Q), the following

are

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Wisbauer

(b) cog(Q) = cop(Q) and Q is w-IIf-quasi-injective; (c) cog(Q) = cop(Q) and Cogs,(Q ) c±~ Q; (d) Cogs,(Q ) = ±~Q,and for every K E cog(Q), Homc(K,Q) mod-S; (e) (i) Q is cog(Q)-injective, (ii) (iii)

±~Qis closed under submodules,

1 for N ~ af[Q], ttomc(N,Q)= 0 = Ext~,tQl(N, Q) implies N = O, and for every K ~ cog(Q), Home(K,Q) ~ mod-S.

Proof. (a)C~(b) and (b)¢~(c) follow from 2.6 and 2.7, respectively. (c)=~(d) In the proof of 3.3, (c)~(d), take N ~±i Q andA a finite index (d)=~(c) Let N E cog(Q) and fa,... ,f~ a generating set for the S-module Home(K, Q). The fi’s yield a canonical monomorphismN -+ Qk. Nowprocede as in the proof of 3.3, (d)=~(c). The remaining implications can also be transferred from 3.3. [] Moreexamples of f-cotilting 4

REFLEXIVE

MODULES

objects will be given at the end of the paper. AND DUALITIES

To avoid technical complications we restrict our study of dualities to module categories. Weinvestigate dualities induced by any left R-moduleQ with S = EndR(Q). 4.1. Canonical functors. D:

Related to ttQs we have the adjoint pair of functors

R-Mod Hom_~-,Q)

Mod-S, D’ :

Mod-S Hom_~,Q)

R-Mod,

and for any N E R-Modand X ~ S-Mod, the canonical (evaluation)

morphisms

¢~N : N .-+ D’D(N), n ~ [/3 (n )/3], ~x : X --+ DD’(X), z ~ [a a( z)]. where Ke~N = Re(N, Q), Ke¢~¢ = Re(N, 4.2. (Semi-)reflexive modules. A module N ~ R-Mod is called (semi-) reflexive if ~N : N -+ D’D(N) is an isomorphism (epimorphism). Similarly (semi-) Q-reflexive objects in Mod-Sare defined. It is straightforward to prove: N ~ a[Q] is semi-Q-reflexive if and only if N/Re(N, Q) is Q-reflexive. The class of all Q-reflexive module~ in R-Mod(in Mod-S)is denoted my RefR(Q) (resp., Refs(Q)). Obviously we have the following 4.3. Basic duality.

For any R-module Q, the functor D = Hom~(-,Q):

Rely(Q)

defines a duality with inverse D’ = Horns(-, Q).

~ Refs(Q)

CotiltingObjectsandDualities

229

For any N E R-Modconsider an exact sequence S(A’) --~ S(^) ~ D(N) --~ in Mod-S. By left exactness of Horns(-, Q) we obtain the exact sequence 0 -~ Homs(D(N), Q) -~ Nowif N E Refn(Q), i.e.,

N ~_ D’D(N), ). we conclude N ~ Copn(Q

4.4. Classes of modules related

to Q. Wehave the following inclusions:

(1) D’(Mod-S) C CopR(Q), D(R-Mod) ). (2) add (Q) c RefR(Q) c Cop~(Q) ) C R-Mod. (3) add (Q) Weinvestigate the properties of the classes considered above in view of certain injectivity conditions. A submodule K C M in called Q-closed in M provided M/K ~ Cog(Q). 4.5. w-II-quasi-injective

modules. Let RQ be w-II-quasi-injective.

Then:

(1) Refn(Q) is closed under Q-closed submodules; in particular, Ref~(Q) is closed under kernels. (2) Every factor module of a Q-reflexive right S-module is semi-Q-reflexive. Proof. (1) Let P ~ Refn(Q) with a Q~closed submodule K C P. Then we have exact commutative diagram, 0 0

~

0

K

-~

P

-~

Y

0 ~ K**

-~

P**

-~

Y**

0 where Y 6 Cog(Q). This shows that K is Q-reflexive. (2) Let X E Refs(Q). For any epimorphism X -~ L we have the exact commutative diagram X ~ L -~ 0 X** -¢ which shows that L is semi-Q-reflexive.

L** -~ 0, ~1

The question arises to which extent properties of Ref~(Q) imply injectivity conditions on Q. Since P~efR(Q)is not closed under products we are not able to get a general assertion converse to 4.5. Howeverthe situation is different if we restrict our considerations to finite products as we will see below. First we formulate a finite version of 4.5. Since for any k ~ SV, Qk e Rely(Q) and Sk e Refs(Q), essentially the same proof yields:

230

Wisbauer

4.6. w-IIf-quasi-injective Then:

modules. Let Q be a w-l-l f-quasi-injective

R-module.

(1) cop(Q) c RefR(Q). (2) Every finitely

generated right S-module is semi-Q-reflexive.

(3) By restriction

we have the functor Hom/~(-, Q) : cop(Q) ~ mod-S.

4.7. Proposition. Let Q e R-Mod. Assume cop(Q) c RefR(Q) and that all finitely generated right S-modules are semi-Q-reflexive. Then Q is w-HI -quasi-injective. Proof. (compare [14, 47.12]) Wehave to show that HomR(-, Q) = (-)* is exact sequences 0-+ K ~ Qk ~ y ~ 0, where Y E Cog(Q). From the inclusion exact rows 0

5 : Im f* -+ K* we obtain the commutative diagram with

~

K

~

0 ~ (Im.f*)*

Q~

-~

(Q *)** -~

Y -~

0

r**

Since (~Q, is an isomorphism and ~ is a monomorphism,~KS*is an isomorphism. NowK being Q-reflexive implies that 5" is also an isomorphism. Wehave the commutative diagram Im f* ~ K* (Im

f*)**~""

K*",

where ~Imf" is surjective (by our ~sumptions) and 6"* is an isomorphism. This clearly implies that ~ is an isomorphism which means that, (-)* is exact on the given sequence. Combiningthe results just derived we obtain: 4.8. Injectivity

and duality.

For Q ~ R-Modthe following are equivalent:

(a) Q is w-IIl-quasi-injective; (b) cop(Q) c Ref/~(Q) and all finitely reflexive;

generated right S-modules are semi-Q-

(c) cop(Q) c RefR(Q) and mod-S fq Cogs(Q) c Refs(Q); (d) Hom~(-, Q) : cop(Q) ~ mod-S f~ Cogs(Q) is a duality. Proof. (b) (c) (c)

(a) ~(b) follows from 4.6. ~(c) is clear by the commentsin ~(a) holds by Proposition ¢~(d) is obvious.

[]

Cotiiting ObjectsandDualities

231

4.9. Remark. Dualizing the notion of s-E-quasi-projective used in Sato [10] we may call an R-module Q s-II-quasi-injective if HomR(-,Q) is exact on sequences 0 -4 K -4 QA-4 QA’, for any (or finite) sets A, A’. With similar proofs one gets that for such modules, Homn(-, Q) induces an equivalence between the kernels of morphismsQ~ -4 Q~, k, l E ~W(a subclass of cog(Q)) and the finitely presented right S-modules (compare [17, 4.6]). By the definition of f-cotilting 4.10. f-cotilting

and duality.

modules, 4.8 yields immediately:

For Q E R-Modthe following are equivalent:

(a) Q is f-cotilting; (b) cog(Q) c Ref/~(Q) and all finitely reflexive;

generated right S-modules are semi-Q-

(c) cog(Q) c Ref/~(Q) and mod-S VI Cogs(Q) Refs(Q); (d) Hom/~(-, Q) : cog(Q) -4 mod-S g~ ) is a d ualit y. Finally we mention somemore examples of f-cotilting objects. Clearly an object Q which is an injective cogenerator in C - or more generally in a[Q] - is f-cotilting. In particular any semisimple object has this property. Cotilting modules Q are f-cotilting, provided for every K E cog(Q), Homc(K, mod-Endc(Q). Sufficient for the latter condition is that Q is noetherian both left R- and right Endc(Q)-module. This situation was considered in Wang-Xu[13, Theorem2 and 3]. If nQ is cotilting and artinian the conditions are also satisfied and this is the situation usually considered in representation theory. It was already mentioned in the remark following 3.9 that over noetherian rings the "finitely cotilting" modulesstudied in Angeleri-Hiigel-Valenta [2] are f-cotilting modules in our sense. Notice that artinian cotilting modules need not be "finitely cotilting" in the sense of [2] since they need not be finitely generated as modules over their endomorphismring (see Example2.3 in [1]). The Morita duality as described in [14, 47.12] is a special case of 4.10. Notice that in [14, 47.12] the class of reflexive R-modulesis closed under factor modules and submoduleswhile in 4.10 this class is only closed under submodules.

232

Wisbauer

REFERENCES Finitely Ill L. ANGELERI-HOGEL, (2000), 2147-2172.

cotilting

modules, Comm. Algebra 28(4)

H. VALENTA, A duality result for almost split sequences, [2] L. ANGELERI-HOGEL, 6o11. Math. 80 (1999), 267-292. A. TONOLO,J. TRLIFAJ, Tilting preenvelopes [3] L. ANGELEP~I-HOGEL, cotilting precovers, Algebras and Repres. Theory, to appear [4] R.R. COLBY, A generalization of Morita duality Comm. Algebra 17(7) (1989), 1709-1722.

and the tilting

and

theorem,

Cotilting bimodules and their dualities, Interactions Between Ring Theory and Representations of Algebras, F. van Oystaeyen, M. Saorin (ed), Marcel Dekker (2000), 81-93.

[5] R. COLPI,

[6] R. COLPI, K.R. FULLEI%Cotilting 192(2) (2000), 275-291.

modules and bimodules,

[7] R. COLPI, G. D’ESTE ANDA. TONOLO,Quasi-tilting equivalences, J. Algebra 191 (1997), 461-494.

Pac. J. Math.

modules and counter

ANDJ. TKLIFAJ,Partial cotilting modules and the lat[8] R. COLPI, A. TONOLO tices induced by them, Comm.Algebra 25 (1997), 3225-3237. [9] J. P~ADA,M. SAORfN,A. DELVALLE,Reflective gories, Glasgow Math. J. 42(1) (2000), 97-113.

and coreflective

subcate-

[10] M. SATO,On equivalences between module categories, J. Algebra 59 (1979), 412-420. [11] B.

STENSTRhM, Rings

of Quotients, Springer-Verlag, Berlin (1975).

[12] T. WAKAMATSU,Pseudo-projectives and pseudo-injectives gories, Math. Rep. ToyamaUniv. 2 (1979), 133-142.

in Abelian c.ate-

MINGYI, XU XONGHUA, ,-modules, co-.-modules and cotilting [13] WANG ules over Noetherian rings, Science in China (Series A) 39(1) (1996),

mod-

[14] R. WISBAUEP~, Foundations of Module and Ring Theory, Gordon and Breach, Reading (1991). [15] R. WlSBAU~.~t,On module classes closed under extensions, Rings and radicals, Gardner, Liu Shaoxue, Wiegandt (ed.), Pitman RN346 (1996), 73-97. [16] R. WISBAUER, Tilting in module categories, Abelian groups, module the.ory, and toplogy, Dikranjan, Salce (ed.), Marcel Dekker LNPAM 201 (1998), 444.

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Static modules and equivalences, Interactions Between Rin9 [17] R. WISBAUEP~, Theory and Representations of Algebras, F. van Oystaeyen, M. Saorin (ed), Marcel Dekker (2000), 423-449. ZHAOYONG, On a generalization of the Auslander-Bridger [18] HUANG Comm.Algebra 27(12) (1999), 5791-5812.

transpose,

Coherent Components of Auslander-Reiten Whose DTr-orbits Are Finite

Quivers

HAILOUYAODepartment of Applied Mathematics, Beijing Polytechnic University, 100 Pingleyuan, Chaoyang District, Beijing,100022, P. R. China, E-mail: [email protected] Dedicated to Professor Shaoxue Lin for His 70th Birthday

ABSTRACT Let k be an algebraically closed field, A be a finite-dimensi0nal algebra over k,rA be the Auslander-Reiten. quiver of A. In this paper we give a necessary and sufficient condition for a coherent componentr Of rA such that the translation quiver obtained from r by deleting the DTr-orbits of projectives is connected and has property that its DTr-orbits are finite. 0

INTRODUCTION

Let A be a connected, basic finite-dimensional algebra over an algebraically closed field, modAbe the category of all finitely generated right A-modules, and FAbe the Auslander-Reiten quiver of A, then FAis given a combinatorial structure making it a translation quiver. Since rA is an important combinatorial and homological invariant of the category modA,a lot of work has been done on describing all possible shapes of the connected components of FA. General theorems for the description of shapes of stable components of rA are due to Happel-Preiser-Ringel [~] and Zhang [12]. The former theorem says that if r is a stable componentcontaining a v-periodic module then r is a stable tube ZAoo/(Vn)with n e + i f r is inf inite; oth erwise, r = ZA/Gwherethe un derl ying graph ~ of A is a Dynkin diagram and G is an automorphism group of ~. The latter theorem says that if r is a stable component of an Auslander-Reiten quiver containing no v-periodic module, then r is isomorphic to ZA where A is a valued quiver without cyclic path.

235

236

Yao

S. Liu has studied the structure of semi-regular components of an AuslanderReiten quiver [4]. His result is that if F is a semi-regular componentcontaining an oriented cycle and no periodic module, then r is a ray tube or coray tube; if r is a semi-regular component without oriented cycles, then there exists a valued locally finite quiver A containing no oriented cycle such that F is isomorphic to a full translation sub-quiver of 7.A which is closed for predecessors and successors. In [5] he gives all possible structures of components in the Auslander-Reiten quivers of tilted algebras. Recently, P. Malicki and A. Skowrofiski have described the shape of an arbitrary coherent and almost cyclic componentof rA (see [8]). Their result is that a connected componentOf FAis coherent and almost cyclic if and only if r is a generalized multi-coil, wherer issaidto bealmost cyclic if allbutat mosta finite number of vertices of r lieon oriented cycles contained entirely inr. A translation sub-quiver of rA is saidto be coherent if the.following two conditions aresatisfied. (cl).Foreachprojective module p in r thereis an infinite sectional path ¯ p "-" Xl -% X2 -% "" " -% ~ri -% ~gi-t-I

-% ~gi-t-2

-% " ¯" ¯

inr,cailed.a raystarting atp. . (c2).Foreachinjective module I in r thereisan infinite sectional path "’" -% Y#-i-2

-% Yj+I -% Y# -% "’’Y2

-% ~I =

I

in r,cailed a corayending in I. . Wesaythata r-orbit inI~ isfinite ifitcontains onlyfinitely manyvertices. The aim of this paper is.to give the structure of a kind of coherent components of Auslander-Reiten quivers with finite DTr-orbits on the basis of results in [11]. Someclasses .of algebras have this type Of AR-componentssuch as tame hereditary algebras,, polynomial, growth trivial extension algebras, and some multicoil algebras, etc. One of the motivations of this article is to see which coils have finite DTr-orbits. Weshall denote by ar the translation sub-quiver of r obtained by deleting the DTr-orbits of all projective modulesin. r. Wehave the following results. THEOREM A. Let r be a coherent component of an Auslander-Reiten quiver. I/OF is a connected subquiver off, then F can be obtained from a stable tube byfinitely manymultiple admissible coray-ray insertions (see section I for its definition)if and only if each DTr-orbit in F is finite. REMARK: Such components in Theorem A exist.This is the case (trivially) for tame concealed algebras,but also for polynomial growth selfoinjective algebras,or for the algebras in the examples in section 4~whichare coil enlargements of tame concealed algebras. COROLLARY B. If r is a component which satisfies then r is a coil with finite DTr-orbits.

the conditions in Theorem A,

COROLLARY C. If 1" is a coil, then each DTr- orbit in r is finite if and only if r can be obtained from a stable tube by finitely many multiple admissible coray..ray insertions.

237

CoherentComponents of Auslander-ReitenQuivers

The concepts and notations which are not explained in the following can be found in [1,2, 9~10]. 1

PRELIMINARIES

From now.onwe let k be an algebraically closed field, A be a representationinfinite, basic, connected, finite-dimensional algebra over k, modAbe the category of finite-dimensional right A-modules, rA be the Auslander-Reiten quiver of A, mad r be a connected component of FA. Weuse r and r- to denote the AuslanderReiten.translations DTr and TrD, respectively. Incase there is no possibility of confusion we do not distinguish between an indecomposable module X in modA and the corresponding vertex [X] in FA. Assume X to be an indecomposable module in m0dA, then X is said to be left stable if rnX 9~ 0 for all positive integers n, and right stable if vnX¢0 for all negative integers n. we use tFA to. denote the full sub-quiver of FAwhich is generated by all r-orbits containing no projective module, whereas rI’A indicates the one generated by those r-orbits containing no injective module. The connected components of ira .are called left stable components of FA while those of rrA are called right stable components of FA. For .a translation quiver r.= (r0, rl, r) we can introduce the concepts of left, right stable vertices and stable vertices in r. Here we still use the notation r and the translation map r. The context will allow no confusion. Let X --~ Y be an arrow in FA, its valuation (dxy, d~x~,) in FAis so defined that :Y occurs dx~. times in the codomain of the source map for X and X occurs d~. times:in the domain of the sink map for Y. Ifdx~. --- d~x~, -- 1.we say that the arrow X ---~ Y ha~ trivial v~luatiom If every arrow in a sub,quiver r of FA has trivial valuation we say that r has trivial valuation. Let r = (ro,rl,r) be a translation quiver, a function l: F0 ~ 1~1 (natural numberset) is, by definition, a length function on r, if (i) For any x which is not projective, l(x) yE~-

(ii) For any x which is projective, l(~) (iii)

For any x which is injective l( x) =

where, as usual , x-(x+)denotes the set of direct predecessors(successors) of r. We quote some Lemmas from [11]. Let F be an infinite connected component of FA in which each DTr-orbit is finite, then we have LEMMA 1.1.

Let 0 -~ X -r ~Yi -~ Z -~ 0 be an Auslander-Reiten

sequence in

modAwith the Yi indecomposable , where X and Z belong to P, then r < 4. In case r = 4, one o] the Yi is projective-injective. LEMMA1.2.

Assume that X is injective

in F and X --~ ~Y~ is a .source map, i=1

238

Yao

then r <_3 and none of Y~ ’s is projective. Dually, assume that Z is projective

in r and ~Y~ -~ Z is a sink map, then

r < 3 and none o] I~ ’s is injective. LEMMA 1.3. F has trivial

valuation.

Letr = (r0,rl,T) be a connected translation quiver andletk(r) beitsmeshcategory. We aregoingto define theso-called admissible operations. DEFINITION 1.1. If SuppHomk(r)(z, -) equals an infinite at x, 2~-~0

--)*

~i

9~2

sectional path starting

--~....

Let t bean arbitrary positive integer ,. and A denote the following translation quiver, isomorphic to the Aus]ander-Reiten quiver of the full. t × t lower triangular matrix algebra

Let F~ be the translation quiver having as vertices those of r , those of A, additi0nally z~j and x~ (where i> 0, j > 1) and having arrows as shown in Figure i:I.

Figure 1.1 The translation T’ is defined as follows: ~’~z~j = Zi_l,j_l , if i _> 2, j > 2, r’z~ : x~_l, if i > l, r’zoj yj-l if j > 2, p = zo~ is projective , r’X~o= ’ ’ if i _> 1, r~(r-~xi) = i~ provided xi i s not i njective i n F, o therwise xi ’ is injective in I’. For the remaining vertices of F or A, r~ coincides with the translation of r or A, respectively. If. t = O, the new translation quiver r ’ is obtained from ][’ by

Coherent Componentsof Auslander-Reiten Quivers

239

inserting the ray consisting of the Ze, i S. DEFINITION1.2. If SuppHomk(r)(z, -)consists of two sectional at x, one infinite and the other finite with at least one arrow,

paths starting

with t >_ 1, in particular, zo is injective, then r I is the translation quiver having as vertices those of r, and additional vertices denoted by p = z~, z~j(i >_ 1, j >_1) and having arrows as shown in Figure 1.2

Figure 1.2 The translation r I is. defined as follows: p is projective-injective

, r’zij =

zi.l,j-l(i _>2, j _>~.), ~’Z~l= ~-1(i _>1), ~’z~j= ~#_l(j_>2), ~’~= z~_~(i> 2)~z~ = ~/t, ~(~-lzi) ---- ~ provided zi is not injective in r, otherwise ~ is injective in r. For the remainingvertices of r ~, .rlcoincides with the translation ~" of r. DEFINITION 1.3. Assume that SuppHomA (~, -) consists of two parallel sectional paths, the first infinite and starting at ~, the second finite with at least, one arrow and starting at a vertex yl such that there isan arrow z ~ y~, not lying on the first path

where t ~ 2, so thaL in p~icul~, zt-~ is inje~ive. Moreover, consider the subquiver of ~ obtained by deleting the ~rows 9i ~ ~-~y~_] ~sume that its connected component r* cont~ning the ve~ex x do~ not contain any of the vertic~ T--gi_1, 2 < i < t. Thenr~ is the tr~slation quiver having ~ re.ices those vertices of F* , ~ditional vertices denoted by z~, z~, z~( where i ~ 1, 1 ~ j ~ t), having ~rows ~ in the Figures 1.3 and 1.4 below. If t is odd

240

Yao

Figure 1.3 If~ is even

Figure 1.4 The translation v’ ofF’ is defined asfollows: x~ is projective, rlzlj = zi-l,j-~ if i _> 2, 2 <_j <.~ ~"z~ = x~_~ifi E 1, v’x~ = yi if 1 < i < t, v’x~ = zi_~,~ ifi E ~+1, r~y~ =--x~_2 if i > t provided x~ is not injective in F , otherwise x~ .is injective in I "~. In both cases, xl_1 is injective. For the remaining vertices of I ’e, r ~ coincides with the translation v of F. For the Convenienceof statements, the ray insertion in definition 1.1 is called a($ + 1)-insertion. The vertex x = x0 is called an a-insertion vertex. Its dual i.e. coray insertion in definition 1.2 is called a*(t + 1)- insertion. In this case, the

CoherentComponents of Auslander-ReitenQuivers

241

vertex xo is called an a*-insertion vertex. The insertionin definition 1.2 is called ~3(t ÷ 1)-insertion, the vertex z0 is called a E-insertion vertex. Its dual is called ~* (t ÷ 1)- insertion. The insertion in definition 1.3 is called 7(t ÷ 1)-insertion, its dual is called 7" (t ÷ 1)-inserti0n. Wenowintroduce a notion as follows¯ Assumethat r is an infinite connected translation quiver and that there is an a-insertion vertex z in r0. Let ml,... ,m~ be natural numbers. So, the support of Hom~(r)(~,-) equals an infinite sectional path starting

Wemake a(ml)-insertion at the vertex z in r to obtain a new infinite connected translation quiver, denoted by rl.Assume that the added projective vertex pl ( as p in definition 1.1) which is a direct successor of x, is a ~*-insertion vertex in rl. So, the support of Hom~(r)(-,p~) consists of the following sectional paths ending at p~, one infinite and other finite.

¯ ""

---@

~i

~ ~i--1

~ "’"

~ ~t2

~--’X0

~ ~tl

:Pl

4r’-’-

Pll

4-"-

So, ~2 = zo is ~ a’-inse~ion vert~. By the definition of that there is a unique m~im~finite sectional path st~ing at The end ve~ex z~ of L~ is still, an ~-i~e~ion vert~ while zx(denoted by zt,~) on the path L~is a 7*-inse~ion vert~ if the suppo~ of Hom~(v)(-, z~,l) is of the following

"

"

"

~

~ ~l,ml~0

~ltml+l

~ ~l,m~:~l

~l,ml--2

~"

.."

~-inse~ion we k~ow px (denoted by L~). the predecessor of m~

"

"

~

~1,2

~ ~1,1

where ~l,i = ~z, ~d zi,m~-I (=pi) is projective. Wem~e a(m2)-insertion a~ z~ in Fi ~o ob~n ~ infinite connected ~r~sla~ion quiver, denoted by F~. Obviously the ~ded projective vert~ ~, which is a dire~ successor ofzl, is still a ~*- insertion vert~. The suppo~ ofHom~( D (-,~) co,isis of the following se~ion~ pat~ ending in ~, one infinite and the other finite. """

~ Di

~ Ui--I

~ "’"

~

Pl ~ "’" ~ ~ ~ ~1 ~ "’" ~

Let L~a denote the nniqne m~imMfinite se~ional path stating at ~ in r~, ~hen the end vert~, denoted by z~, is still ~ a- insertion Vert~. The dire~ predecessor of z2 (denoted by ~,~) on the p~thL~is still ~ ~*-insertion ve~ex if m2~ 3,. ~d the predecessor z~ (deno~d by ~,ma) of ~ is still ~ a* inse~ion ve~ex. IncluSively, we obtain an infinite conne~ed translation quiver ~ by a(m~)-inse~ion at the vert~ z~_~. The ~ded proje~ive ve~ex p~, whi~ is ~ direct successor of za_~, is ~ ~*-insertion vertex. So, the support of Hom~(r,)(-, pu) consists of the following se~ionM paths. One infinite ~d the other finite, ¯

" " ~ ~i

~ ~i-I

--’-~---’~

Pl ~ " ¯ " ~ P~ 4----

P~,I

4--

¯ ."

~--

Let L~+ denote the unique maximal finite sectional path starting at p~ in I’u, the end"vertex denoted by z~, is both an a-insertion and a*-insertion vertex.

Yao

242

The direct predecessor of z~ (denoted by z~,i ). on the path L~+~.is still a if*insertion vertex if rn~ >_ 3, and the predecessor z~-i (denoted byz~,m~) of p~ an a*-insertion vertex. The support of Hom~(r~)(-, x~,l) is in the following if rn~ > 3. "

where Y~,1 = rz~, zp,~_, = p~ is injective, the path I/~,~n._~ --~ "’" ¯ .. ~-~ yp,1 .and ... --~ zp,2 --~ z~,1 are sectional and every sectional path at z~,~ (corresponding at Y~,l ) is a sub-path of one of the paths ~/~,~ ~ z~,,1 ¯ " --~ x~,a ~ x~,l (respectively, of ~/~,m~_l--~ ..- ---+ I/~,~ --~ Y~,l). Wemake the following insertions (not necessarily all of them ) (1) ~*(m~)-insertion at the vertex p~, where (2)

~/*(m~)-insertion at the vertex z~,~,where

¯ (3) a* (m~)-insertion at the vertex z~ or x~,,~, = z/~-i or other coray vertex, where m~ may not be equal to rap. So, we obtain a translation quiver r~. Fromthe definition of/~* (rn~.)- insertion, if* (m~)-insertion and a* (m~)-insertion we learn thatthe ~*-insertion vertex p~:’s r# correspond to the new~*- insertion vertices( of course, they are still projective), still denoted by pi’si for i = 1,... ,/~ - 1, and 7*-insertion vertices correspond to the new -~*-insertion vertices (Horn(-, xi,l) is the same as translation sub-quiver asis in r~) for i = 1,,.. ,#- 1, and a*:insertion vertices are still a*-insertion vertices for i = 1,2,... ,/~- 2. If we do not make a*(m~)insertion at the vertex x~,m~ = z~-i in r~, then ~,m# also corresponds to anew c~*-insertion vertex in r~ , still denoted by ~#,m~(=. z~). Weagain makethe following insertions (not. necessarily all of them ) in I?~. . (1) ~*(m*.. 1),insertion (2) 9"(m; l)-insertion

at the vertex P/~-I, where m~_ 1 = m~_l. at the vertex ~.-l,l,

where

(3) a*(m~_~)-insertion at the vertex z~-2 = z~_l,m,_~ or the other coray vertex, where m~_l may not be equal to So, we obtain a translation quiver r’ The/~*-insertion vertices pi’s,. insertion vertices x~,l’s correspond to the new/~*-insertion vertices (still denoted by pi’s) and the new ~,*-insertion vertices ( still denoted by ~i,l’s) , respectively, i ~ 1, 2,---, ~ - 2. The a* -insertion vertices x~,m~’s(=z~_l) ~e still a*-insertion ve~ices for i = 1, 2,... ,p - 3. X~_l,m._~ Msocorresponds to a new a*-insertion ¯ ve~ex (still denoted by ~_l,~_l) in r’ if we do not make a*(m~_~)-insertion at the ve~ex z~-~,~_~in r~.. Indu~ively, by the p*-th step, where p* may not be equMto p, we obtain a tr~slation quiver F~ , whi~ ~ said to be obtained from F by a multiple admissible

Coherent Componentsof Auslander-Reiten Quivers

ray-coray

insertion

243

if ~ rn~ = ~ m~. The vertex x is called

ray-coray insertion vertex. Du~ly, the definition

of multiple

~missible

EXAMPLE.Consider a tube rz = BZA~/(1),we rz to obtain a translaton quiver r2.

~ O.

a multiple

admissible

i~e~ion

can be m~e.

coray-ray

make an a(3)-insertion

at z

O~

:

r2

wherethevertical dottedlineshaveto be identified in orderto obtaina stabletube and ray tube (similarin the below).We againmake an c~(3),insertion at z~ in to obtaina translation quiverr3. We makean 8" (3)-insertion at p2 in r3 to obtain a translation quiverr4.

¯

Y~

~’

"’ ,

"’ ¯

~,;

z"

~:~zu’ ’ ’~ ~

1

;

r3

we also make a 7* (3)-insertion at i~z in r3 to obtaina translation quiverr~ follows.Thus,r~ is obtainedfrom the stabletube rz by one multipleadmissible ray-coray insertion.

r5

SOME

LEMMAS

244

Yao In this section we will establish somepreliminary lemmasto prove the theorem

A. Let Y be a translation quiver without multiple arrows. A mesh with exactly three middle terms will be called exceptional and a projective middle term in an exceptional mesh will be called exceptional projective. Other meshes and projectires will be called ordinary. The set of vertices which are the starting or ending vertex of a mesh in Y with unique middle term will be called the mouth of F. Let l be a length function in Y. A ray starting at x will be denoted by [~, Dually, a coray ending in y will be denoted by (oc, y]. Firstly, we quote several lemmasfrom [8]. LEMMA 2.1. Assume F contains a translation

sub-quiver of the form

with B1 projective and t > 2. then at most one of the modules At and Bt can be injective. Moreover, if it is the case, then Bt is injective if t is odd, whaeA~ is injective if t is even. []

Proof. This is Lemma4.2 in [8]. LF, MMA 2.2. Assume F contains a translation

sub-quiver of the form

where t > 2, AI or B1 is projective , and At or Bt is injective . Then A-~ = {Di+l}=Bi + for l < i < t-1, and A~- ={Di}=B~- for 2 < i < t. Here and.B~ are sets of successors of Ai and Bi, respectively; A:~ and B[ are sets of predecessors of Ai and Bi, respectively.

Coherent Componentsof Auslander-Reiten Quivers

245

Proof. This is Lemma4.5 in [8]. LEMMA 2.3. F contains no translation

sub-quiver o.[ the .form

with B1 projective and Q projective-injective. Proof. This is Lemma4.4 in [8]. LEMMA 2.4. F contains no translation sub-quiver which is created by identi~ing the sectional path At --~ Dr+! ~ "’" ----t, Ft ~ Et+l ~ F#+I in the Figures A and B, and D~ ~ ... --~ M ---~ N in the Figures B and C below At

A~

A~

Figure A

F/gure B

246

Yao

FigureC where De+x ~-~ Cx --~ Ca --+ ... --+ Ck is a sectional path , nl, B~ are projective, Ae or Be is injective, A~. or B~ i~ injective , possibly some (or all ) C~:, C3,"’, CAare projective, t >_ 1, s >_ 1, k >_ 1 and possibly B or Ea does not exist. Proof. This is Lemma4.7 in [8]. LEMMA 2.5. r contains no translation

Ot

= D~

sub-quiver of the following form.

"""

’"

z%

"1~

,,

,,/

where Bx is projective and P is projective-injective. Proof. Assumethat F contains such a sub-quiver. Using the length function l we obtain the following inequalities I(G,+~) + I(Es+~) > I(D,+~) + I(P) + I(E1) /(ai)

+/(Di+l) > l(Ai) +/(Bi) +/(ai+l)

1 <

(1) (2)

Thus, we have ICCx) + Z ICDi+a) -> l(a,+a) + ~_~(ICA~) + l(Bi)), i=1

(3)

i----I

Since I(A2j_~) + l(A2j) > I(D2j), (2 < 2j < s)and I(B~) +/(Bz~+x) > l(D~j+~), (3 < 2j + 1 < s), we obtain that

CoherentComponents of AuslanderoReiten Quivers

247

C~CAO + ~C~,))> ~C~,)+ ~(~). /=2

/=1

where M= Ba if s is even, and M= A, if s is odd. So we get from (1), (3) (4) that /(G1) -~ l(E,+l) )_ l(P) + l(E~) + l(Bz) This is impossible because l(P) )_ l+/(Es+x) and l(B1) )_ l+/(Gx). This completes the proof. [] LEMMA 2.6. 1~ contains no translation

sub-quiver in the followin 9 form

,. where P is o~inary projective, Bx is p~jective and At is injective if t is even, and Bt is injective if t is odd. P~of. Using the.len~h ~nction we have the following inequalities:

t(P) + t(Dx)t(S) + t(E), t(n) +t( F~) ~t(P) + t(S) + t(E~)~ t(~x)+t(F~),t(F)+ ;(A~)~ t(J) t(A)+ ~(~)~ t(P)+ Combining the above inequ~ities

we get that

t(n) + t(E1) + t(A~) l( A) ~ l( P) + l( Fo) + l( D~) + Since P is proje~ive we have I(P) ~ 1 + l(A) + l(B). So, we obt~n l(E~) + l(A~) ~ 1 + l(Fo) + l(Da)

(.)

In c~e t = 1, i.e. B~ is projective-injective, then E~ = Cx. So, we have l(Dx) 1 + l(A~) +/(C1). ~om(,) we get 0 ~ 2 l( Fo) + l( J). This is ~ contradiction.

248

Yao In case t > 2, we have l(Di) + l(Ei+l > l( Ai) + l( Bi) + l( Ei), 1 < i Then

we have EI(D~) > E[/CAI) +/(B~)] l( E1) where D, +I = i=l

i=1

If t is odd , le~ t = 2s + 1, ~hen B~ is injec~ive. inequalities:

Wehave the following

l(A2j)+l(A2j+l)> l(Dj+1), < Then we get l(D1) + I(D~+I) > I(A1) + l(Bt) + I(E~). Since Bt is injective we have I(B~) > 1 + I(D~+I). Combining these inequalities with (,) we obtain > 2 + l(J) + t(Fo). A contradiction. Similarly, if t is even, then At is injective, let t = 2s(with s _> 1) , then have

ICA .j+

I(B2j-1)

) > 1 <

I( B2j)_> l( D2j), 1

Wege~ I(D~) ÷ t~D~+~) _> I(AI) + l(At) + I(~i). Since As is injec~ive l(At) > I(~) ÷ l(Dt+~) ÷ I, combining these inequalities with (,) we obtain 0 > 2 + l(Fo) + I(~) + l(J). A contradiction. This completes the proof. LEMMA 2.7. F contains no translation

[]

sub-quiver of the following form

where B1 is projective and I is injective. Proof. Using the length function l we have inequalities as follows l(Di) + l(Ei+l) >/(A,) + l(Bi) l( Ei), 1 < i < s Thus, we obtain s--1

s--1

l(m,) ~’~ l( Di) _>~-~’~[/(Ai) +/( Bi)] +/( i=1

i=1

(1)

CoherentComponents of Auslander-ReitenQuivers

249

Since l(A2j-1) + l(A~j) >_/(D2j) (with 2 _< 2j < s - 1) and l(B~) + l(B~+l) > l(D~i+~) (with 3 < 2j + 1 < s - 1), we then have s--I

s-1

l(D~) + E(I(A,)

+ l(Bi))

>_ El(D,) + l(Bl)

+ (2)

where M= A8_1 if (s - 1) is odd and M= Bs_~ if (s - 1) is even. So, we get the inequality from (1) and (2) as well l(B ~) = 1+ l( l(E,) >_ 1 + l(E,) +

(3)

On the other hand, we have l(D.) + l(A’) >_l(A.) + l(B.)

(4)

l(I) > 1 + l(A’) + l(E,)

(5)

So, we have the following inequality from (3), (4) and l(D,) > 2 ÷ l(A,) ÷ I(B.) ÷

(6)

As l(A._x) + l(A.) > l(D.) and l(Bo_~) ÷ l(B.) > l(Da) as well as M= A._xor B8-1 we get a contradiction from (6). This completes the proof. LEMMA 2.8. F contains no translation

sub-quiver of the following form.

-.-.""

where B1 is projective and Fr is injective; An is injeetive if n is even, and Bn is injective if n is odd. The path between Dn+xand Hr+x is sectional. Proof. In fact, it follows from the proof of Lemma2.7 that the statement of Lemma 2.8 is true if r = 0. So, we mayassume that r > 1. A similar calculation to the one in the proof of Lemma2.7 yields

t(E.+l) _>1 t( E ) + t( M) where M = An if n is odd, and M= Bn if n is even.

Yao

250 On the other hand, we have the following inequalities l(D,+l) + l(Kr) >_ l(E,,+x) + l(H,) + l(F,+~) >_ l(F,) + l(H,+~). Thus, we obtain that l(D,~+l) +/(gr) +/(Fr+l) >/(Fr) + l(Hr+l) l( E,~+l).

(2)

Wethen get from (1) and (2) l(D,+l) -I- [(gr) + [(Fr+l) _> l( Ex) + I( M)l(Hr+~)+ l(Fr)

(3)

By induction on r one can show that I(M) +/(Hr+l) > l(D,+~). So , using the inequality l(F~) > 1 +/(Kr) l( Fr+x) wewil l get a co nt radiction from (3). The proof is completed. [] LEMMA 2.9. F contains no translation

sub-quiver of the following form

where D~, Px and P2 are projective , Ds+~, I~ and I2 are injective. Px, P2, Ix and I~ may not exist. If Px(P2) does not exist, AI(Bx) is projective. If lx(I2) not exist, A,(B,) is injeetive. Proof. Assumer contains such a translation sub-quiver. Using the length function l we have /(Di) +/(Ei+2) _>/(Ai) +/(Bi) l( Ei+i), 1 < i < So we obtain

+

_> i=1

+

+

i----1

Wealso have l(A2j_x) + I(A~) >_ l(D2~) (1 _< 2j - 1 < s) and l(B~) + l(B~j+~) l(D~+x) (2 < 2j < s). Thus, we get from (1) l(Es+2) + l(Dt) > l(E~) + l(M) + l(B~).

(2)

CoherentComponents of Auslander-ReitenQuivers

251

where M= A8 if s is odd, and M = B, if s is even. Now, we have l(G) + l(E~) > l(D:) + l(E:).

(3)

In case s is odd, then M= As. Thus, we have l(A.)+ l(I~)>_/(Ds+l).

(4)

l(P,) + l(B,)> Therefore, we get the following inequalities from (2), (3), (4) and I(Es+~).+ I(G) + l(P1) +/(I1) _) I(E~) + I(D.+I) +/(D1).

(6)

Since l(Dz) > l(P2) + l(P~) + l(G) + 1 and l(D,+:) >_ l(I~) + l(I2) + we get from (6) that E 2 ÷ l( El). Th is is a c ontradiction. In case s is even, then M= B, and we can get a contradiction similarly. This finishes the proof. [] LEMMA 2.10. F contains no translation

sub-quiver of the following form

whereP is ordinary projective and Fr is injective, the path Gx --} P1 = K~---} ... -} Kr+~is sectional. Proof. Without loss of generality we mayassume that there is no projective on the path/(2 -} "" -~ Kr. Weconsider two cases. CaseI. Fr is noton thepathP = Ki -~ ... -} Kr+ior on theraystarting at P. Wethen have l(P) + l(E,.) >_ l(E~) + l(K,.) and I(K,.) +/(F~+I) _> l(K,.+l) + l(F,.), as well as l(G:) + l(E:) >_ l(P) + Thus we obtain that /(G1) -t-/(Fr÷l) Since P is projective,

l(Er) )_/( +/(Fr) + l(Kr÷l)

A is not injective.

So, v-A = K2 by Lemma2.9.

(1)

252

Yao

By induction on r one can show that I(A) + l(Kr+l) >_ I(P). Thus, we obtain that 0 _> 2 + l(G,) since l(Fr) >_ 1 + l(Fr+l) + l(Er) and l(P) >_ 1 + l(G1) + l(A). This is a contradiction. Case II. Fr = Kr or Fr is on the ray starting at P. Firstly assume that Fr = Kr. It is obvious that r ¢ 1 since P is projective. Without loss of generality we may assume that there is no projective or injective on the path K2 ~ ... ~ Kr-1. Since Kr = Fr is injective, Kr+l is not projective. Hence we have the following form.

If rG or

rg +2does not

exist,

we agree l(rG) = orl(r Kr+=) = OThen we have

l(rg2) +/(gr+~) _> l(rK~+2) + l(P)

l(rEd + = l(rG) So, we obtain l(rK~) + t(rE~) + l(E~) + I(Kr+I) >_ t(P) + t(K~) + I(TK~+~) This is impossible since l(P) >_ 1 + I(TK~) + t(TE1) and/(Kr) _> 1 ÷/(Kr+~)

l(Er). If the injective Kr is on the ray starting at P, we can similarly showthat it is impossible. This completes the proof. [] PROPOSITION 2.11. Let r be a coherent connected translation quiver with finite r-orbits. Assumethere is a length function I in r. If or is a connected sub-quiver of r, then the mesh category k(r) contains no oriented cycle of projectives.

CoherentComponents of Auslander-ReitenQuivers

253

Proof. Firstly, by Happel-Preiser-Ringel’s theorem [3] we knowthat Br is a stable tube. Let P0 --A Pl --+ "’" --~ Pt = Po be a cycle of projectives in k(r). Weclaim that there exists another such a cycle of projectives with each p, either exceptional or else such that there exist arrows q --~ x ~ p,, with q injective. Let us consider Po and define a vertex z as follows. If p0 is exceptional and not injective, let x be its unique direct predecessor, and c = ~-tx be such that one of the direct predecessors of c is injective. As r is coherent, there exist a ray [Po, oo) and a coray (oo,~--t+l~]. Set ~ =[po, OO) f~ (oo,~--t+lz]. Let z denote the dire successor of z~ on the ray [Po, oo). There is a sectional path between z and c by Lemma2.7. If P0 is exceptional and injective, let z = c. If P0 is not exceptional, set z = Po. Now,let us consider the set of vertices y on the sectional path from z to the mouthsuch that there exists a ray [9, o~). This set is not emptysince it contains z. Let 9 be a maximalelement in this set ( that is, closer to the mouth)and let [z, denote the sectional path from z to 9. There exists a ray Iv, oo) for each v on [z, 9] by Lemma2.7, 2.8 and 2.10. Let 7~ denote the mesh-complete translation sub-quiver consisting of all vertices lying on these rays. Then T£ C_ SuppHom~(r)(P0,-): is clearif z = Po andz = c. Otherwise it follows fromthef~ctthatc andexactly twoOf itsdirect predecessors (namely, theonelyingon [z,9] , andtheinjective direct predecessor) belong to SuppHomk(r) (P0,

We learnthat7~ contains no exceptional meshfromLemmas2.1-2.6. We claimthat[9,oo)contains eitheran arrowq ---*x, withq injective and z a direct predecessor of a projective p~ , or elsea vertex ¯ beingthedirect predecessor of an exceptional projective//. Assumethatthisis notthecase.Since Hom~(r)(P0,p~) ¢ 0 andtherayswithv on [z,9]existwe musthavethatp~ longsto [c,9]fromLemma2.7,2.8 and2.10.As Homk(r)(pi,pi+~) ~ 0 for 1 < i < t - 2, and7~consists of ordinary meshes. Weconclude thatIvy,. ¯ ¯ ,pt_~ all belong to [c,y].Fromthedefinition of 9 andtheassumption that[9,co)contains no vertices as required, we haveHom~(r)(pt-l,p0) = 0. Thisis a contradiction. Notethatp~,as defined intheprevious claim, istheunique projective vertex having a direct predecessor on[9,oo). For,if thisis notthecase, theexistence of theray~t,oo)implies thatwe havetwocasesin thefollowing.

254

Yao

(i)

(2)

Since there exist rays [l/,oo) and [p’, cx~), the projective p" is exceptional. So, will consider p" in two cases. Case I. f’ is projective-injective. By using length function I we have l(x) +l(w) l(u)+l(p~)+l(p")+l(v) in the diagram(I). Since l(p’) =/(x)+l, and l(p") we obtain that 0 = 2 + l(u) + l(v), a contradiction. A similar contradiction can obtained in the diagram (2). Case II. p" is projective but not injective. Then both of cases in the diagram (1) and (2) can be formed into the case in the following sub-quiver.

with At or Bt injective. Firstly, we have the inequality l(z) + l(Al) > l(p’) + l(D1). Since p’ is projective, we get l~’) > 1 + l(x). Hence we obtain

l(Al)> Z+ Secondly, we have the following inequalities ICD~)+ l(Ei+~) l(Ai) + I (Bi) + I (Ei), 1 < Summingup these inequalities

we get t

~ ICDd) >_ ~-~[/(A~) +/(B,)] ICE~) i=l

~=I

(,)

255

CoherentComponents of Auslander-ReitenQuivers

because D~+i--If t is odd, let t = 2s ÷ 1 for somes _> 1. Wehave the following inequalities.

l(A2j)+ l(A2j+l)> + So, we obt~n that l(D~) +/(Dt+~) ~ l(A~) +/(Bt) + l(E~). ~om Lemma2.1 we . know that Bt is inje~ive ~d hence we have l(B~) ~ 1 + l(Dt+~). ~om (.) we get that 0 ~ 2 + l(E~). Th~ is a contradiction. If t is even, let t = 2s( s ~ 1), then At is inje~ive by Lemma2.1. Using the following inequalities /(A21) l( A2j+l) ~ l( D2j+l), 1 ~ j ~ s l(B2j-~) + l(B2~) ~ l(D2i), 1 ~ we get/(Dx) +/(Dr+l) l( A~) +/(At) +l(E~). Since l( At) ~ 1 +l(Dt+~), wehave that l(D~) ~ l(Ax) + l(E~) + 1. ~om(*) we obt~n 0 ~ 1 + l(~x), a contr~iction. Therefore, p’ is the unique proje~ive vert~ with a dire~ predecessor on the ray Let i be su~ that the projectives p~, .-., pi belong to ~ while p~+~ ~ ~, then either Pi+x = P~ or the morphism pl ~ pi+~ f~tors through p~. If p~+~ = p~ we repl~e Pl,"" ,Pl,P~+~ by f, while if p~ ~ Pi+l factors through p~, we repl~e px,... ,p~ by f. In this way we repine inductively the given cycle by a cycle of proje~iv~ satisfying the required property. A~u~ly we have obt~ned a cycle of the following form "’"

~Ps

~’"~

Zs

~’"~ys

~’"~Ps+l

~""

,

where we have sectional paths ~,, zs] pointing to infinity , [z,, y,] pointing to the mouth, and [y~, x,] (where zs is a dire~ predecessor of ps+~ on this cycle) pointing to infinity. For ea~ s, let u, denote the dire~ predec~sor of z, on the ray [Ys-~, ~). Then we have a cycle

where paths correspond to sectional paths, [us, y,] pointing to the mouth, [y,, us+l] pointing to infinity. Wewill get contradiction by carrying over the corresponding argument verbatim in the proof of Proposition 4.5 in [1]. This completes the proof. [] REMARK. It follows from the proof that all projectives in P lie above some cyclical path, and consequently F has only finitely manyprojectives. 3

THE

PROOF

OF

THEOREM

A

Firstly, we introduce several kinds of cancellations in a translation quiver r -(r0, rl, r). Let k(r) denote its mesh-category. 1). Let p be an ordinary projective vertex in r. If SuppHom~(r)(p,-) consists of a mesh-complete translation sub-quiver of F of the form of the vertex x~~ and z~j

256

Yao

Let 7~ be the set of the vertices in F lying on the sectional paths from the mouth to infinity passing through z0,, z02, .. ¯, Zo~and x~. Let F’ be the translation quiver obtained from F by deleting SuppHom~(r)(p, -) and replacing the sectional paths xi --~ zil ~ "’" --+ ~i~ --~.~ ~ ci( if they exist) by arrows zi ~-~ ci, i _> we define z’c~ = xi_l for i = 1, 2, .., and for the other vertex x in F’ we define z’x = rx. Wethen say that F’ is obtained from F by a(t + 1)- cancellation at the vertex p. 2). Let p be a projective-injective vertex in F. If SuppHom~(r)(p, -) consists of the vertices z~ and z~ of a mesh-complete translation sub-quiver of the form.

Denoteby r’ the translation quiverobtained by deleting SuppHom~(r) (p, -) replacing the sectional pathsxl ~ .."~ ci-~( If theyexist) by arrowsxi ci-~fori _> 2. Define ~"ci= zi fori _> 2 , x0 is injective and~’~= ~-zforthe otherx in rt We thensaythatr’ is obtained fromr by fl(t+ 1)-cancellation thevertex p. 3).Letp be exceptional projective butnotinjective inr. If SuppHom~(r) (p, consists ofthevertices x~andzi~of a mesh-complete translation sub-quiver of F of the form

Coherent Componentsof Auslander-Reiten Quivers

if t is odd,or

if t is even.

257

258

Yao

Denote by T~ the set of vertices in F of the forms x~, i _> 0, z~j, i _> 1, 1 < j < t and by F’ the translation quiver obtained from r by deleting 7£, and replacing the sectional paths x~ --+ ... ~ Y~+x by arrowsx~ --~ Yi+~ (0 < i _< t-l). The sectional paths yl ~ zii ~ yi+~ by arrows Fi ---+ Yi+l (1 < i < t- 1) and the sectional pathsx~ ~-~...~ z~ --~ci-i( if theyexist) by arrowsx~ --~c~-1,i E t + 1. Hencer’ is of the rightform.We thensaythatr’ is obtained from r by7(t+ 1)-cancellation at thevertex

REMARK. i).Thedualof a(t+1)-cancellation,/~(t+ 1)-cancellation and~,(t+ cancellation willbe called a*(t+ 1)-cancellation and/~*(t+ 1)-cancellation 7*(t+ 1)-cancellation, respectively. ii).Assume thatthere is a length function I ina translation quiver r,letr’be a translation quiver obtained fromr by oneof theabovesixkindsof cancellations, then if set l’(z) = l(z) for any z in r’, l’ is a length function in r’. For the convenience of statement we introduce the following notations. Let r be a translation quiver, SuppHomk(r)(p,-) consist of vertices x~, i and zij, i E O, 1 < j < t of a mesh-completetranslation sub-quiver of r, if we have the form of r as follows,

’ i>_0, z~j,z "_>O,l<_j 1. define ~-~ci = xi_~ for i >_ 1, and ~-~x = ~-x for any other x in F~. Wethen say that I ’~ is obtained from r by ~(t + 1)-cancellation at

CoherentComponents of Auslander-ReitenQuivers

259

In fact, I" is obtained from I’ by (t-1) times successively, and one time of a(2)-cancellation at p, finally. The dual of an ~(t + 1)-cancellation is called an if* (t + 1)-cancellation. Wenow set up to prove Theorem A and its corollaries. Corollary B is the immediate consequence of Theorem A.

Proof. of Corollary C. If r is a coil, then F is coherent by theorem 4.2 in [1]. If each DTr-orbit in P is finite, then OFis a stable tube, and so , 0P is connected. Therefore, we get corollary C from theorem A. [] In order to prove TheoremA , it is sufficient by Lemma1.1, 1.2 and 1.3.

to prove the following theorem

THEOPd~M 3.1. Let F be a coherent connected translation quiver with a length .function I. l.f OI’ is connected, then F can be obtained from a stable tube by finitely many times of multiple admissible coray-ray insertions if and only if each r-orbit in F is finite. ttEMARK. The idea of the following proof is similar to 4.6 in [1]. Here we use the methodso called ’cancellation’. Proof. Necessity. From the definition of multiple admissible coray-ray insertions we learn that each ~--orbit in 1~ is finite if I’ is obtained from stable tube by only one time of multiple admissible coray-ray insertion. Hence , by induction on the number of multiple admissible coray-ray insertions one can show that each r-orbit in [’ is finite. Sufficiency. Wehave learned that 1" contains only finitely many projective vertices from the remark of Proposition 2.11. So we will show the theorem by induction on the number of projective vertices in r. By Proposition 2.11 there exists a projective vertex p E r0 such that SuppHom~(r)09, -) contains no projective vertex. Wewill consider p in two cases. Case I. Assume that SuppHom~(r)(-,p) contains no projective vertex. Ifp is an ordinary projective vertex, consider the sectional path from p pointing to the mouth p = a0 --~ al --~ ... ~ at( denoted by Lp+) with at lying on the mouth. Let s be the largest index such that there exists a projective vertex p~ E P0 and a sectional path [p~, as] pointing to infinity. Obviously we can choose p~ so that its successors in ~, ae] are not projective. Observe that SuppHom~(r)(p~, -) contains no projective vertex. Indeed, by the definition of s, no projective vertex lies on a sectional path pointing to infinity and passing through at, s < r _~ t. Moreover, by the assumption on p, the sectional path [at, o~) contains no direct predecessor of a projective vertex. Also, p~ is ordinary. In fact, if p ~ p~,then Hom~(r)(p,p~) -- 0. Thus p~ lies above the sectional path [p, at] (denoted as L+~). On the other hand, there exists a maximal sectional path (denoted by L~-) from mouth to the projective p = ao. By Lemma2:10, any vertex in SuppHom~(r)(p, -) is on one of rays starting a point on [P, at]. Similar to the proof of Proposition 2.11 one can show that

260

Yao

SuppHom~(r)(p, -) does not contain an exceptional mesh. Thus , if p’ is exceptional, the injective vertex P in the v-orbit of p~ = b0 must be above the sectional path L~+. Since p = ao is ordinary and OF is a connected sub-quiver of F, ’the vertices (roughly speaking, i.e. vertices between the paths L~- and Lp+) on the sectional paths ending in the vertices ai and pointing to infinity are not v-periodic for i = 0, 1,... ,t. So, the sectional path ending in the injective vertex I ~ must be finite. This contradicts the condition that r is coherent. Thus, p~ is ordinary. Therefore, F contains a mesh-complete translation sub-quiver of the form:

As each v-orbit in r is finite, each v-orbit of a projective vertex contains an injective vertex and each v-orbit of an injective vertex contains a projective vertex. Thus, by Lemma2.9 , we have projective vertices bo, bl,... ,bin and injective vertices Uo, U~,"" ,ur such that nCL+uo)+ ... + nCLu+,) = nCL~o) +... + nCL~,,,), where nCL~) and n(L~) are the numbers of vertices on Lu+, and L~for 0 < i < r and 0 _~ j _< m, respectively.Here Lu+~is a maximalsectional path starting at ui and pointing to the mouth and each vertex in L~+i is injective for i = 0,... , r and L~ is a maximalsectional path from the mouth to bj, and each vertex on L~ is projective. Without loss of generality we may assume r = 0 and m = 0 i.e. n(L+uo) n(L~o), denoted by no(if r ~ 0 or m ~ 0, the proof is similar as in case r =: and m = 0, but the statement will be tediously long). So, we can make an cancellation at p~ = bo to obtain a translation quiver of r * = (Fo, Ft , v ), whichis by the definition of a(no)-Cancellation, left stable except for finitely manyC-orbits containing both projectives and injectives. Then , we make an a*(no)-cancellation at uo in ~ t o o btain a tr anslation quiver F" = (rg, r~,v"), which is stable, except for finitely manyv’-orbits conraining both projectives and injectives, by the definition of a*(no)-insertion and a(no)-insertion. So, there must exist the following sub-quiver in

CoherentComponents of Auslander-ReitenQuivers

261

whereviandxi belong to r~ fori = I, 2 andthereisa positive integer r suchthat ~-rx2= vl,andanyothervertices in theabovesub-quiver liein r0\r .Moreover, " vl andxl areY-periodic. So,we havea sub-quiver in r" as follows

and T"~2= 1)1. Hence, v~ and x~ are ~"-periodic in r" since vl and x~ are Y-periodic in r.( In fact, let the r-period of x2 is n, then r"-rvi = xi. So, (r")"-r+~x2 = (r’)~-rvi ~,-rvl = x~) Thus, from Happel-Preiser-Ringel’s Theorem Is] and the condition that Dr is connected we learn that each r"-orbit in r" is finite. Obviously, r" is coherent by the definition of a(n0)-insertion and a* (n0)-insertion. It is also evident that r" has at least one projective less than r and still there is a length function l" in r’. If p is exceptional we have two cases to consider. (i). Suppose that p is injective, then SuppHom~(r)(p, -) and SuppHom~(r)(-,p) are mesh-completetranslation sub-quivers of r of the following form.

262

Yao

So, we make a f~(n0)-cancellation at p = x~ in r to obtain a translation quiver -~ (ro,rl,-r) where no = t + 1. Thus, r’ is left stable except for finitely many v’-orbits containing both projectives and injectives. Again we make an a*(n0)-cancellation at xl in r’ to obtain a translation quiver r" = ~-o,-1, rr,- ~where no = t + 1. Thus, F" is stable except for finitely manyv’- orbits containing both projectives and injectives , by the definition of a(no)-insertion and fP(no)-insertion. Similarly, there must exist the following sub-quiver in r

where vi and x~ belong to r~~ for i = 1, 2, and there is a positive integer r such that vrx2 = vl, and any other vertices in the sub-quiver lie in Fo\F~. Moreover, vl and x2 are v-periodic. So, we have sub-quiver in F" as follows

and1TI’.

~2 "-~ I)

Hence, vl and x2 are v’-periodic in F" since they are v-periodic in F. Then, by Happel-Preiser-Ringel’s Theorem and the condition that OFis connected we learn that each v’-orbit in F" is finite. Obviously, F" is coherent by the definitions of a*(no)- and/~(n0)-insertions. F" has at least one projective less than F, and still there is a length function 1" in (ii) Suppose that p is not injective, then by the Lemmas2.1-2.4 and 2.7-2.8, infer that F has the following translation sub-quivers.

Figure 3.1 if t is odd, or

CoherentComponents of Auslander-ReitenQuivers

263

Figure 3.2 if t is even.

We make a 7(no)-cancellation, where no = t+ 1, at the vertex xo in F to obtain a translation quiver ’ ’ ’ r’ = (l~o,I~l,r). Hence , r’ is of the right form. By the definition of 7(no)insertion we knowthat F’ is left stable except for finitely manyr’-orbits conraining both projectives and injectives.

Weagain make an i~(n0)-cancellation, where no = t + 1, at the vertex to obtain a translation quiver r"= ~-0 t~" ,-1, ~" v"), which is stable except for finitely manyr"-orbits containing both projectives and injectives. Similar to the above, each ~"-orbit in r" is finite and there is a length function l" in r". Obviously, r- is coherent, and r" has at least one projective less than r. Case II. There is no projective p such that SuppHomk(r)(-,p) contains no jective. So we assume that there are projectives Pl,P~,.." ,P,~ such that p~ E SuppHom~(r)(pi-1,-) for i = 2,... ,m and both SuppHom~(r)(-,pl) SuppHom~(r)(p,,, -) contain no projective. Wemake one of the following cancellations at the vertex Pm in F. (1). If P,n is ordinary, we consider the sectional path from p,~ to the mouth Pm=ao ~ a~ .-~ ... ~ at, with at lying on the mouth. Because of the Case I and without loss of generality we mayassumethat no projective lies on the sectional

264

Yao

pathstarting fromthemouthandending in ai(i= i, 2,..., t). Letnm = n(L~-~,), whereL~-.,is a sectional pathfromthemouthto thevertex Pro.Sinceeachv-orbit inF is finite andwithout lossof generality, we mayassume thatthereexists an injective Im suchthatnm= n(L+~,,,) whereL~+,,is a sectional pathfromIm to themouth, andn(L+~,~) (orn(L’~,~)) is thenumberof vertices on the pathL~.~( or L~-,).Hencewe makean a(nm)-cancellation at the vertex to obtain a translation quiver r(I)(I)r0) v(~) ~which is lef t sta ble exc ept for finitely manyv0)-orbits containing bothprojectives andinjectives, bythedefinition of a(nm)-insertion. (2).If Pm is exceptional andinjective, without lossof generality, we may assumethatno projective lieson thesectional pathsstarting fromthemouthand passing through thevertices on L,~p.), wheres(p~)is theuniquedirectsuccessor of p~. So,we makea ~5(n~)-cancellation at the vertexp,~,wherenm = n(L~p~)), to obtain a translation quiver r(I) -- ~om the definition of /~(~m)-insertion we learnthatr(1)is leftv(1)-stable except forfinitely m~ny orbits containing bothprojectives andinjectives. (3).Ifproisexceptional butnotinjective thenby Lemma2.1-2.4 and2.7-2.8 inferthatY hasa trans:ationquiver as intheTigure 3.’_ant_"Tigure 3.2intheCase I. So we makea 7(~m)-cancellation at thevertex~o in r to obtaina translation quiverto) = ~r(~) r(~) ~(~)~whi&is le~~(D-stable, exceptfor finitely ~O)-orbits cont~ning both proje~iv~ ~d inje~iv~, by the de~nition of ~(nm)inse~ion where n~ = t + 1. From the definition of a(nm)- and ~O(nm)- and 7(nm)-cancellation we that p~,... ,P,n are still projectives in tO) and they satisfy the property that p~ SuppHom~(ro D (p~,-) fori ----2,...,m- 1, andSuppHom~(r~) ) (-,p~)contains projective andSuppHom~(ro) (Pro-i, -) contains no projective. ) Inductively we makea corresponding cancellation atthevertex p~in thetranslation quiver r(~-~)according asp~ is ordinary or exceptional. So,at thera-thstepwe obtain a translation quiver r( whichis leftV(m)-stable exceptforfinitely manyv(m)-orbits containing both jectives andinjectives. Thereisa length function ~(m)in is coherent. Wewill makeone of the following cancellations in F(m). (I).Ifplis ordinary in r thenthereareinjectives Ii,I~,... ,I~,suchthat n(L~) +...+n(L~+~) = n(L~ ) +..-+n(L~) since each v-orbit is finite in r. loss of generality we may assume u = 1 and r = 1. So we have n(L~) = n(L~.). Thus, n~ = n(L+r, ). As we have made an a(n~)-cancellation in (’~-1) t o o btain F(m), we have a translation sub-quiver in the following form in "(m)

CoherentComponents of Auslander-ReitenQuivers

265

Figure 3.3 Thus we make an a*(nl)-cancellation at I1 in (m) t o o btain a tr anslation qu iver (m) ~r(m). = ~(m) o , ~r1 ,0r(m)). Obviously (m) is coherent and ther e is a len gt h (m) (m) function l in @r (2). If pl is exceptional and injective, we also have the translation sub-quiver as Figure 3.3. So we also make an a*(ni)-cancellation at the vertex I1 to obtain a translation quiver @r(m) -- (~r(0~), m),Dr(m)). Obviously Dr ( m) is coherent and there is a length function Dl(m) in ~gr(m). (3). If pl is exceptional and not injective , there is a translation sub-quiver in the form as in Figure 3.1 and Figure 3.2 in Case I. So we have the following translation sub-quiver in r(m).

So we make ~* (nl)-cancellation in (m) : (Dr(o m), Dr ~m), Dr(m)) to obtain a t ranslation quiver Dr(m). Obviously, Dr(m) is coherent, and there is a length function Dl(m) in Dr(m). Inductively, we make corresponding cancellation in D(~-l)r (m) ac(m), cording as pi is ordinary or exceptional in r to obtain a translation quiver D(0r and finally at the m-th step we obtain a translation quiver ~9(m)r(m), which is coherent and has a length function D(m)/(m). Furthermore we know that D(m)r(m) is D(m)-stable except for finitely many~’(m)- orbits containing both projectives

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Yao

injectives by the definitions of a(nl)-cancellation and a*(n~)-cancellation cancellation and a* (ni)-cancellation as well as 7(ni)-cancellation and ~* (n~)cellation. It is easy to see that there is at least one O(m)~(m)_periodicvertex O(m ) (r( m) ). Thus , each O(m)~(’~)-orbit is finite by applying Happel-Preiser- Ringel’s theorem to the translation quiver O(m)r(m). If we still write O(m)r (m) as r ~, then I" has at least One projective less than r. By the definitions of cancellations we knowthat or ~ obtained by deleting C-orbits of projectives is a connected translation subquiver of r’. So, by the assumption of induction r’ is obtained from a stable tube by finitely manymultiple admissible coray-ray insertions. Since r can be obtained from r ~ by one time of multiple admissible coray-ray insertion. Wecomplete the proof. ¯ []

4

AN EXAMPLE

In this section we will give an exampleof a finite-dimensional algebra A over an algebraically closed field which has a connected componentsatisfying the conditions in the main theorem. Let A be given by the quiver

bound by ~zw = O, oq = O, #A = O, up = 0 and ~)~ = ~z~p. Then FA has connected component as follows:

CoherentComponents of Auslander-Reiten Quivers

267

whereindecomposable modules arerepresented by theirdimension-vectors and one identifies along thedashlines. ACKNOWLEDGEMENT The authorwouldliketo expresshis gratitude to the University of Sherbrooke forherhospitality during hisvisit.Theauthor wouldalsoliketothankprofessor I.Assem forhismanyusefulsuggestions anddiscussions. Finally, theauthorwould liketo thankthereferee for thecomments. Thisworkis partially supported by BeijingYouthFund. REFERENCES Multi-coil algebras, Proceedings of ICRAVI, [I]I. Assemand A. Skowrofiski, Canadian Math.Soc.Conference Proceedings, 14(1993), 29-68 [2]G. D’EsteandC. M. Ringel, Coherent tubes,J. Algebra 87(1984), 150-201 U. Preiser andC. M. Ringel, Vinberg’s characterization of Dynkin [3]D. Happel, diagrams usingsubadditive functions withapplication to DTr-periodic modules.Lecture Notesin Mathematics 832Springer, Berlin, 1980,280-294. components of an Auslander-Reiten quiver.J. London [4}S. Liu,Semi-stable Math.Soc.47(1993), 405-416 components of theAuslander-Reiten quiverof a tilted [51S. Liu,Theconnected algebra, J. Algebra, (2)161(1995), 505-523 [6] S. Liu, Shapes of Connected components of the Auslander-Reiten quivers of Artin Algebras. C. M. S. Conference Proceedings, Vol. 19(1996) 109-137 [7] P. Malicki, Generalized coil enlargements of algebras, Colloq. Math. Vol. 76, No. 1(1998) 57-63 [8] P. Malicki and A. Skowrofiski, Almost cyclic Auslander-Reiten quiver, Preprint

coherent components of an

[9] C. M. Ringel, Tamealgebras and integral quadratic forms, Lecture Notes in Mathematics 1099, Springer, Berlin, 1984 [10] A. Skowrofski, Cycles in module categories, Finite Dimensional Algebras and Related Topics, NATOASI Series, Series C (Kluwer Academic Publishers), 424(1994), 309-346 [11] H. Yao, Infinite connected components of an Auslander-Reiten quiver in which each DTr- orbit contains only finitely manypoints, comm.Algebra, 1999, Vol 27 (11),5167-5189 [12] Y. Zhang, The structure 672.

of stable components, Can. J. Math. 43(1991),652-

Twisted Hopf algebras PU ZHANG Department of Mathematics, University of Science and Technology of China, Hefei 230026, P R China, E-mail: [email protected]

LI-BIN LI Department of Mathematics, University of Science and Technology of China, Hefei 230026, P R China, E-mail: [email protected]

ABSTRACT The aim of this paper is to introduce the concept of a twisted Hopf algebra, and then to discuss some properties and to give some constructions of twisted Hopf algebras. Such a twisted Hopf algebra turns out to rise naturally, for example, the positive part U+ of a quantized enveloping algebra U, and the Ringel-Hall algebras. INTRODUCTION Throughout this paper, let K be a field, c a non-zero element in K, and I a set. Denote by ZI the free abeli~n group with I as basis, whose element is written as x = (x~)~el with x~ E Z and x~ = 0 for almost all i E I, and by l~10I the subset { x ~- (xi)iex ~ EI I xi ~ l~10 }. Let (X1,X2)be a pair of Z-valued bilinear forms ZI. The aim of this paper is to introduce the concept of a (K,c, I, (X1,X~))-Hopf algebra (or simply, a twisted Hopf algebra), and then to discuss its properties and constructions. It is well known that a quantized enveloping algebra U = U+ ® U0 ® U_ has a Hopf structure, and that the Hopf operations for U are not closed inside the positive part U+(see e.g. [J]), so we naturally hope that inside U+there is a "nearly" Hopf algebra structure, this is a motivation of introducing a twisted Hopf algebra. This is particularly natural whenwe use Ringel-Hall algebras to study U+(cf. [R2], [G]).

The authors

gratefully

acknowledge

the support

of K. C. Wong Education

269

Foundation,

Hong Kong.

270

Zhangand Li

The axiomatic difference between a twisted Hopf algebra from a Hopf algebra lies on the axiomof the comultiplication, i.e., the comultiplication 6 : A ---+ A®Aof a twisted Hopf algebra A is an algebra homomorphism,not for the componentwise multiplication on A ® A, but for the multiplication on A ® A given by a twisted rule via a pair (X1, X2) of bilinear forms. This idea of a twisted multiplication should not be considered as a disadvantage, on the contrary, it can be used for any algebra and any coalgebra, as Lusztig (ILl) and Ringel (JR1, R2]) did, see [G]. Moreover, as we will showin Example2.6, for any datum k, c, I, X1, X2, there always exists a (K, c, I, (X1, X2))-Hopfalgebra. It is proved in Theorem2.3 that (K, c, I, (X1, X2))-bialgebra is always a (K, c, I, (X1, X2))-Hopfalgebra. In 2.4 we point out that the antipode s of a (K, c, I, (X, 0))-Hopf algebra A gives NoI-graded algebra anti-isomorphism s : A --~ Axr and an NoI-graded coalgebra anti-isomorphism s : A~ ---~ A. Given a twisted Hopf algebra, by twisting its multiplication or comultiplication, we construct some new twisted Hopf algebras and discuss some relations of their antipodes in §3. Throughout this paper, let X : ZI ® ZI ~ Z be a bilinear form. Such a bilinear form is not symmetric if no otherwise is stated. By XT we denote the Z-valued bilinear form on Z given by xT(x, y) = X(Y, 1

TWISTING

1.1.

AN ALGEBRA

AND A COALGEBRA

Consider an NoI-graded K-algebra

A = (A,m,e),

i.e.,

A = (~ Ax with XENol

Ao = K is a direct decomposition of K-spaces, such that A~Au C_ A~+u. If a E A~, then we denote by x by lal. Define a new multiplication * on A by (1)

a ¯ b = cX(lablbDab for homogeneous elements a, b in A. Denote this multiplication we have

map by mx. Then

LEMMA. ([R1]) x =(A, re x, e) is again an NoI-graded K-a lgebra. 1.2. By definition (see e.g. [R3], p.206), an NoI-graded K-coalgebra A = (A, 6,¢) is a direct decomposition of K-spaces A = (~ A, with A0 = K, such that xENol

,

(i) there is a K-linear map 6 : A ---~ A ® A which is coassociative,

i.e.

(id ®6)6= (6®id)6; (ii) the projection ¢ from A onto Ao = K is a counit, i.e. (¢ ® id)6; (iii) 6 respects the grading, i.e. 6(Ad)C= ~ A~ ® Au. x+y----d

Weneed the following easy fact (see also [LZ]) LEMMA. Let A = (A, 6, ¢) be an NoI-graded K-coalgebra. Then

(i) 6(1)= 1 (ii)

For a E Aa with O ~ d ~ NoI we have 5(a)=a®l+l®a+

"Z

ax®bu

x+y=d;z,y~O

(id ® ¢)6 = id

TwistedHopfAlgebras

271

where ax E A~, b~ ~ Ay. In particular, we have 5(a) = a ® l + l ® a for a ~ Ai, i ~ 1.3. Let A = (A, 5, ¢) be an NoI-graded K- coalgebra. Define a new K-linear map ~f~ : A --~ A ® A by (i x (a) = E cX(la’l’la21)(at

® a2)

(2)

where 5(a) = ~ al ® a2 is Sweedler’s notation with all factors at, a2 homogeneous. By a direct verification we have the following (see also [LZ]) LEMMA. Ax = (A,5~,¢)

is again an NoI-graded K-coalgebra.

Note that the construction of gx has been introduced by Lusztig for ’f and f in ILl, p.6. LEMMA 1.4. Let x : ZI x ZI --4 Z be a bilinear form, and A = (A,5,¢) be NoI-graded coalgebra. Then xA = (A, xS, ~) is again an l~oI-graded coalgebra, the K-linear map xS: A ---~ A ® A is defined by

=c- (1 11,121)(as®at),

(3)

where 5(a) = ~ at ® as with all factors homogeneous. Proof. Note that x5 is exactly TS_x, where T is the twisted map given by T(a®b) b ® a. Thus, the assertion follows from Lemma1.3 and the fact that if (A, 5, ~) an NoI-graded coalgebra, then so is (A, TS, ¢). 1.5. Let A be an NoI-graded algebra. Then A ® A is an (l~IoI)~-graded algebra with componentwise multiplication, where (A ® A)(,,u) = A~ ® Au for x, y ~ Note that we can identify (ZI) 2 = ZI ~ ZI with the free ab~lian group with F as basis, and (NoI)~ = 510I’, where I’ is a set with IIq = 211I. Thus, by Lemma 1.1, in order to twist the tensor algebra A ® A, one needs to have a Z-valued bilinear form X on (ZI) ~, i.e., a mapX : (ZI) ~ ~-~ Z satisfying

=

+ xt,. +

+

(4)

+ X( ¯ 1,X ~,Yt,Y~)" ~ ~

(5)

and X(xt,x~,yt

~ + Yt,Y2 + Y~) = X(Xt,x2,Yl,Y~)

However,in order to deal with a twisted bialgebra and a twisted Hopf algebra, this is not enough. As observed by Ringel ([R3], p. 227), one should consider a bilinear form on (ZI) 2 given by a pair X = (Xt,X~) of bilinear forms on ZI, again denoted by X, i.e., X(zt,x:,y~,y~) = Xt(x~,y~) X~(x~,y~). 4 Such a map X : (ZI) --4 Z satisfies not only (4) and (5), but X(Xl

-{-Xi,~2,yl

+y~,y2

)

= X(Xl,x2,Yl,y2)

+X(x~,x2,yl,y2)

(6)

(7)

272

Zhangand Li

and ~ X(Xl,X2 + x~2,yl,y2 + Y~2)= X(Xl,X2,y~,Y2) + X(x~,x2,Yl,Y2) .I Moreover, any mapX : (ZI) 4 ~ Z with properties (4), (5), (7), (8) is given

(8)

The reason we need not only (4) and (5), but also (7) and (8), is that really used in Example2.6, which guarantees the existence of a twisted Hopf algebra. 1.6. Let A be an NoI-graded algebra, and X = (Xx, X2) be a pair of bilinear forms on ZI. Using (6) and applying Lemma1.1 to the tensor algebra A ® A, we obtain the (NoI)2-graded algebra (A ® A)x with multiplication ¯ given by (a~ ® a2) * (bl ® b2) =c~(l’~21’lb~l)+~(lall’lbal)(albl

(9)

for homogeneouselements a~, a2, bl, b~ E A. 1.7. Dually, let A = (A, 5,e) be an NoI-graded coalgebra. Then A ® A is (NoI)2-graded coalgebra with comultiplication (id ® T ® id)(5 ® 5), again denoted by ~ if no confusion caused, where T is the twisted map. Thus, if 5(a) = ~ a~ ® and (i(b) = ~ b~ ® b~, then 5(a ® b) = ~ al ® bl ® a2 ® b2. Nowusing (6) and applying Lemma1.3 to the tensor coalgebra A®A, we obtain the (NoI)2-graded coalgebra (A @A)x = (A ® A, 5x, e ® e) with comultiplication Jx(a ®b) : ~ C.X~([b~i’i’~al)-t’~’a(la~l’lb~l)(al ®bl ®a2

(10)

where (i(a) = ~ a~ ® a~ and 5(b) = ~ b~ @b2 with all al, a2, bl, b2 homogeneous. 1.8. In many situations, we shall choose X2 = 0 in the pair X = (X~,X2). In this case, for convenience, we replace X~ by X, i.e., let X be a Z-valued bilinear form on ZI. Thus, the algebra structure on (A ® A)x is given by Lusztig’s rule (al ® a2) * (bl ® b2) cX(la2I’lb~l)(alb~ ® a2b2)

(11)

for homogeneouselements al, a~, bl, b2 E A. Also, the coalgebra structure

on (A ® A)x is given

5x(a ® b) = cx(Ibtl’la~l)(at ® b~ ® a2

(12)

where 5(a) = ~ al ® a2 and 5(b) = ~’~ b~ ® b2 with all al, a2, bl, b~ homogeneous. 2

TWISTED

HOPF

ALGEBRAS

2.1. Let A = (A,m,e) be an l~IoI-graded algebra and A = (A,5,¢) be an No/graded coalgebra. Then it is clear by definition that ~ : A --~ K is an algebra homomorphism, and e : K ---+ A is a coalgebra homomorphism.Let X = (X~,)/2) be a pair of Z-valued bilinear forms on ZI. Then we have the following fact

TwistedHopfAlgebras

273

LEMMA.5 : A -~ (A ® A)x is an algebra homomorphism if and only rn (A ® A)(xT,x2) --~ A is a coalgebra homomorphism,where the algebra structure (A ® A)x and the coalgebra structure on (A ® A)(xr~ ) are"given via (9) and (10) in 1.6 and 1.7, respectively. Proof. It is clear that ~m= e ® ~, and 6(1) = 1 ® 1. For any a, b E A, let ~(a) ~ al ®a2, 8(b) = ~ bl ®b2, with all factors homogeneous. Now, ~ : A --~ (A®A)x is an algebra homomorphism if and only if 6(ab) = 8(a) * ~(b), and if and only if

= (m

®b),

which is exactly the claim that m : (A ® A)(~r,~) ~ A is a coalgebra homomorphism. [] Let X = (X~, X2) be a pair of Z-valued bilinear forms on ZI.

2.2.

DEFINITION:(i) An NoI-graded K-algebra A = (A, rn, e) together with an graded K-coalgebra A = (A, ~,e) is called a (K, c, I, x)-bialgebra, or simply, x-bialgebra, provided that ~ : A ~ (A ® A)x is an algebra homomorphism, where the algebra structure on (A ® A)x is given by (9) in (ii) A (K, c, I, x)-bialgebra A = (A, m, e, 6, e) is called a (K, c, I, x)-Hopf bra, or simply, a x-Hopf algebra, provided that there is a K-linear map s : A ~ A satisfying m(id ® s)6 = ee = m(s ® id)~.

(1)

The map s is called an antipode of A. The notion of a x-bialgebra has been introduced by Ringel in [R3], and a x-Hopf algebra has been introduced in [LZ] (but in [LZ] we only consider the case X = (X1,0) with X~ symmetric). Notice that, to say A = (A, m, e, 6, e) a (K, c, I,x)-bialgebra, is equivalent to say it is (K,c-~,I,-x)-bialgebra, where

= Recall that for any K-algebra A = (A, m, e) and any K-coalgebra C = (C, the K-space HomK(C,A) become an K-(associative) algebra with identity ee the convolution * defined by

(e) for f, g E Homg(C,A) and x ~ C, see IS]. Thus, for a (K, c, I, x)-bialgebra (A, m, e, 6, e), it is easy to see that existence an antipode above is equivalent to the existence of a K-maps with the following property in the convolution algebra Hom~(A, A) s,id=id,s=ee. It follows that a (K, c, I, x)-Hopf algebra has a unique antipode.

(3)

274

Zhangand Li

Wehave the following basic property of a twisted bialgebra THEOREM 2.3. Let X = (X1,X2) be an arbitrary pair of Z-valued bilinear forms on ZI, and A = (A, m, e, 5, ~) be a (K, c, I, x)-bialgebra. Then there is a graded K-maps : A --~ A such that A = (A, m, e, 5, ~, s) is a (K, c, I, x)-Hopf algebra. Proof.

Let A = (~ A~. Define

a K-map Sr : A ~ A inductively.

Define

x~Nol

s~(1) = 1. For a ~ A~, 0 ~ d 6 NoI, by LemmaL2 we have 6(a)

=a@l

+ l@a+

~ ax@bu

x+y=d;x,y~O

where a~ ~ Ax, bu ~ An. By induction sr(bu) has been well defined, it follows that we define st(a) = -a ~ axsr(by). Thus, we have m(id @sr)$ = ee, i.e., sr is a right inverse of id in the convolution algebra Hom~(A,A). Similarly, one has a left inverse s~ of id. It follows that Sr = s~ = s, i.e., s is the antipode of A. 2.4. Nowconsider an important speciM c~e, i.e., X = (X~,0). In this case, shall use (11) and (12) in 1.8, and we have the following basic property THEOREM. Let X : ZI × ZI ---~ Z be a bilinear form. Let A = (A, m, e, 5, e, s) be a (K, c, I, x)-Hopf algebra. Then we have (i) s : A ~ A~" is an NoI-graded algebra anti-isomorphism, where xT (x, y) X(y,x) for x,y E l~oI, and the algebra structure on A~r is given via (1) in 1.1. (ii) s : A~ --+ A is an NoI-graded coalgebra anti-isomorphism, where the coalgebra structure on Ax is given by (2) in 1.3. (iii) xrA = (A,m,e, xrS, e) is a (K,c,I,-xT)-Hopf algebra with antipode s-~, where the coalgebra structure on xrA is given by (3) in 1.4. Proof. By Theorem 2.3 we know that s is graded. In order to prove (i), we first prove that s : A ~ Axr is an algebra antihomomorphism. Since s(1) = 1, it suffices to prove s(ab) = s(b) ¯ s(a) for a, b E A, or, sm = mxr (s ® s)T, where T is the twisted map and mxr is the multiplication in Axr. Since e(e ® e) is the unit of the convolution algebra HomK((A® A)~r, A), follows that it suffices to prove that there holds (sin), m = m * (m~r (s ® s)T) e(e ® ~) in the convolution algebra HomK((A® A)xr, Let a, b E A be homogeneous,5(a) = ~ a~ ® a2, and 5(b) ~’~ b~® b: with all factors homogeneous. Since 5 : A ---~ (A ® A)x is an algebra homomorphism, follows that (i(ab) = ~c~(l~l’lb~l)(a~b~ ® a2b2), and hence by applying (12) in 1.8 we have ((sm).m)(a = m(sm ® m)tf~r (a ® b) : ~ c~(Ibal’la~l) s(albl)a2b~ = ~ c x(la~l’lb~l) s(a~b~)a2b2 = m(s ® id)ti(ab) = ~(ab) = (e(~ ® ~))(a

TwistedHopfAlgebras

275

On the other hand, we have

it follows that if b ~ A0, then e(b) = 0, and hence (m, (mx~.(s ® s)T))(a ® b) = 0 = (e(e ® e))(a and if b e A0, then x(la2[, [b[) = 0, and hence (m * (mxT (s ® s)T))(a

=~-~als(a2)e(b) =m(id®s)6(a)e(b) =(e(e@e))(a®b).

This proves that s : A ~ Axr is an algebra anti-homomorphism. In order to prove (ii), we first prove that s : x ---~ Ais a c oalgebra ant ihomomorphism.It is clear that es = e. It remains to prove that 6s = T(s ® x. s)6 Since (e ® e)e is the unit of the convolution algebra HomK(A, (A ® A)x), it follows that it suffices to prove (6s), (i = 6, (T(s ® s)6x) = (e ® e)e in the convolution algebra HomK(A,(A ® A)x). Let a be a homogeneous element in A. Since 6 is algebra homomorphismfrom A to (A ® A)x, it follows that

((58) .6)@= Z:6s(al)* = ~ 6(s(al)a2)

= = = (e

= 5m(s ® id)5(a)

®

Let 6(al) = ~ al~ ® a~2 and 6(a~) = ~ a~ ® a~ with all factors homogeneous. Then

and we have (5 * (T(s ® s)5~))(a) ×(l~tl’l~l) (a~l ® a~2) * (s( a~2) ® s(a : ~ Cx(la~ I’la=~l)+x(la~l’laa~l)alls(a22)® al2s(a21). On the other hand, by using the coassociativity

of 6, we have

(5 ® 6)6 = (6 ® id ® id)(id ® 6)6 = (6 ® id ® id)(6 = (id® 6 ® id)(5 ® id)5 = (id® 6 ® id)(id® and hence we have

276

Zhangand Li Za11

® ~12

~ a21

® ~22

:

Z ~I ~ °’211

@ °’212

(4)

® a22

where 5(a21) = ~ a211 @ a212 with all factors homogeneous. Nowdefine a K-linear map L : A ® A ® A ® A --> A @ A ® A ® A by L(al ® a2 ® a3 ® a4) -- cx(la21+laai’la4l)(al ® a2 ® a3 a4) for all al, a2, a3, a4 homogeneous. By applying the K-linear map 0 = (m®id)(id®T)(id®m®id)(id®id®s®s)L to the both sides of (4), and applying the definition of antipode, we get ~ C~(~a~

I+la~’

I’la~l)alls(a22)

a1 2s(a21 )

= ~ c~(la~’l+la~’al’la~l)als(a22) @a2~s(a~) = ~ cx(laa’l’la~l)als(a22) @a211s(a212) = ~ cx(laal-la~at’laa~l)a~s(a22) ~ a2~is(a2t2) = ~ c~(la~l-la~l’laa~Dals(z(a21)a22) Weclaim that

In fact, if la~] # 0, then z(a21) = 0, and hence the both sides are 0; if then la~l - 1a22] = 0, and hence the claim follows. In this way, we see that

la~l = 0,

~ c~(la~l+la~xl’la=~l)a1~s(a22) @a~2s(a~)

= e(a) where we have used the property of the antipode and the counit property. Altogether, we have proved that (6s)* ~ = ~ (T(s ~ s) Sx). This pr oves th s : Ax ~ A is a coMgebra anti-homomorphism. Nowit remains to prove that s is invertible, and the assertion (iii). By Theorem2.3, in order to prove that ~rA = (A, m, e, xrS, e) is a (K, c, I, _xT)_ Hopfalgebra, it suffices to prove that it is a (K, c, I, -xT)-bialgebra. By Lemmai.4 we know that xrA = (A, x~5,e) is an NoI-graded coalgebra. only need to prove that xr5 : A ~ (A @A)_xr is an algebra homomorphism,i.e., xrS(ab) = xrS(a) * xrS(b) for homogeneous elements a,b ~ A, where * denotes the multiplication in (A @A)_~. In fact, since ~ : A ~ (A @A)~ is an algebra homomorphism,it follows that ~(ab) = ~ cX(la2l’lbll)alb~ ® a2b2, and hence = ~ CX(la21,lbl

I)-xr(la~

I÷lbl

[,la2l÷lb2

I)a2b2

alb~

(5)

TwistedHopfAlgebras

277

On the other hand,

~r(f(b) = C-Xr(Ibl$’lb21)b2 ® bl = EC-x( Ib2[’lb~l)b2 @bl, and hence in (A ® A)_xr we have

T6(a) = ~ C-~(la21,1a~l)-~(Ib21,1b~l)-~r(la~l,lb21)a2b2 ® albl -~ ~ C-X([a2[’lal[)-~([ba[’lbt[)-x(lb2l’[axl)a2b2 ®

(6)

By comparing (5) and (6) we see that ~TS(ab) = Finally, let s’ be the antipode of (K, c, I, -xT)-Hopf algebra xrA = (A, m, Weprove that s’s(a) = ss’(a) for any a ~ Adbyusi ng induct ion on d. This i s clear for a ~ Ao. By Lemma1.2 we have for d ~ 0 ~(a)

=a®l+l®a÷

E

a~®b~

~+y----d;z,y~O

with a~ ~ A~, b~ ~ Au. Then by (3) in 1.4 we have ~rtf(a)

-- a ® 1 ÷ 1 ® a +

By using m(id ® s)ti(a) s(a)

----

-a

Z

-x(~’~) c~ @c~.

= e6(a) = 0 and m(id ® s’) xrS(a) ee(a) = 0 weget

- E axs(by);

s’(a)

=-a-

x + y=d; x,y:~O

-~ (y’~)

xA-y----d;z,y~O

By (i) we have known that s’ : A ---+ A_~ is an algebra anti-homomorphism, follows from induction that s’s(a)

a s’s(bu) * st (~)

= -s’(a) z-~y-~.d;x,y~O

= a -k

~ x~-y=d;x,y~O

c-X(u’~)bus~(a~) -

x+y=d;x,y~O

c-X(u’~)bus’(a~ )

.-~ a.

Dually, we have ss’(a) = This completes the proof.

[]

REMARK: Wedo not know what is the corresponding result for a pair X = (X1, X2) of Z-valued bilinear forms on ZI, with X2 ~ 0. The following lemmais useful in verifying a given map s : A ---+ A being the antipode of a (K, c, I, (X, 0))-bialgebra LEMMA 2.5. Let X be a Z-valued bilinear ]orm on ZI, A = (A, m, e, 6, e) be a (K, c, I, x)-bialgebra, and s : A --~ Axr an NoI-graded algebra anti-homomorphism, where xT(x, y) = X(Y, x) for x, NoI. Assume that A is generated as al gebra by a subset X consisting of homogeneouselements o] A, such that (1) in ~.~ holds for all a ~ X. Then s is the antipode of x-Hop] algebra A.

278

Zhangand Li

Proof. By assumption, it suffices to prove that if m(id ® s)g(a) = t(a) and m(id s)g(b) = ~(b), then m(id ® s)~(ab) = e(ab); and if m(s ® id)5(a) = ~(a) and m(s ® id)6(b) = e(b), then m(s ® id)6(ab) = e(ab), where a, b are homogeneous elements in A. Let g(a) = ~al ® a2, 5(b) = ~bl ® b2, with all factors homogeneous. 5(ab) = Z cX(la21’lbll)a~bl ® a2b2, and hence m(id ® s)5(ab)

it follows that if b ~ Ao, then ¢(b) = 0 and m(id ® s)5(ab) = 0 = ~(ab); and b E A0, then m(id ® s)g(ab) al s( ag.)e(b) = m(id® s)t i( a)e(b) = e(a )e (b) = e(a Also we have m(s ® id)5(ab)

and by the same argument we know that m(s ® id)(i(ab) = e(a)e(b)

Let X = (X1, X2) be a pair of bilinear forms on ZI. Wewill show the existence of a (K, c, I, (X1, X2))-Hopfalgebra. EXAMPLE 2.6. Consider the free K-algebra ’F with 1 with generators 0i, i E I. For each x = (xi)iel NoI wi th l = ~ xi , le t ~F ~ be the K-space with basis i~I

all words 0~ ... 0i~ such that for any i ~ I the number of occurances of i in ~he sequence il,... ,i~ is equal to xi. This is just the grading induced by the weight function w : k(Oi, i ~ I) --~ ZI where w(Oi) is i-th coordinate vector in ZI. Then ’F = (~) ’F~ is an NoI-graded algebra. x~Nol

Let 5 : ~F ~ (~F ® ~F)~ be the unique algebra homomorphism such that 5(0~) = ~ ® 1 + 1 ® 0~, i ~ I, and ~ : ’F --~ ’Fo = K be the projection. Then is a (K, c, I, X)-bialgebra. For a proof see [R3], p.228, in particular, we emphasize that (?) and (8) in 1.5 is needed. It follows that it is a (K, c, I, x)-Hopf algebra Theorem 2.3.

TwistedHopfAlgebras

279

If in addition X2 = 0, then we can determine the antipode s of ~F. In fact, let s : ’F ---+ ’Fxr be the unique algebra anti-homomorphism such that s(1) = 1 and s(Oi) = -01 for all i E I, where xT(x, y) = X(Y, for x, y E ZI. Then s is exactly the antipode of ~1e by Lemma2.5, since ’we have m(id® s)5(0~) = 0 = m(s ® id)5(O~), However, we do not know what is the antipode of ’F for X2 ~ 0. EXAMPLE 2.7. Let A = (a~j) be an n × n generalized Cartan matrix with symmetrization (d~)l<~<,~, in the sense of [K]. Let U be Drinfeld-Jimbo’s quantized enveloping algebra of the Kac-Moody algebra of type A. Then we have

u =u+ ®Uo®U_. Let v be an indeterminate, (I, .) be the Cartan datum determined by A. Denote by ’f the free (Q(v), v, I, .)-Hopf algebra defined above, with antipode s, and the ideal of ~f generated by the quantumSerre relations: viva, i#j, (7) t Jd, V~ o where [1-~a’~]d ’ is Gaussian binormal coefficients. Then the positive part U+ and the negative part U_ are isomorphic to f = ’f/J as algebra, see ILl. Since s : ’f ---+ ~fx with X -- ¯ is an algebra anti-isomorphism,it is easy to verify that s(J)C=J, it follows that U+-~ f is a (Q(v), v, I, .)-Hopf algebra. 3

SOME CONSTRUCTIONS

OF TWISTED

HOPF

ALGEBRAS

The aim of this section is to construct some new twisted Hopf algebras from a given twisted Hopf algebra. PROPOSITION3.1. Let ¢,¢,XI,X~ : ZI x ZI ---~ Z be bilinear forms. Let A = (A, m, e, 5, ~) be a (K, c, I, X)- bialgebra, where X = (X1, X2). Then (A, m¢, e, ¢5, ~) is a (K, c, I, ~)-Hopf algebra, where ~ = (~1, ~2) with

Proof. First, by Lemma1.1 we know that A = (A, mo,e) is an NoI-graded Kalgebra. Second, by Lemma1.4 we know that (A, ¢5, e) is an NoI-graded coalgebra. Now, we need to prove that ¢5 : A¢ ---~ (A¢ ® A¢)~ is an algebra homomorphism, i.e., ¢5(a.1 b) ¢5(a) *2 ~5(b) for homo geneous elements a, b ~ A, where denotes the multiplication in A¢, and *2 denotes the multiplication in (A¢ ® A¢)~. In fact, let 5(a) = ~] a~ ® a2 and 5(b) = ~ bl ® b2 with all factors homogeneous. Since 5 : A --+ (A ® A)~ is an algebra homomorphism,it follows that 5(ab) = C~ (la~l’la~l’lb~l’lbal)a~b~ ® a~b2, and hence ¢5(a "1 b) = ¢(l~l’lbl) ¢~(ab) =E C~(~"a~’b~’ba)a2b2® albl,

280

Zhangand Li

where

t(al,a=,bt,bs)=¢(lal,Ibl)+x(l~tl,lasl,Ibtl,Ibsl)- ¢(latI +IbtI, lasl+Ib21). Onthe other hand,~5(a) = c- ¢(lall,la=l)as ®al, ¢5 (b) = ~ -e(Ibll,lb21)bs ® bl, and hence in (A¢ ® A¢)~ we have ¢~(a) *s ~(b) = ~ t’(a’’a2’b’’b2)asb

2 ®atbt,

wheret~ ( al , as, bt , b2) is

-¢(lall,la2l)- ¢([bll,Ibsl)+¢(lasl,Ibs[) 4- ¢(laxl, Ib~l) +:~(lasI, latl,Ibsl,Ibll). Note that

¢(lal,Ibl)=¢(la~l, Ib~l)+¢(la~l, Ibtl)+¢(la~l, Ibsl)+¢(lasl, x(la, I, la~l,Ib~l,Ib~l) =X,(lasl, Ib,I) +xs(latl, Ib~l), and and that by comparing(1) and (2) we see that t(at, a2, bl, b2) : t(al, as, b l, b 2), a nd hence

~(~,~ ~) = ~z(~),~ ~(~). Thus, A = (A,m¢,e, x6,e) is a (K,c,I,~)-bialgebra Theorem2.3 it is a (K, c, I, :~)-Hopf algebra.

algebra,

and hence by

REMARK: If X2 = 0, ¢ = 0, ¢ = X~T in Proposition 3.1, then Theorem2.4(iii) gives the relation between the antipodes of the two twisted Hopf algebras in Proposition 3.1. But we do not knowthe general situation. COROLLARY 3.2. Let X : ZI x ZI ---+ Z be a bilinear form, A = (A, m, e, ~, ~) be a (K, c, I, (XT, X))- bialgebra. Then A = (A, m, e, ~, e) is a Hopf algebra. Proof. Note that a (K, c, I, 0)-Hopf algebra is exactly a K-Hopfalgebra. Now,the assertion follows directly from Proposition 3.1. [] Since T6 = o6, by Proposition 3.1, we have COROLLARY 3.3. Let A = (A, m, e, 6,e) be a (K, c, I, (Xl, A = (A,m,e, Tb, e) is a (K,c,I, (Xs,X1))-HopI algebra.

X2))"

bialgebra. Then

PROPOSITION3.4. Let ¢,¢,X~,Xs : ZI x ZI --~ Z be bilinear forms. A = (A,m,e,6,e) be a (K,c,I,(x~,xs))bialgebra. Then A = (A,m¢,e,6¢,e) (K, c, I, ~)-Hopf algebra, where ~ = (~lt, :~2) with

Let

281

TwistedHopfAlgebras

Proof. By Corollary 3.3, A = (A,m,e, T6,e) is a (K,c,I,(x2,X1))-Hopf algebra, now applying Proposition 3.1 to it, and note that _¢r(T6) = 6~, we then get the [] desired result. COROLLARY 3.5. Letx : ZI×ZI --~ Z be a bilinearform. LetA = (A,m,e,~i,e) be a (K, c, I, (X, XT))- bialgebra. ThenA = (A, m, x, e) is a Hopfalgebra In general, we do not knowa relation between the antipode s of A = (A, m, e, 5, e) and the antipode s’ of A - (A, me, e, 5¢, e). However, if ¢ = -¢ in Proposition 3.4, then we have the following PROPOSITION3.6. Let ¢,X1,X2 : ZI × ZI ~4 Z be bilinear forms. Let A = (A, m, e, 6, ¢, s) be a (K, c, I, (X~, X2))- Hop] algebra. Then the antipode (A, m_~,e, 6~,e, s) is also Proof. By Proposition 3.4, A -- (A, rn_¢, e, ~, e) is a (K, c, I, (X1, ¢-¢T+X~))-Hopf algebra, say, with antipode s’. Let a E Ad, 0 ~ d ~ l~oI. Then by Lemma1.2 we have x+y=d;x,y~O

and 6~(a) = a® 1 + 1 ®a + ~

¢(~’u)

a~ ®by,

where ax 6 A~,b~ ~ A~. By using m(id @s)6(a) = ee(a) s’) tic(a) = ee(a) = weget s(a) = -a -

Z

a~s(bu);

and m_¢(id ®

= -a -

x+y=d;x,y~O

x+y=d;x,y~O

and hence s’ = s by induction. PROPOSITION3.7. Let ¢ : 7.1 x ZI --~ Z be a bilinear form. Then A = (A, m, e, 6, e) is a (K, c, 1, (0, -¢))- Hopfalgebra if and only if A = (A, me, e, ~, ~) is a (K, ~, I, ( X, O))-Hopf algebra, where Xis thesyrnmetrization ore, i.e., X(X, y ¢(z,y) + Cr(x,y) = ¢(x,y) + ¢(y,x). Proof. Weonly need to prove that 5 : A --~ (A® A)(0,-¢) is an algebra homomorphism (with respect to c) if and only if 5¢ : A¢ ---+ (A¢ A~)(x,o ) is an algebra homomorphism(with respect to x/-~). Let *~, *2, and *a denote the multiplications in (A ® A)(0,-¢), in A¢, in (A¢ ® A¢)(~,0), respectively. For homogeneouselements a,b ~ A, let 5(a) ~ a~ ® a~, if(b) -- ~ b~ ® b2 with all factors homogeneous.Then 5(ab) -- 5(a) *~ (using c) means that (i(ab) -- Z c-¢(la~l’lb~l)a~b~ ® a2b~. While 5¢(a *2 b) = 5¢(a) *a 5¢(b) (using v/~) means Vf~¢(lal’lbl)5¢(ab) = ~ ~(a~’aa’b~’~:) a~b ® a~b 2,

(3)

282

Zhangund Li

where t(al, a2, bl, b2) ¢(lal I, la21)÷¢(Ibll, Ib~l)-~-x(la~l, Ibl l)÷¢(lal I, IblI)÷ ¢(la~l, Ib~l or simply, means that Jo(ab)--- ~V~’~/-~¢(]a~l’lauD4-~P(Ib~l’lbul)"i’¢(Ibll’la20-¢(la~[’[b~])a. ~ bl @a2b2.

(4)

It remains to prove the equivalence of (3) and (4). By definition of 5~ (remember that we should use V~ to weight alb~ ® a~b~), this is clear. [] ACKNOWLEDGEMENT The authors thank the referee for the helpful comments. REFERENCES J. A. Green: Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120(1995), 361-377. J. C. Jantzen: Lectures on quantum groups, Graduate Studies in Math., vol. 6, Amer. Math. Soc. 1995. [K] V.G. Kac: Infinite dimensional Lie algebras, Progress in Math. 44, Birkh/~user, Boston.Basel.Berlin 1983.

ILl G.

Progress

[LZ] L. B. Li and P. Zhang: Twisted Hopf algebras, Green’s categories, preprint.

Ringel-Hall

Lusztig: Introduction to quantum groups, Birkh~iuser, Boston.Basel.Berlin 1993.

[R1] C. M. Ringel: Hall algebras revisited, ceedings. Vol.7 (1993), 171-176.

Israel

in Math. 110, algebras and

Mathematica Conference Pro-

[R2] C. M. Ringel: Hall algebras and quantum groups, Invent. Math. 101(1990), 583-592. [R3] C. M. Ringel: Green’s Theorem on Hall algebras, Canad. Math. Soc. Conf. Proc. Vo1.19(1996), 185-245; AMSProvidence RI 1996. IS] M. Sweedler: Hopf Algebra, Benjamin, NewYork, 1969.