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C toip : E —> C is a. morphism / : D —• E of C such that ipf = ip. The composition of morphisms of C/C is the same as that of C. (1.1.3) Let W b e a universe (see, e.g., [9]) and C a category. We say that C is a W-category if C(C, C) £ U for objects C, C" of C. We fix once for all a universe U, and we only consider this universe U unless otherwise specified. An element of U is sometimes referred as a small set. Unless otherwise specified, a category means a ZY-category. However, when we form a new category from some W-categories, exceptions (see (1.1.5) and (1.4.7)) may apply. If we need to emphasize a category is a W-category, we use the expression 'category with small horn sets.' The category Set means the category of elements oiU, and the category Grp means the category of small groups (i.e., groups in U), and so on, and
2
I. Background Materials
these concrete categories are also ^-categories. We say that a category C is small if ob(C) 6 U. We say that a category C is svelte (or skeletally small) if there exists a small full subcategory V of C such that the inclusion V <-> C is an equivalence. (1.1.4) Let C be a category, and R a commutative ring. We say that C is an R-category if C(M, N) is an /^-module for M,N € C, and the composition C(M,N) x C(L,M) -> C(L,N) is /?-bilinear. A Z-category is also called an Ab-category, or a preadditive category. A preadditive category with finite direct products (in particular, with a terminal object) is called an additive category. A finite direct product in an additive category is naturally isomorphic to the coproduct, and in particular, the category has a null object. Let .4 and B be /^-categories, and F : A ^> B a. functor. We say that F is an R-linear functor if the canonical map F : EomA(a,b)
-> HomB{Fa, Fb)
is .R-linear for any a, b € A. A Z-linear functor is called an additive functor. (1.1.5) For categories A and B, we denote the set of functors from A to B by Func(A,B). For F, G 6 Func(,4, B), we denote the set of natural transformations from F to G by Nat(F, G). Note that Func(^4, B) is a (not necessarily small) category with Nat(F, G) its horn set. For A 6 A, when we define y{A) := Hom^(?, ^4), we get a functor y : A -> Func(^4op ,Set). The following is well-known as Yoneda's lemma. Lemma 1.1.6 Let A be a category, and T : Aop —> Set a functor. Then we have that the natural map Y :T —• Nat(y(?),T) given by (Y(t))(
(for A, Be ob{A), t € T(A),
zKercokery> is identified with ia Coker ker iip —> ia Ker coker iip, since the counit ai —> Id is an isomorphism and a preserves both kernels and cokernels. This is an isomorphism as V is abelian. As i is faithful, this shows that S is also abelian. Now it is clear that a is exact. Again by Lemma 1.2.6, i preserves injective objects. The category <S satisfies (AB5). In fact, the inductive limit in S is nothing but the composite alim(z?), and hence it is left exact if filtered. As it is easy to see that (aty is a small family of G-generators of S if is a small family of G-generators of V, we have that 5 is Grothendieck.
•
1. From homological algebra
23
(1.8.11) Let T and V be sites. We say that / : T -> V is a continuous functor if / : CatT -> CatT' is a functor such that for any {£/j ->•[/} G CovT, we have {/[/* -> / [ / } G CovT" and the canonical map
is an isomorphism for any i, j . If / : T -> T" is a continuous functor and .F G sh(T',.4), then / P (.F) is a sheaf. Hence, we obtain a functor fs = fp : sh(T',.4) -> sh(T,.4). Note also that fs := afP : sh(T,A) -> sh(T',.4) is left adjoint to / s . Example 1.8.12 Let B be an abelian category. When we define that {B{ -+ 5 } , 6 / is a covering if / is a finite set (we allow the case / is empty) and ©jBj -» B is an epimorphism (resp. isomorphism), then B is a site, which we denote by B\ (resp. Bo). The category <Si := sh(B\,A) (resp. So := sh(fi0, A)) is nothing but Sex(S op , ^4) (resp. the category of contravariant additive functors from B to A). In particular, as we are assuming that A is Grothendieck and satisfies (AB3*), Sex(6 0p ,.A) is also Grothendieck and satisfies (AB3*). Moreover, Id e : Bo -> B\ is clearly continuous. (1.8.13) Let T be a site with fiber products, and T a subcategory of f such that ob(T) = ob(T). We assume that any isomorphism of T is a morphism of T, and any base change of a morphism of T is again a morphism of T. Moreover, we assume that t/j —» U is a morphism of T for any {U{ -> U\ G Cov(t). For X G f, the full subcategory of f/X consisting of morphisms of T (with the fixed codomain X) is denoted by Tx- Note that Tx is a site with the topology of T. If
X is a morphism of T, then a continuous functor tp# : T/X -» T/X' is defined by M of A is called a right X-approximation of M if X e X, and for any X' £ X and any g e A(X',M), there exists some h € A{X',X) such that fh = g. It is equivalent to say that A(?,X) € Func(A', Ab) is representable, and A(7, / ) : -4(?, X) —>• A(f, M) is an epimorphism of the functor category Func(A',Ab). Left X-approximation is the dual notion. A right (resp. left) minimal right (resp. left) A'-approximation is simply called a right (resp. left) minimal ^-approximation. A right minimal Xapproximation of M is unique up to isomorphisms as an object of A/M, if it exists. Let B be a full subcategory of A which contains X. We say that X is contravariantly finite (resp. covariantly finite) in B (or B has right (resp. left) Xapproximations) if any object in B has a right (resp. left) ^-approximation. If any object in B admits a right (resp. left) minimal A'-approximation, then we say that B has right (resp. left) minimal -Y-approximations. Lemma 1.12.1 Let be an exact sequence in A such that X € X, and assume that Ext\(X, Y) = 0. Then p is a right X-approximation of M. Proof. For any X' 6 X, we have that the sequence A{X', X) -> A(X', M) -4 E x t ^ ( * ' , Y) = 0 N a morphism of A. Set i : K —» M to be the kernel of f. Consider the following two conditions. 1 The morphism p is right minimal. (x) = 0. If Y be a morphism of finite type of noetherian schemes. We say that -> dim^x) is locally constant. If this is the case, the value e = dimv,(x) is constant on each connected component XQ oi X, and we say that the relative dimension of ip over Xo is e. (2.7.3) If ip : X —> Y is a flat morphism of schemes locally of finite presentation, then tp is an open map [65, IV.1.8.4]. If ip : X —¥ Y is a morphism of finite type between noetherian schemes all of whose local rings are equidimensional, and Y is equidimensional (of finite type), then tp is flat. This follows from the corresponding statement for local rings [110, Theorem 23.1] and the dimension formula [110, Theorem 15.6], see [65, (IV.13.3.6)]. (2.7.4) Let R —> S be a ring homomorphism of commutative rings. We denote the kernel of the multiplication map m : S <8>K S -> 5 (a ® i H ab) by / . The 5-module I/I2 is denoted by CIS/R- U R -* R' is a. ring homomorphism of commutative rings, then we have fis'/w = R' ®R ns/R S is a local homomorphism if (m) C n. Let ip : (R, m) -> (5, n) be a local homomorphism of local rings. Theorem 2.7.8 ([20, (1.1)]) Let ip : (R,m) -> (5,n) be a local homomorphism of noetherian local rings. If S is complete, then there exist some complete noetherian local ring T and local homomorphisms T : R —¥ T and a : T —> S such that r is fiat, T/mT is regular, a is surjective, and GOT — tp. We call such a decomposition a or of tp a Cohen factorization of tp. Ker a is a perfect ideal (resp. Gorenstein ideal, complete intersection ideal, ideal of finite projective dimension) for some Cohen factorization if and only if this is true for any Cohen factorization (because of the comparison of Cohen factorization, see [20, (1.2)]). If a Cohen factorization of the composite map r , f . n Y of locally noetherian schemes is Cohen-Macaulay (resp. Gorenstein, complete intersection, of finite flat dimension) at x £ X if the local homomorphism Oy,f(x) —> Ox,x is CohenMacaulay (resp. Gorenstein, complete intersection, of finite flat dimension). We say that / is Cohen-Macaulay (or locally Cohen-Macaulay) (resp. Gorenstein (or locally Gorenstein), local complete intersection (l.c.i., for short), locally of finite flat dimension) if / satisfies the corresponding property at any point of X. A ring homomorphism R -¥ S of noetherian rings is said to be locally Cohen-Macaulay (resp. locally Gorenstein, local complete intersection, regular, locally of finite flat dimension) if the corresponding property is satisfied by the associated morphism Spec 5 -» Spec R. Note that a local homomorphism of local rings p : R —> 5 is locally of finite flat dimension if and only if it is of finite flat dimension. We say that a noetherian ring R is a G-ring if the completion Rp —» Rp is regular for any P £ Spec R. It is known that R is a G-ring if and only if the completion Rm —» Rm is regular for any m € Maxi? [110, Theorem 32.4]. Note that a complete local ring is a G-ring. The ring Z is also a G-ring. A ring essentially of finite type over a G-ring is also a G-ring. Let ip : R —> S be a local homomorphism of noetherian commutative local rings. If R is a G-ring, then is locally Cohen-Macaulay if and only if it is Cohen-Macaulay as a local homomorphism. This is not true without the G-property in general. Similarly for the Gorenstein and c.i. properties. A locally noetherian scheme X is Cohen-Macaulay (resp. Gorenstein) if and only if X is Cohen-Macaulay (resp. Gorenstein) over Spec Z. We say that a noetherian local ring (A, m) is a complete intersection if A is complete intersection over Zmn z- We say that a locally noetherian scheme X is a local complete intersection (l.c.i.) if X is l.c.i. over SpecZ. X is l.c.i. if and only if Ox,x is a complete intersection for any closed point x of X. If a local homomorphism tp : (R, m) —¥ (S, n) is flat, then ip is c.i. (at n) if and only if S/mS is a complete intersection. Note that a regular scheme is l.c.i., and an l.c.i. scheme is Gorenstein. Yz is flat for any z £ Z, then Y be morphisms between locally noetherian schemes. We assume that l.c.i. => Gorenstein => Cohen-Macaulay. It is not overstating to say that this hierarchy has been playing the central role in the modern theory of commutative rings. , and ip o m,A = ™,A' ° {f ® ¥")• An i?-coalgebra map is defined as follows. B ' is an .R-bialgebra map, then we have tp o SB = SB' ° ¥>• Exercise 3.2.4 In Example 3.1.4, G is a group if and only if RG is an .R-Hopf algebra. If G is a group, then S(g) = g~x gives the antipode. Conversely, if 5 is an antipode of RG, then we have g • S(g) = S(g) • g = 1 by the definition of antipode, and it is easy to see that g is invertible in G. In particular, considering the case that G is the trivial group, R is an /J-Hopf algebra. In general, if B is an .R-Hopf algebra and b € B is group-like, then we have 5(6) = b-1. Hence, the set of group-like elements X{B) of B is a subgroup of B*. If G is a group and R has no non-trivial idempotents, then we have X(RG) = G, and we can recover G from the Hopf algebra RG. Let A and A' be commutative .R-algebras, C an ,4-coalgebra, and C" an /l'-coalgebra. Then letting c ®a C a coalgebra map. For M € M B , letting TV is a universal map if for any X-algebra A, ipA : MA -> NA is & natural transformation, and for any X-algebra map f : A-> B, p'f o {(B®A?) GL(Af) such that, for any commutative X-algebra f : SpecA—*>X, VA{9\, B be as in (5.1.3). Let M and N be jB#[/-modules, and V and W be ^4#t/-modules. Then the standard maps (5.1.5) (5.1.6) (5.1.7) (5.1.8) (5.1.9) (5.1.10) (5.1.11) (5.1.12) (5.1.13) (5.1.14) (5.1.15) (5.1.16) (5.1.17) (5.1.18) (5.1.19) B be as in (5.2.1). Let M be a B # t / module, and V an A#t/-module. Then the £#[/-module Li(M(gi/i?)(Vr) is isomorphic to Torf (M, V) as an .A-module, since any v4#[/-projective module is A-flat. So we denote the 5#I/-module L i (M® j4 ?)(V) by Torf (M, V). If F is an 4-flat resolution of M in B # t / M , then we have Hi(F ®A V) = Torf (M, V) in B #yM. In particular, if B is .A-flat, then we have l)-modules V, W, the maps in (5.1.5)(5.1.19) are (G, S)-homomorphisms natural with respect to M, N, V and W, provided the first argument(s) of Hom^ (resp. Horns) are of finite presentation as >l-module(s) (resp. B-module(s)). For example, in the map (5.1.11), we require that V is of finite presentation as an j4-module, and M is of finite presentation as a 5-module. However, as for (5.1.8-5.1.13), when we only assume that the first argument of Horn inside another Horn is of finite presentation, the map in question is an isomorphism of B-modules, and the set of G-linear maps is mapped bijectively onto the set of G-linear maps. In particular, we have isomorphisms (5.3.6) G,/IM. If V is a (G, .4)-module, then V ®A B is a (G, B)-module, which we denote by B. (1.1.2) X' JP*L are isomorphisms, the left square is also a fiber square. Hence, by Theorem 1.2.7.11, we have axl{U) = U{F, Spec/?, is G-stable open. By Lemma 2.1.9, we have that U = 0 or U = G. When we take the generic point 7 of an irreducible component of G, then C G , 7 is artinian, and hence is CohenMacaulay. This shows that 7 G U and hence U ^ 0, and we have U = G. D Remark 2.1.12 More is true. Under the same assumption as in the corollary, we have that G is l.c.i. over R. To verify this, we may assume that R is a field again. Then by Lemma 1.4.2.8, G is a closed subgroup of some P = GLn(R). By [43, III.3.5.4], we have that the quotient P/G is a scheme. VC(A) is extended to a map denned on Vc(A) for any / € Endc(Lc(A)). This shows that soc is surjective. Similarly, top is also proved to be bijective. D F. We set G := Cokeryj. Then G/mG = 0 is obvious. By Lemma 1.2.1.4, J(m) is an isomorphism, since top P(m) —> top J(tn) is an isomorphism by Lemma 1.1.12.5. By Lemma 1.2.1.4, tp is injective, and we have P = J. D Corollary 4.1.4 Let it be a finite poset ideal of X+, and X its maximal element. Then Ac (A) is SC(TT) -projective. Proof. The case R is a field is obvious by Definition 1.1.4 and Theorem 2.3.17. The general case follows from the lemma. • Corollary 4.1.5 Let n be a finite poset ideal of X+, let V e Mc", and assume either X is a maximal element of n or X £ X+ \ it. Then we have Ext l c (A c (A), V) = 0 for i > 0. Proof. We set p \— TT U (—OO, A]. Then p is a finite poset ideal of X+, and A is a maximal element of p. By Corollary 4.1.4 and Theorem 2.3.17, we have (i > 0). Ext^(A c (A), V) a Exf Cp (A c (A), V) = 0 4. Assume the contrary, and let T be a non-zero direct summand of X contained in Imz. As T G w^ro, replacing T by its direct summand, we may assume that T = Tg(\) = TC(X) N an R-linear map. Then the following are equivalent. 1 tp is semipure. ). As G has a filtration of the desired form by induction assumption, P also has such a filtration. The last assertion is now obvious. D Corollary 1.1.5 If M is a (G, A)-module which is rank-one projective as an A-module, then there exists a rank-one R-projective G-module A such that M = A A -4 M is .4-pure by the proof of the lemma. As M is rank-one .4-projective, we must have that A is of rank one and Cokery; = 0. The last assertion is obvious by (A = t - 1} (if £ = 1, then we consider that /o = 5 as a convention). Note also that U is an open subset of Y, and it is non-empty, since there is an m x n matrix of rank (t - 1). The restriction 7r : 7r~1(U)(W) -> U(W) has the inverse (a(tp),ip((p)), where a(y>) is the exact sequence 0 -> Kery) ->• / ' V -> Im 0, and ^(v3) is Imy) "-> f'W*. In fact, the image of a bundle map whose rank is (t — 1) everywhere is a quotient bundle of rank (t — 1). Conversely, the only rank-(< - 1) quotient through which a bundle map of rank (t — 1) factors is its image. By Yoneda's lemma, we have that ?r : 7r -1 (U) -> U is an isomorphism. In particular, n is birational. Finally, we prove that Y is mn — (m — t + l)(n - 1 + l)-dimensional. As is well-known, we have dimG = (t — l)(m — t + 1). As Z is a vector bundle of rank (t - l)n over G, we have dim Y = dim Z = (t — l)(n + m — t + 1) = mn- ( m - £ + l ) ( n - t+1). D
24
1.9
I. Background Materials
Noetherian categories and locally noetherian categories
(1.9.1) Let A be an abelian category, and M G ob(.4). We say that M is a noetherian (resp. artinian) object if the set of subobjects of M satisfies the ascending (resp. descending) chain condition. An abelian category is called noetherian (resp. artinian) if it is svelte and any object in it is noetherian (resp. artinian). An abelian category is called locally noetherian (resp. locally artinian, locally finite) if it satisfies (AB5), and has a small family of G-generators consisting of noetherian (resp. artinian, finite length) objects. Note that the full subcategory of noetherian objects in an abelian category is very thick. Any noetherian category C is embedded in V := Sex(Cop, Ab) by the Gabriel-Quillen embedding. By Example 1.8.12, V = Sex(Cop, Ab) is locally noetherian and satisfies (AB3*). Moreover, C is equivalent to the full subcategory of T> consisting of its noetherian objects via the Gabriel-Quillen embedding. In fact, any skeleton of C is a noetherian generator of T>. (1.9.2) Conversely, for a given locally noetherian category T>, its full subcategory Vf of V consisting of noetherian objects of V is noetherian, and the Gabriel-Quillen functor T> —> Sex(D^p, Ab) is an equivalence. In particular, any locally noetherian category satisfies (AB3*). Moreover, noetherian categories and locally noetherian categories are in one-to-one correspondence as above. We always use the symbol Vf to denote the full subcategory of noetherian objects of a locally noetherian category T>. For example, /{M is locally noetherian if R is a noetherian commutative ring. In this case, RM^ is nothing but the category of infinite modules. Hence, RM —> Sex(ftM^p, Ab) is an equivalence. More generally, if X is a noetherian scheme, then the category Qco(X) is locally noetherian, and Qco(X)f is nothing but the category Coh(X) of coherent Ox-modules, see [58]. Let V be a locally noetherian category. Note that the presheaf direct product, projective limit, and filtered inductive limit of objects in Sex(Cop, Ab) are left exact again, and hence are the direct product, projective limit, filtered inductive limit in Sex(Cop, Ab), respectively. In particular, we have: L e m m a 1.9.3 Let V be a locally noetherian category, Y S Vf, and let (X{) be a filtered inductive system in T>. Then the canonical map
is an isomorphism. In particular, a filtered inductive limit (e.g., a direct sum) of injective objects in V is again injective.
1. From homological algebra
25
By the lemma, we have the following Lemma 1.9.4 (Gabriel) Let V be as above. Then for F £ V, viewed also as an object of Sex(T>y, Ab), the following are equivalent. 1 F is a contravariant exact functor on V;. 2 F is an injective object ofV {in other words, V(?,F)
is exact on V).
3 For any M € Vs, ExtJ,(M, F) = 0. Lemma 1.9.5 Let V, Y, and (Xi) be as in Lemma 1.9.3. Then the canonical map
is an isomorphism for j > 0. Proof. As V is Grothendieck, there is a functorial injective resolution Jf in V, see (1.7.7). By Lemma 1.9.3, lim is an isomorphism of complexes of abelian groups. Since limJ(Xj) is an injective resolution of lim Xi by the (AB5) condition and the last assertion of Lemma 1.9.3, the assertion follows immediately, taking the cohomology. D
1.10
Semisimple objects in a Grothendieck category
(1.10.1) Let A be a Grothendieck category, and B a full subcategory of A closed under direct sums, subobjects, and quotient objects. Note that B is abelian, and the canonical embedding i : B '-> A is exact. Note also that B is Grothendieck, which is less trivial. In fact, if U is a generator of A, B € B and / € A(U,B), then we have I m / c B and hence we have I m / € B. So the set (V;) i€/ of quotient objects of U which lies in B (by Lemma 1.7.3, we can take / to be small) is a small family of G-generators of B. As the inductive limit in A of an inductive system consisting of objects in B lies in B by assumption, it is also an inductive limit in B. Hence the (AB5) condition holds in B. As is easily seen from the proof, B is locally noetherian if A is. Lemma 1.10.2 The embedding i : B <—¥ A has a right adjoint j . The unit u : Id —> ji is an isomorphism.
26
I. Background Materials
Proof. For A € A, the set of subobjects of A is indexed by a small set (Lemma 1.7.3). So we may form the sum j(A) of all subobjects of A which lie in B. As B is closed under inductive limits in A, we have j(A) G B. Note that j(A) is the largest subobject of A which lies in B. For B € B and / £ ^4(5, J4), we have that / factors through j(A), as I m / G 5. Hence, we have an isomorphism
In particular, for A, A' £ A and g S A(A, A'), the restriction of g to J(J4) factors through j(A'), and we have an induced morphism j(g) € B(j(A),j(A')). It is easy to verify that j is a functor with this definition, and <&B,A is natural on B and A As i is fully faithful, the last assertion follows from Lemma 1.2.6. a As j has an exact left adjoint i, it preserves projective limits (in particular, kernels) and injective objects. Lemma 1.10.3 Let j be the right adjoint of i as above. If A is locally noetherian, then the canonical map f:\imj(Ax)
->
j(limAx)
is an isomorphism for any filtered inductive system (Ax) in A. Proof. Note that if B € Bf, then i(B) € A/. Hence, using Lemma 1.9.3, we have isomorphisms of functors on Bf
B(?,\hnj(Ax)) S* lu S A(i(?),\imAx) = B(?,j(\imAx)). As the canonical functor y : B -» Sex(So P ,Ab) is an equivalence and the composite isomorphism above is nothing but y(f) : y(\unj(A\)) —> y(j(\imAx)),
f is an isomorphism.
D
(1.10.4) Let A be a Grothendieck category. We say that A € A is a simple object if there are exactly two subobjects of A. In other words, A is simple if and only if A ^ 0, and any monomorphism into A is either zero or an isomorphism. We say that A £ A is semisimple if A is isomorphic to a direct sum of simple objects. The following is well-known, and is proved in [123] in the case of modules over an algebra. L e m m a 1.10.5 The full subcategory AsS of A consisting of semisimple objects of A is closed under inductive limits, subobjects, and quotient objects in A.
1. From homological algebra
27
By the lemma, the sum of all semisimple subobjects of A G A is the maximum semisimple subobject of A. This object is called the socle of A, and we denote it by soc A. Thus, soc : A —> A s is a right adjoint functor of the canonical embedding AgS •-> A. Note that soc A is also the sum of all simple subobjects of A. As it is a right adjoint, soc preserves projective limits (e.g., kernels). Assume moreover that A satisfies the (AB3*) condition. Then for A £ A, we set
radA:=
f]
B
BcA, v4/£?:semisimple
and call rad^l the radical of A. We denote A/radA by top .A, and call it the top of A. Note that top^l is not necessarily semisimple. However, if A is an artinian object, then top A is semisimple, and hence is the largest semisimple quotient of A. Any non-zero artinian (resp. noetherian) object admits a simple subobject (resp. quotient object). Hence, we have Lemma 1.10.6 Let Abe a Grothendieck category which satisfies the (AB3*) condition. If A is an artinian (resp. noetherian) object of A and soc A = 0 (resp. top A = 0), then we have A = 0. The following is also trivial. Lemma 1.10.7 Let V be a locally noetherian category, and (Dx) a filtered inductive system in V. Then the canonical map limsoc(D^) —> soc(limD.x) is an isomorphism. If A is a locally artinian category and 0 ^ A € A, then we have soc A ^ 0. (1.10.8) We say that a ring A is a division ring if A ^ 0 and any nonzero element of A is a unit. The following is well-known as Wedderburn's theorem [123]. Theorem 1.10.9 Let A be a ring. The following are equivalent. 1 The A-module &A is a semisimple object o/^M. 1* The right A-module AA is a semisimple object ofMA. 2 A is a finite direct product f l ^ i Mat nj (Dj) of matrix rings over division rings. If the conditions above are satisfied, then A is called a semisimple ring. A semisimple ring is both left and right artinian.
28
I. Background Materials
1.11 Full subcategories of an abelian category Let A be an abelian category, and X a subset of ob(,4). We define some full subcategories of A. 1 We denote the full subcategory of A consisting of objects isomorphic to a direct summand of a finite direct sum of objects of X by add-^. If X = 0, then add(A") consists of null objects of A. Obviously, add(X) is a Karoubian additive category. If X is closed under extensions in A, then so is add(A"). 2 The full subcategory of A consisting of objects A € A such that there exists some r > 0 and a filtration 0 = A) C Ax C • • • C Ar = A such that Ai/Ai-x is isomorphic to some object in X for i = 1,2,..., r is denoted by F{X). Note that T{X) is closed under extensions in A. Note also that F(X) is not closed under direct summands in general even if X is so. 3 Let A1 b e a n additive full subcategory of A. The full subcategory of A consisting of A € A such that there exists some exact sequence (1.11.1)
0->Xh^>
> Xx^> XQ-+ A->0
with Xi € X is denoted by X. An exact sequence of the form (1.11.1) is called a finite X-resolution of A. The smallest non-negative integer i such that Xi+l = 0 is called the length of the -^-resolution (1.11.1). For A € X, we call the minimum length of /^-resolutions of A the Xresolution dimension of A, and denote it by #-resol.dim A If A £ X, then we define <¥-resol.dim>l :— oo. 3* Let X be an additive full subcategory of A. Then we define X := (X°P)°P. In other words, X consists of A G. A such that there exists some exact sequence (1.11.2)
0^A^X°-^X1^
>Xh^0
with X1 £ X. An exact sequence as in (1.11.2) is called afiniteXcoresolution-oi A. The minimum non-negative integer i such that Xl+1 = 0 is called the length of the /f-coresolution (1.11.2). For A € X, the minimum length of A"-coresolutions of A is called the X-coresolution dimension of A, and we denote it by X -cores.dim A. If A ^ X, then we define X -cores.dim A := oo.
1. From homological algebra
29
4 For A G A, we define *-inj.dim A := sup({z > 0 | Ext^A", A) ^ 0} U {0}), and we call #-inj.dim^l the X-injective dimensionoi A. If A'-inj.dimyl = 0, then we say that A is X-injective. The full subcategory of .4 consisting of AMnjective objects in A is denoted by XL. Similarly, X-projective dimension is defined, and we denote it by A'-proj.dimA We also define an X-projective object in a similar way, and the full subcategory of Xprojective objects of A is denoted by LX. Note that X1 is closed under extensions, direct summands, and monocokernels. Note also that LX is closed under extensions, direct summands, and epikernels.
1.12 .^-approximations and the Auslander-Buchweitz theory Let A be an abelian category. We say that a morphism p : M -> N of A is right minimal if
30
I. Background Materials
is exact.
D
Lemma 1.12.2 (Wakamatsu's lemma) Let
o->r Ux ^M be an exact sequence in A such that p is a right minimal X-approximation of M. If X is closed under extensions, then Ext\(X,Y) = 0. For the proof, see [146, Lemma 2.1.1]. (1.12.3) Let A be a ring. The radical rad^/l of A as a left A-module and the radical rad AA of A as a right Amodule agree, and it is simply denoted by rad A and called the Jacobson radical of A. Note that rad ,4 is a twosided ideal of A li 0 ^ M £ AM is A-finite module, then M ^ (rad A)M (Nakayama's lemma). For basics on Jacobson radicals, see [123]. Lemma 1.12.4 Let A be a ring, and I = rad A The following are equivalent. 1 Any finitely generated A-module has a projective cover (in the category AM).
1* Any finitely generated right A-module has a projective cover (in the category ^opM). 2 The ring A/1 is semisimple, and for any idempotent e of A/I, there exists some idempotent e of A such that e modulo I equals e. We say that a ring A is semiperfect if the equivalent conditions in the lemma are satisfied. Lemma 1.12.5 Let A be a semiperfect ring, I = rad A, and p : P —» M an A-linear map between A-modules. Assume that M is A-finite. Then p is a projective cover if and only if P is A-finite projective and the induced map P/IP —> M/IM is an isomorphism. We say that a ring A is local if A/ rad A is a division ring. A ring A is local if and only if A ^ 0 and the set of non-units of A is closed under addition. We say that a commutative noetherian local ring R is Henselian if any /J-module finite algebra is semiperfect. If moreover the residue field of R is separably closed, then it is called strictly Henselian. Note that a complete local ring is Henselian [110, 111]. If a semiperfect ring A ^ 0 does not have any non-trivial idempotent, then A/ vad A is a division ring by Theorem 1.10.9.
1. From homological algebra
31
Let C be a Karoubian additive category, and C an object of C. We say that C is indecomposable if C ^ 0, and for any split monomorphism i: C" —» C, it holds either i = 0 or i is an isomorphism. If A := Endc C is local, then C is indecomposable, since A does not have any non-trivial idempotent. Conversely, if A := Endc C is semiperfect and C is indecomposable, then A is local. The following Krull-Schmidt theorem holds. Lemma 1.12.6 LetC be a Karoubian additive category. Then the following are equivalent. 1 For any M € C, there exists some decomposition M = Mi © • • •ffiMT (r > 0) such that Endc M{ is local for each i (in particular, each M< is indecomposable). 2 For any object M € C, we have Endc M is semiperfect. Moreover, if these conditions are satisfied, M € C, and there are two decompositions M = Mi © • • • © Mr = A^i © • • • © Ns such that Mt and Nj are indecomposable, then we have r = s, and there exists some permutation a £ S r such that Ni = Ma{ for any i. Here we only sketch the proof of 1=^2. Take a decomposition M = M\ © • • • © Mr and set E := End c M. For each i, set a{ : M{ <-^> M to be the inclusion, and TTJ : M —> Mj to be the projection. First prove that J := {
32
I. Background Materials
2 The objects K and M do not have any direct summands in common through i. In other words, if MQ £ A, I e A(M0, K), and •n £ A(M, Mo), then ir oiol is not an isomorphism. Then 1=>2. / / moreover, End^(M) is semiperfect, then there exists some decomposition M = Mo © My such that Mo C Im i and Mx —> N is right minimal. In particular, 2=>1 in this case. Proof. Assume that M = Mo © Mj, Mo ^ 0, and Mo Clmi. Then when we define ip to be the projector to M\, then ip is not an isomorphism, but we have pip = p. Hence 1=>2. Next, assume that E := End^(M) is semiperfect. We denote the functor A(M, ?) : A —> £oPM by eg. We have an exact sequence of right E-modules
As the right ^-module C := I m e ^ p ) is finitely generated, there is a projective cover IT : P —> C of C. As E is also £-projective, the map es(p) : E —> C lifts to p : E —> P so that Trp = e£;(p), and p is surjective by the definition of projective cover. Hence, p splits, and there exists some a : P -> E such that pa = Ip. The map e := op 6 EXKIEEE = E is a projector, and hence is an idempotent. When we set Mo := Im(l — e) and Mi := Im e, then M = M o © Mj. As pe = e E (p)(e) = IT pap = irp = p, Mo C Kerp = Imi. We prove that pj : M\ -> iV is right minimal, where j : Mj <—> M is the canonical inclusion. Let tp be an endomorphism of M\ such that pjtp = pj. Applying eg, we have that eE(pj)eE{i>) = esipj). As es{pj) factors through the projective cover eE <-»• E —> C of C, we have CE(V') is an isomorphism. As es '• addM —> add fig is an equivalence [12, Proposition II.2.1], rp is an D isomorphism. Hence, pj : Mi —> N is right minimal. Corollary 1.12.9 Lei X be summands, and assume that is semiperfect. If M e A has has a unique right (resp. left)
a full subcategory of A closed under direct the endomorphism ring of any object of X a right (resp. left) X-approximation, then M minimal X-approximation.
Let A be an abelian category, and A" be a full subcategory of A. We say that a full subcategory w of X is a cogenerator of X if for any X 6 X, there exists some short exact sequence of A
1. From homological algebra
33
such that W £w and X' £ X. We say that u is an injective cogenerator of X if it is a cogenerator of X and w C X1. The following theorem is due to Auslander and Buchweitz [10]. Some part is taken from [11]. Theorem 1.12.10 Let A be an abelian category, and X, y, and ui be full subcategories of X such that AB1 X is closed under extensions, epikernels and direct summands in A. AB2 y is closed under monocokernels, extensions and direct summands in A, and we have y C X. AB3 w = X ny, and w is an injective cogenerator of X. Then the following hold: i & = y. 2 Ifcj'cXis
an injective cogenerator of X, then add a/ = w.
3 For M £ X, the following hold. i (X-approximation) There exists some exact sequence of A O^Y^X
^M
->0
such that X EX andY ty. ii (y-hull) There exists some exact sequence of A 0-> M UY such that X £ X and Y
->X
^0
ey.
4 For M £ X, the following are equivalent. iM£X ii' Ext},(M, y) = 0
ii ExtJi(M, y) = 0 (* > 0) iii Ext^(M, w) = 0 (i > 0).
Hence, the morphism p in the short exact sequence in 3, i is a right X -approximation of M. 5 For M £ X, the following are equivalent. iMey ii Ext^X, M) = 0(i>0) iii A'-inj.dimM < oo, and M £ or1.
ii' E x t ^ * , M) = 0
Hence, the morphism L in the short exact sequence in 3, ii is a left yapproximation of M.
34
I. Background Materials
6 For M 6 X, we have A'-resol.dim(M) = y -pro j . dim (M) = w-proj.dim(M). 7 For Y £ y, we have w-resol.dim(y) = A'-resol.dim(y). 8 / / 0 —> Mi -> M2 -> M 3 -» 0 is an exact sequence of A and two of Mi, M2, and M 3 belong to X, then the third also belongs to X. We call an exact sequence as in 3, i an X-approximation. It is called minimal if p in 3, i is right minimal. We call an exact sequence as in 3, ii a y-hull. It is called minimal if L in 3, ii is left minimal. We say that a triple (X, y, u>) of full subcategories of an abelian category A is a weak Auslander-Buchweitz context in A if the conditions AB1—3 in the theorem are satisfied. If moreover X = A, then it is called an AuslanderBuchweitz context. Ii (X,y,cj) is an Auslander-Buchweitz context in A, then any of X, y, and w determines the others. We list some useful lemmas related to (weak) Auslander-Buchweitz contexts. Lemma 1.12.11 Let A be an abelian category, and a :
0 -> M o ->• Mi -> M 2 -> 0,
0o : 0 -> Yo -> X o -> M o -> 0, 02 : 0 -> Y2 -> X2 - ^ M 2 -» 0
exact sequences in A. If Ext2A(X2, Yo) = 0, then we have a commutative diagram 0 0 0
I
I
0
- > Vb - >
0
->
I Xo
I Yi
I ->
->
Mo
I
-¥
-¥ 0
X2
->
0
M2
-•
0
4- V1
-i-*
Y2
I Xi
40
-¥
Mi
I
0
->
I 0
0
exact rows and columns such that the first and third columns agree with 0o and 02, respectively, and the third row agrees with a. If moreover we have ExtiA(X2,Y0) = 0, then such a diagram is unique up to isomorphisms of diagrams. Proof. Set a' : 0 -» M o ^
M[ -> X2 -> 0
1. From homological algebra
35
to be the exact sequence atp, the pull-back of a by ip. From a', we have a long exact sequence Ext1A{X2,Y0)
-
If we have an exact sequence # : 0 -¥ Yo -> Xi ->• Af{ -»• 0 such that #(/3J) = /?o, then letting Y\ be the kernel of the composite morphism X\ -> M[ -» Mj, the construction of a diagram in question is easy. Conversely, if a diagram as in the lemma exists, then taking M[ to be the cokernel of the composite morphism Yo —> Y\ -» -X"i, we obtain /?J such that (p#(P[) = /?o, and giving /3J is the same thing as to construct such a diagram. Such a /3[ does exist if Ext^(Xj, Vo) = 0, and it is unique up to equivalence if Hence, the assertion of the lemma follows.
D
Proposition 1.12.12 Let A be an abelian category, and X, y, ZQ, and Z be full subcategories of A. Assume that X and y are closed under extensions, and Ext^(A", y) = 0. Moreover, assume one of the following: a Z = T(Z0) b Z = add F(ZQ) and Z C X. If for any Z £ Z§ there exists some exact sequence
o->y ->• x -> z ->o such that Y E y and X £ X, then there exists some exact sequence of the same type for any Z € Z. If for any Z £ ZQ there exists some exact sequence such that Y £ y and X £ X, then there exists some exact sequence of the same type for any Z £ Z. Proof. Assume a. The first assertion follows easily from Lemma 1.12.11, and the last assertion is the dual assertion of the first. The case where b holds is proved easily from a, and is left to the reader. •
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I. Background Materials
Corollary 1.12.13 Let A be an abelian category, and XQ a full subcategory of A. We set X := add.7r(,fo)- Let LJ be a full subcategory of XC\XX closed under direct sums. If for any X 6 Xo there exists some exact sequence
such that W € LJ and X' £ X, then u> is an injective cogenerator of X. Lemma 1.12.14 Let X andy be full subcategories of A which satisfy A B 1 , AB2. We assume that y C XL. Let ui be an injective cogenerator of X such that ui C X ny and addw = u>. Then (X,y,to) is a weak AuslanderBuchweitz context in A. Proof. By assumption, X fl y is an injective cogenerator of X. Hence, we have that (X,y,X n y) is a weak Auslander-Buchweitz context. By Theorem 1.12.10, 2, we have w = X D y. D Remark 1.12.15 Some important terminology in category theory is not unified in its usage. The definition of Grothendieck category in these notes is the same as that in [33]. In Freyd's book [56], a Grothendieck category means an abelian category which satisfies the (AB5) condition. For the definition of an exact category, we follow Quillen [126]. There is some old literature in which the same expression 'exact category' is used for different meanings. For the definition of generator and cogenerator, we follow Auslander-Buchweitz [10]. A family of generators (resp. generator) in the sense of Grothendieck [63] is called a family of G-generators (resp. G-generator) in these notes for clarity. A family of G-generators and a generator (in our sense) are one and the same thing for an abelian category. Moreover, the words thick and epaisse, and sheaf and faisceau are used for different meanings in these notes. Epaisse is used only for triangulated categories, and a faisceau is a sheaf in the fppf topology. The definition of Grothendieck topology and a site is a little more restrictive than that in [9]. There, a Grothendieck topology in our sense is called a pretopology, and a site in our sense is a site whose topology comes from a pretopology. The definition of sheaf in (1.8.4) is the same as that in [9, (II.6.1)], but we put an unnecessary restrictive hypothesis on the value category C (i.e., the existence of limits) for simplicity. Similarly for the definition of continuous functor (1.8.11), see [9, (III.1.6)]. The definition of contravariant and covariant finiteness of X in (1.12) may be different from that in [13], if
Notes and References. This section is merely a survey of keywords for later use, and there is no new result. For basics on category theory, relative homological algebra, and cobar resolutions, we refer the reader to [144, 105,
2. From commutative ring theory
37
106, 56, 63, 58]. For exact categories, see [126, 141, 120] for more. For triangulated categories and derived categories, see [69, 143, 130, 121]. For Grothendieck topology and sheaf theory, see [8, 111, 9]. For AuslanderBuchweitz theory, see [10, 11, 112, 146, 13].
2
From commutative ring theory
This section is devoted to giving a summary of commutative ring theory used later.
2.1 Flat modules and pure maps An .R-module M is said to be R-fiat if ? ®R M is an exact functor. It is said to be R-faithfully flat if ? ®R M is faithful exact. An .R-algebra A is called i?-flat or fl-faithfully flat if the same holds for A as an .R-module. For any multiplicatively closed subset S of R, the localization Rs is i?-flat. The following is well-known. Lemma 2.1.1 Let M be an R-module. Then the following are equivalent. 1 M is R-flat. 2 For any R-module N and any i > 0, Torf (N, M) = 0. 2' For any finitely generated ideal I of R, Torf (R/I, M) = 0. 3 M is an inductive limit of an inductive system of R-finite free modules parameterized by a directed set. 3' M is a filtered inductive limit of R-flat modules. 4 For any commutative R-algebra S, S <8>H M is S-flat. 4' For any m € Max/?, Mm is Rm-flat. The proof of 1=>3 is due to Lazard [100]. By Lazard's proof, the following holds. Lemma 2.1.2 Let R be a noetherian ring, and F a countably generated Rflat module. Then F is an inductive limit of an inductive system of R-finite free modules parameterized by the ordered set N. The following lemma is also well-known. L e m m a 2.1.3 Let M be an R-module. Then the following are equivalent. 1 M is R-faithfully flat.
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I. Background Materials
2 M is R-flat, and M ^ mM for any m € Max R. An 7?-linear map / : N —> M of iZ-modules is said to be pure if lv ® / : y ® / V — ^ V ® M i s injective for any .R-module V. If / is a split monomorphism, or / is injective and Coker / is .R-flat, then / is pure. Note that a pure .R-linear map is injective. Let TV be an 7?-submodule of M. We say that TV is a pure submodule of M, if the canonical injection TV •-> M is R-pure. Lemma 2.1.4 Let R be noetherian, P and F be R-flat modules, and f : P —» F be an R-linear map. Consider the following conditions. 1 f is injective and Coker / is R-flat. 2 f is pure. 3 For p £ Spec R, /c(p)
0 -> P/K -> F -> C -» 0,
we get a long exact sequence 0 ->• Torf (R/I, C) -¥ R/I ® P/K -> F / 7 F -> C/IC -»• 0. As R/I® f : P/IP -> F/IF is injective, we have R/I®P^ R/I
2. From commutative ring theory
39
Note that K = 0 leads to a contradiction as we already know that C is .ft-flat. We prove K = 0. If R is not an integral domain, then we have R/p ® K — 0 for p € Spec /?, and /? admits a finite filtration
of .R-modules such that for any i, Ri/Ri-i is isomorphic to R/p for some p € Spec R. This shows that K = R
f
is an R-linear map (an A-linear map if A is commutative) which is natural with respect to M, N and H. It is an isomorphism if one of the following holds. a H is R-flat and M is A-finitely presented. b M is A-projective and H is finitely presented. c H is R-finite projective. d M is A-finite projective.
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I. Background Materials
Proof. We only prove a. As both Hom,i(?, N) ® H and Homj4(?, N ® H) are left exact, we may assume that M = An by the five lemma, which case is trivial. D Corollary 2.1.8 Let R be noetherian, and assume that I is an injective R-module, and F a flat R-module. Then I <8> F is R-injective. Proof. Note that the category RM of /^-modules is locally noetherian, and RMJ is nothing but the full subcategory of finitely generated .R-modules. We have an isomorphism Hom(?, /
2. From commutative ring theory
41
Proof. As P is a direct summand of an R-free module, we may assume that P is an .R-free module with a basis B. As M is fl-finite, there exists some finite subset Bo of B such that M is contained in the fi-span Po of Bo. If we denote the fl-span of B \ Bo by Pu then we have P/M ^ Po/M © Pi. Hence, replacing P by Po, we may assume that P is infinite free. Then P/M is ^-finitely presented and i?-flat, and hence is /?-projective. D Similarly, the following holds. Exercise 2.1.12 Let R be noetherian, and / : M -> P be an .R-linear map. Assume that P is infinite projective and M is infinite. If /
2.2
Mittag-Leffler modules
We review the theory of Mittag-Leffler modules after [128]. Throughout this subsection, A denotes a directed set. (2.2.1) We say that a projective system of .R-modules V — (P\, /A^)A€A,/J>A indexed by A satisfies the Mittag-Leffler condition if for any A € A, there exists some /x > A such that for any 7 > fi, we have Im /A 7 = Im f\^. Lemma 2.2.2 (Grothendieck) LetO-*V'^>V^>V"^O be an exact sequence of projective systems of R-modules indexed by A. Then we have 1 IfV' and V" satisfy the Mittag-Leffler condition, then so does V'. ilfV
satisfies Mittag-Leffler condition, then so does V" •
3 Assume that A has a final countable subset. If P' satisfies the MittagLeffler condition, then the sequence 0 -> limP' -> limP -> limT5" -> 0 is exact. For the proof, see [64, Proposition 0.13.2]. From now on, until the end of this subsection, any projective or inductive system is assumed to be indexed by a directed set. Lemma 2.2.3 Let V be a projective system of R-modules which satisfies the Mittag-Leffler condition. If F : RM -¥ Ab is a right exact functor, then F(V) satisfies the Mittag-Leffler condition. In particular, for any R-module M, the projective system M ®V satisfies the Mittag-Leffler condition.
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I. Background Materials
Proof. Obvious.
D
Lemma 2.2.4 Let (P\) be an inductive system of finite free R-modules such that the protective system {P{) satisfies the Mittag-Leffler condition. If M is an R-module, then the "protective system (HomR (Px, M)) satisfies the Mittag-Leffler condition. Proof. Obvious by Lemma 2.2.3 and Lemma 2.1.7, d.
•
Definition 2.2.5 We say that an i?-module M is R-Mittag-Leffler if there exists some inductive system (FA) of finite free .R-modules such that M = lim FA and the projective system (F£) satisfies the Mittag-Leffler condition. By Lemma 2.1.1, an .R-Mittag-Leffler module is i?-flat. We list some properties of Mittag-Leffler modules. For the proof, see [128]. From Lemma 2.2.4, Lemma 2.1.2 and Lemma 2.2.2, we have the following. Proposition 2.2.6 Let R be noetherian. If M is a Mittag-Leffler R-module of countable type, then M is R-projective. The following criterion for the Mittag-Leffler property of an .R-module is due to Raynaud-Gruson [129]. Proposition 2.2.7 For an R-flat module M, the following are equivalent. 1 M is R-Mittag-Leffler. 2 For any inductive system (F\) of finite free R-modules such that lim F\ = M, the projective system (FJ) satisfies the Mittag-Leffler condition. 3 For any finite free R-module Q and any x € Q ® M, there is a smallest R-submodule Q' of Q such that x £ Q'
is Mittag-
Corollary 2.2.10 An R-projective module is R-Mittag-Leffler. Lemma 2.2.11 Let (M\) be an inductive system of R-Mittag-Leffler modules consisting of R-pure maps. Then lim M\ is R-Mittag-Leffler.
2. From commutative ring theory
43
Lemma 2.2.12 Let (M\) be a family of R-modules. Then for any finitely presented R-module N, the canonical map
is an isomorphism. Proof. As both sides are right exact with respect to N, we may assume that N = R, which case is trivial. D Corollary 2.2.13 / / moreover R is noetherian in the lemma, then there exists some isomorphism
Tor? (AT, I ] ^A) = I I T o r ? (N, Mx) A
A
for i > 0. Corollary 2.2.14 Let R be noetherian, and (M\) a family of R-modules. Then 17A ^ A is R-flat (resp. Mittag-Leffler) if and only if the same is true of M\ for any A. An argument similar to above shows the following. Proposition 2.2.15 Let R be noetherian, (P\) a projective system of Rflat modules (resp. R-Mittag-Leffler modules) indexed by a countable directed set which satisfies the Mittag-Leffler condition. Then limPx is R-flat (resp. R-Mittag-Leffler). A projective module over a noetherian commutative ring is characterized as follows: Theorem 2.2.16 Let R be noetherian, and P an R-module. Then the following are equivalent. 1 P is a direct sum of countable Mittag-Leffler R-modules. 2 P is R-projective. Proof. 1=>2 follows from Theorem 2.2.6. 2=>1 is well-known as Kaplan• sky's theorem [92]. Exercise 2.2.17 Let R be a noetherian, and F a countably generated flat .R-module. Prove that we have proj.dim^F < 1. Let F :
> Fn -> F n _! -> • • •
be a chain complex of .R-modules. We say that F is an i?-free (resp. Rprojective, /?-flat) complex if each Fn is i?-free (resp. .R-projective, .R-flat). A free complex F is said to be finite free (resp. finite projective) if each Fn is /^-finite free (resp. /J-finite projective) and F is bounded. Sometimes an .R-nnite projective complex is referred as a perfect complex.
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I. Background Materials
2.3 Faithfully flat morphisms and descent theory (2.3.1) Let / : A —> B be a homomorphism of commutative rings, and assume that / is faithfully flat. Then by definition, F := / # = B<8>A? : ^M —>• B M is faithful exact, and it has a right adjoint G := / * = Hom^-B, ?) : BM —• yjM. Then for an ,4-module V, the cobar resolution (1.6) Cobar F (V) of V with respect to the adjoint pair (F, G) is as follows:
where i+l
® • • • ®fy® u) = ;=0
and e(i>) = 1 <8> u. More generally, if tp : Y -> X is a faithfully flat morphism of schemes, then
0 -> T(X, M) -> T(Y,
xx Y, ip'M),
where ijj : Y Xx Y —>-Xis the canonical map. (2.3.3) functor
For a commutative ring R and an /t-scheme X, the representable
is a faisceau of i?-algebras (1.8.14). We denote it by Ox or simply by Ox (by abuse of notation). We say that an O^-module faisceau M is quasicoherent if for any X-morphism of affine schemes over X of finite type Spec B -> Spec A, the canonical map B ®A A4(Spec ^4) -> A1(Spec 5 ) is an isomorphism. (2.3.4) The restriction of a quasi-coherent O x -module M. to the Zariski site of X is a quasi-coherent Ox-module in the usual sense. Conversely, if M is a quasi-coherent Ox-module, then defining an Ox-module W(M) by W{M){f) :- r ( y , / * M ) for any / : Y -> X, W(M) is a faisceau by (2.3.2), and is clearly quasi-coherent. With this correspondence above, we identify a (usual) quasi-coherent Ox-module and a quasi-coherent O x -module. If X = Spec A is affine and M = N, then W(M) is also denoted by Na. For a quasi-coherent Ox-module M and any X-scheme / : Y —> X, we sometimes denote T{Y, f*M) simply by T(Y, M).
2. From commutative ring theory
45
(2.3.5) Let p : Y -» X be a faithfully flat morphism of schemes. Let Pi : Y xx Y -> Y be the zth projection, pj{ : Y xx Y xx Y -> Y xx Y be the morphism given by (2/1,2/2,2/3) >-> (j/i.J/j), and ft : F xx Y xx Y -> y be the zth projection. We call a pair (M,<j>) a descent datum of a quasi-coherent sheaf with respect to p, if M. is a quasi-coherent Oy-module, and
and v a
h2 : ptM
' the unit
then TV agrees with Ker(/ii - /i2) by (2.3.2). Thus, M is recovered from the (2.3.6) For descent data {M,<j>) and (M',
46
I. Background Materials
Theorem 2.3.9 Let f : Y -> X and g : X' —» X be morphisms of locally noetherian schemes. Assume that g is faithfully flat and quasi-compact. If we denote by f : Y' -> X' the base change of f by g, and if f is quasi-compact (resp. of finite type, proper, an open immersion, affine, finite, quasi-finite, flat, smooth, etale), then so is f. For the proof, see [65, IV.2.6-2.7]. For the definitions of the properties of morphisms listed here, see (2.7). Exercise 2.3.10 Let A —> B be a faithfully flat homomorphism of commutative rings, and M an ,4-module. If M ®A B is finitely generated (resp. countably generated, flat, Mittag-Leffler, finitely presented, coherent) as a B-module, then so is M as an v4-module. If A is noetherian and M <8u B is B-projective, then so is M as an ^4-module.
2.4
The /-depth
Let R be a commutative noetherian ring. The reference for this subsection is [110, Chapter 6]. (2.4.1) Let M ^ 0 be a finite .R-module. The dimension dimM of M means the Krull dimension of the ring R/ annM, where annM := {r € R | rM = 0} is the annihilator of M. Note that dim M is the same as the dimension of the closed subset supp M :={P e Spec R | MP ± 0} = Spec(ii/ ann M) of Spec R. Let / be an ideal of R. If IM ^ M, then we define depth R (7, M) = depth(7, M) := min{i | Ext l R (i?//, M) ^ 0}, and call depthR(I, M) the I-depth of M. We define depth(7, M) := oo if IM = M, as a convention. For a finitely generated i?-module M, we have IM ^ M if and only if depth(7, M) < oo if and only if depth(7, M) < dimM. (2.4.2) We say that a sequence of elements a\,..., an of R is a poor Msequence if the multiplication a* : Mj_i —> Mj_x is injective for i = 1 , . . . , n, aj)M for i = 1 , . . . , n — 1. If moreover where M = Mo, and M; = M/(ai,..., M ^ ( a i , . . . ,an)M, then we call it an M-sequence. Theorem 2.4.3 ([110, Theorem 16.6]) Let R be a noetherian ring, M a finitely generated R-module, and I an ideal of R. Assume that M ^ IM. For n > 0, the following are equivalent.
1 depth(7, M) > n.
2. From commutative ring theory
47
2 For any finitely generated R-module N such that supp N C supp R/I, we have Ext'fi(JV, M) = 0 (i < n). 2' For some finitely generated R-module N such that supp N = supp R/I, we have ExVR(N, M) = 0 (i < n). 3 There is an M-sequence a\,..., I.
an of length n consisting of elements of
(2.4.4) An M-sequence ai,...,an consisting of elements of / is called a maximal M-sequence if it cannot be extended to a longer M-sequence a i , . . . ,an,a with a £ I. By the theorem above, the length n of maximal M-sequences is independent of the choice of a sequence, and it agrees with depth(/,M). (2.4.5) For an i?-finite module M, ht denotes the height of an ideal. scheme and M. a coherent sheaf of inf l € s u p p xcodimo X l Mx. If / : Y define codimx Y := codimx f,Oy-
we set codim^M := h t a n n M , where More generally, if X is a noetherian X, then we define codim^ M to be -> X is a finite morphism, then we
(2.4.6) For an fl-finite module M, we define grade M := depth(ann M, R). By the theorem above, we have grade M = min{z | Ext'H (M, R) ^ 0}. In general, if M ^ 0, then we have (2.4.7)
proj.dim R M > codimM > grade M.
If a.\,..., aT is an ii-sequence consisting of elements of ann M, then we have codimM > ht(aj,... ,aT) = r, and the second inequality follows. The first one is a consequence of the New Intersection Theorem proved by P. Roberts, stated below. Theorem 2.4.8 ([133]) Let (R, m) be a noetherian local ring, and F a finite free R-complex. If F is not exact, and the homology groups of F are of finite lengths, then the length of F is greater than or equal to the Krull dimension dim R of R. Let h := codimM, and take a minimal prime P of a n n M such that h t P = h. Then Mp is a non-zero artinian ylp-module, and the length of the minimal free resolution (see (2.8)) of MP is at least h = h t P = dim AP by the theorem. Hence, we have proj.dimfl M > proj.dim fip MP >h = codim M, as desired, and the proof of the inequalities (2.4.7) is complete.
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(2.4.9) We say that an i?-finite module M is a perfect R-module if M ^ 0 and proj.dimR M = grade M. If M is perfect, then the inequalities in (2.4.7) are equalities of finite numbers, and we call codim M the codimension of M. Let 5 be a noetherian commutative /Z-flat algebra such that S ® M ^ 0. Then we have gradefi M < grade s S
2.5
Cohen—Macaulay, Gorenstein, and regular rings
(2.5.1) Let (R, m) be a noetherian local ring. For an ii-finite module M, we denote depth(m, M) by depth M, and call it the depth of M. If M ^ 0, then we have depth M < dim M < dim R. If depth M = dim M, then we say that M is Cohen-Macaulay. If depth M — dimi?, then we say that M is maximal Cohen-Macaulay (MCM, for short). As a convention, we define that 0 is both Cohen-Macaulay and MCM. We say that R is a Cohen-Macaulay local ring if the .R-module R is a CohenMacaulay module. (2.5.2) mula.
The next theorem is well-known as the Auslander-Buchsbaum for-
Theorem 2.5.3 ([110, Theorem 19.1]) Let R be noetherian local, and M ^= 0 be an R-finite module. //proj.dim^M < oo, then we have proj.dimR M + depth M — depth R.
2. From commutative ring theory
49
By the theorem above, if M is an MCM i?-module of finite projective dimension, then we have R is Cohen-Macaulay local, and M is i?-free. Let R be a Cohen-Macaulay local ring. Then we have Ass R = Min R. Moreover, for a non-zero fl-module M, we have (2.5.4)
dim M 4- grade M = dim M 4- codim M = dim R.
In particular, R is equidimensional (i.e., dim/? = A\mR/P for any P £ Mini?), see [110, Theorem 17.4]. If R is a Cohen-Macaulay local ring and M is an i?-finite module of finite projective dimension, then M is Cohen-Macaulay if and only if M is perfect. This is an immediate consequence of Theorem 2.5.3 and (2.5.4). (2.5.5) The following corollary to Theorem 2.4.8 had long been known as Bass's conjecture until Theorem 2.4.8 was proved by P. Roberts. Theorem 2.5.6 Let (R, m) be a noetherian local ring, and assume that there exists some non-zero finite R-module of finite injective dimension. Then R is Cohen-Macaulay. For the implication from Theorem 2.4.8 to Theorem 2.5.6, we refer the reader to [83]. Conversely, a Cohen-Macaulay local ring has a non-zero finite module of finite injective dimension [110, p. 151]. (2.5.7) A noetherian local ring (R, m) is called a Gorenstein local ring if the .R-module R is of finite injective dimension.
Theorem 2.5.8 ([110, Theorem 18.1]) Let {R,m) be a d-dimensional noetherian local ring. Then the following are equivalent. 1 R is Gorenstein. 1' inj.dim R ii = d. 2 Fori±
d, we have Ext l fi (#/m, R) = 0, and ExtdR(R/m, R) ^
R/m.
3 There exists some i> d such that Ext^(i£/m, R) = 0. 4 Ext l fi(i?/m, R) = 0 fori < d, and ExtdR(R/m, R) = R/m. 4' R is Cohen-Macaulay, and ExtdR(R/m, R) ^
R/m.
(2.5.9) For a noetherian local ring (R, m), we denote the minimal number of generators dimfl//m m/m 2 of m by emb.dim R, and call it the embedding dimension of R. We have emb.dim R > dim R in general. If this inequality is an equality, then R is called a regular local ring. The following theorem is well-known as Serre's theorem.
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J. Background Materials
Theorem 2.5.10 For a d-dimensional noetherian local ring (R,m), the following are equivalent. 1 R is a regular local ring. 2 gl.dim/? < oo. 3 gl.dimR = d. 4 proj.dim fl i?/m < oo. 5 proj.dinift R/m = d. As the assertion 5 shows that Extj^l(R/m, R) = 0, so a regular local ring is Gorenstein. Moreover, any regular local ring is a UFD (proved by Auslander-Buchsbaum). (2.5.11) If R is a Cohen-Macaulay (resp. Gorenstein, regular) local ring and P is a prime ideal of R, then the localization RP is again CohenMacaulay (resp. Gorenstein, regular). If R is a noetherian local ring and M is an MCM .R-module, then MP is again MCM for any P G Spec R. (2.5.12) Let R be a noetherian ring which is not necessarily local. We say that R is Cohen-Macaulay (resp. Gorenstein, regular, normal) if Rp is Cohen-Macaulay (resp. Gorenstein, regular, an integrally closed domain) for any P £ Spec R. A locally noetherian scheme X is called Cohen-Macaulay (resp. Gorenstein, regular, normal) if the same is true of Ox,x for any x € X. It is equivalent to say that so is Ox,x for any closed point x of X. (2.5.13) Let R be noetherian, and i > 0. We say that an .R-finite module M satisfies Serve's (Si) condition if depthMp < i implies that Mp is MCM as an /? P -module for any P £ Spec R. A finite .R-module is MCM if and only if (Si) condition is satisfied for any i > 0. We say a ring R satisfies (Si) condition if the .R-module R does. We say that R satisfies Serre 's (Ri) condition if dim Rp < i implies that Rp is regular local for any P G Spec R. For a noetherian ring R, R is reduced if and only if R satisfies (RQ) and (Si). R is normal if and only if (i?i) and (S2) are satisfied (Krull-SerreNagata's theorem).
2.6
Local cohomology
The references for this subsection are [26] and [70].
2. From commutative ring theory
51
(2.6.1) Let X be a topological space, Y a closed subset of X, U := X — Y, and T an abelian sheaf on X. Then we denote the kernel of the natural restriction map Y{X,T) -*• T{U,T) by TY(X,T). Note that TY{X,?) is a left exact functor. We denote its ith right derived functor by HY(X, ?), and call it the ith local cohomology functor with support in Y. What we are interested in here is the following case. R is a noetherian ring, X = Specfl, / is an ideal of R, Y = V(I) = Spec R/1, M is an Rmodule, and T = M. In this case, we denote TY(X,F) by F/(M). As is easily verified, we have T](M) = {m € M | 3n Inm = 0} = lim EomR{R/In, M), and hence F/(M) is an ii-module in a natural way. As J is flabby for an injective .R-module J [71, Proposition 3.4], and we have HY{X,J-) = 0 (i > 0) for any flabby sheaf T [70, Proposition 1.10], the derived functor of F/(M) in the category of i?-modules agrees with the local cohomology, which we denote by H)(M). As RM satisfies the (AB5) condition, we have H)(M) S limExt' fi (R/r, M). If dimR = n, then we have H)(M) = 0 (i > n) [70, Proposition 1.12]. Lemma 2.6.2 Let R be a noetherian ring, I a proper ideal of R, M an R-finite module, and n > 0. Then the following are equivalent. 1 H){M) = 0
(i
2 depth(7, M) > n. For the proof, see [70, Theorem 3.8].
2.7
Ring-theoretic properties of morphisms
The references for this subsection are [65], [17], [18], [19], [15] and [20]. (2.7.1) Let k be a field. We say that afc-algebraR is geometrically regular (resp. normal) over k if K <S>k R is regular (resp. normal) for any finite algebraic extension K of k. If k is a perfect field, then R is geometrically regular (resp. normal) if and only if R is regular (resp. normal). Let ip : X —> Y be a morphism of locally noetherian schemes. We say that ip is regular (resp. normal, flat) at x 6 X if Ox,x is flat over Oy^,) and Ox,xlmY,
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I. Background Materials
(2.7.2) Let (/? : X —> Y be a morphism locally of finite type of locally noetherian schemes. For x £ X, we define dim^x) to be inixeU dim(£/ D ip~l{
= S' ® s QS/R,
Moreover, for any multiplicatively closed subset F of 5, where 5' = R'^RS. we have Qsr/R — Sr ®s ^S/R [HO, P- 198]. Hence, as the patching process goes well, a quasi-coherent Ox-module CIX/Y is defined for a morphism of schemes X —> Y. We call QX/Y the sheaf of Kahler differentials. If X —> Y is a morphism locally of finite type between locally noetherian schemes, then Q,X/Y is coherent. Proposition 2.7.5 Let
2
2. From commutative ring theory
53
For the proof, see [66, (II.5.5)]. In particular, if tp is smooth, then tp is equidimensional and its relative dimension is rankfix/y over each component. Let tp : X —> Y be a smooth morphism of locally noetherian schemes. We denote the invertible sheaf A top ^ x / y by u>x/yTheorem 2.7.6 Let X —>• Y -^-> Z be a sequence of morphisms locally of finite type of locally noetherian schemes. Then we have 1 There exists an exact sequence of the form f*nY/z
-> Vx/z -> ftx/y -> 0.
2 //, moreover, f is smooth, then there is an exact sequence 0 -> f'nY,z
-> fix/z -> fix/y -» 0.
3 If f is a closed immersion, and I is the defining ideal sheaf of X, then there exists an exact sequence of the form
i/i2
-> f*nY/z
-> nx/z ->• o.
4 //, moreover, X is smooth over Z in 3, then there exists some exact sequence of the form
o -> J / J 2 -> / * n y / z -> n X/2 -> o. If gf in 4 is an isomorphism, then we have fCly/z
— X/X2.
(2.7.7) So far we have defined Cohen-Macaulay and Gorenstein properties of rings. We extend these notions to morphisms of finite flat dimensions. For two commutative local rings (R, m) and (5, n), we say that a map
completion
A
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I. Background Materials
satisfies the condition, we say that ip is a Cohen-Macaulay (resp. Gorenstein, complete intersection (c.i., for short), of finite flat dimension) local homomorphism. As we have flat.diniR5 = flat.dirriRS < oo <£=> proj.dim T 5 < oo, flat.diniH S < oo if and only if ip is of finite flat dimension. If S is ii-flat, then
Theorem 2.7.10 Let
2. From commutative ring theory
55
be a sequence of morphisms of locally noetherian schemes. Then the following hold. 1 If ip and if are Cohen-Macaulay (resp. Gorenstein, l.c.i, regular), then so is ipip. 1' IfY and
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2.8
Betti numbers, Bass numbers and complete intersections
(2.8.1) Let (R, m) be a noetherian local ring, and M an .R-finite module. We denote dimfl/m Torf {R/m, M) by /3/*(M), and call it the zth Betti number of M. We denote dimfi/m Ext'fi(i?/m, M) by ^'R(M), and call it the zth Bass number of M. Note that /?f(M) < oo and A4(M) < oo for any i £ Z. We call the power series P$(t) := Ei>o PpiM)? £ Z[[i\] the Poincare series (or Settz series) of M. The power series Q%(t) := Ei>o/4( M ) 4 i 6 Z[[t]] is called the Bass series of M. We say that an .R-free complex (2.8.2)
F:
> Fi % F^ -> • • •
is minimal if di(Fi) C mFj_i for each i. Lemma 2.8.3 Let (R, m) 6e a noetherian local ring, M a finite R-module, and F (2.8.2) an R-free resolution of M. Then the following are equivalent. 1 F is minimal. 2 Fo —> M and Fi —> Im d{ (i > 1) are projective covers. 3 Fori>
0, we have rank F* = /3f(
TViere exists some minimal free resolution of M, uniquely up to isomorphisms of complexes. Proof. Follows easily from Lemma 1.12.5 and Lemma 1.12.4.
D
If F is a minimal free resolution of R/m, then we have fSR(R/m) = dim f l / m // i (Hom R (F, J R/m)) = dimR/m(R/m
® R Fi)* = rankFi = /3*(fl/m).
Proposition 2.8.4 Let (R, m) be a d-dimensional noetherian local ring, and e = emb.dim R. Then we have
2 0?(R/m) > Q + e - d. 3 The following are equivalent.
a 0*(R/m) = Q + e - d
2. From commutative ring theory b' There exist some e' > 0 and d > 0 sucA «/m£ P£/m(<) = (1 + t)e'/(l
57 -
ey. c R is a complete intersection. For the proof, see [14, Theorem 7.3.3]. What is important is that the complete intersection property of R is determined only by the Betti series of R/m. Lemma 2.8.5 If M is a non-zero finite R-module, then the following hold: 1 depthfi M = inf{i | /4(M) ^ 0}; 2 inj.dim R M = sup{i | fJR{M) ± 0}. (2.8.6) Let M be a Cohen-Macaulay /?-module. Then we call iidR(M) the Cohen-Macaulay type (or sometimes simply the type) of M, and denote it by typeM, where d = dimM = depth M. We say that an .R-module M is Gorenstein if M is a Cohen-Macaulay .R-module with typeM = 1. Do not confuse a module of Gorenstein dimension 0, which is sometimes erroneously referred to as a Gorenstein module in the literature, with a Gorenstein module.
2.9 Resolutions of perfect modules Let R be noetherian. Lemma 2.9.1 Let M ^ 0 be a perfect R-module of codimension h, and N a finite R-module. Then ExtR(M, R) is perfect of codimension h, and E x t ^ M , N) S Tor«_i(VxtR(M, R), N) for i > 0. Proof. Let F be a finite projective resolution of M of length h. Then as Ext^(M, R) = 0 (i =fi h), the complex F*[/i] is a finite projective resolution of ExthR(M, R) of length h. Hence,
and it is 0 for i ^ h, and is M for i = h. As M ^ 0, we have gradeExthR(M,R) = h> proj.dim R Ext£(M,R) > gradeExt£(M,R), and Ext^(M, R) is also perfect of codimension h. Moreover, M, N) S ^(Hom R (F, iV)) = / T ( F
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Corollary 2.9.2 Let R be a noetherian ring, I a Gorenstein ideal of R of codimension h, and N an R-module. Then we have
ExthR(R/I, R) ®R/I Tor?(R/I, N) * Ext{f (fl/7, N) for any i e Z . The next theorem is called the depth sensitivity of a resolution of a perfect module. Proposition 2.9.3 Let R be a noetherian ring, M a perfect R-module of codimension h, and N a finite R-module. Assume that M
>A1^J^>fl->0,
where the boundary map is given by di(eu A • • • A e,,) = ^ ( - l ) ^ 1 ^ ^ , A • •l- • • A e,,). i=\ For an i?-module M, we denote Kos(ai,..., a/,)<8>J{M by Kos(ai,..., O/,; M). The next lemma, which is standard [110, Theorem 16.8], is called the depth sensitivity of a Koszul complex. Lemma 2.9.5 Let R be a noetherian ring, ai,...,ah € R, and M be a finite R-module. Set I := ( a i , . . . , a^) C R. If M ^ IM, then we have depth fi (7, M) = h- inf{z | fT i (Kos(ai,..., afc; M)) ^ 0}. In particular, Kos(oi,...,%; M) is a resolution of M/IM if and only if depth(7, M) = h if and only ifa\,..., a/, is an M-sequence. Assume that a i , . . . , a/, is an ii-sequence so that 7 := ( a i , . . . ,a/,) is a complete intersection ideal. Then F := Kos(ax,..., ah\ R) is a resolution of R/I. As we have F*[/i] = F, we see that 7 is a Gorenstein ideal. The next lemma is called the rigidity of a Koszul complex. Lemma 2.9.6 Let R be noetherian, I a complete intersection ideal of R of codimension h, and M an R-module. If i > 0 and Tor?(7?/7, M) = 0, then we have Torjl(7?/7, M) — 0 for any j > i. In particular, we have depth(7,M) > h-i.
2. From commutative ring theory
59
Note that even the following holds. Theorem 2.9.7 (Lichtenbaum [101]) Let R be a regular ring, M and N finite R-modules, and i > 0. If Tor?(M, N) = 0, then Torf (M, N) = 0 for j > i-
2.10
Dualizing complexes and canonical modules
For dualizing complexes, see [69] and [80]. (2.10.1) Let X be a noetherian scheme. We say that a complex of quasicoherent Ox-modules /* is a dualizing complex of X if /* is bounded, each term of 7* is an injective Ox-module, each cohomology group of /* is coherent, and the canonical map
is a quasi-isomorphism. Usually, a dualizing complex is regarded as an object of the derived category D + ( x M ) , and hence any object isomorphic to a dualizing complex in D+(xM) is also called a dualizing complex. Note that a quasi-coherent Ox-module 7 is an injective Ox-module if and only if its stalk Ix at x is an injective Ox,i-module for any x £ X [69, Proposition II.7.17]. If /* is a dualizing complex of X, for any complex F* of Ox-modules with coherent cohomology groups, the canonical map F* -> H o m ^ H o m ^ F ' , /*),/•) is a quasi-isomorphism [69, Proposition V.2.1]. The dualizing complex is unique, in the following sense. Theorem 2.10.2 ([69, Theorem V.3.1]) Let X be a connected noetherian scheme, 7* a dualizing complex of X, and I" a complex of Ox -modules bounded above with coherent cohomology groups. Then 7'* is dualizing if and only if there exists some invertible sheaf L and some integer n such that I'' is isomorphic to I* ®o x ^[ n ] * n D(xM). In this case, L and n are determined by
(2.10.3) A complex 7* of quasi-coherent Ox-modules is called a fundamental dualizing complex if 7* is bounded with coherent cohomology groups, and
©7'^©J(z) i€Z
x€X
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I. Background Materials
is satisfied, where J(x) denotes the constant sheaf (ix),{Eox X(K(X)))~, where EOXX(K(X)) denotes the injective hull of the C?x,x-module K{X), and ix : Spec Ox,x —• X is the canonical map. A fundamental dualizing complex is a dualizing complex, and if there is a dualizing complex of X, then there is a fundamental dualizing complex of X [69, V.2.3, V.7.3]. (2.10.4) If there is a dualizing complex of X, then X is finite dimensional. A bounded-below complex /* of quasi-coherent Ox-modules with coherent cohomology groups over a locally noetherian scheme X is called pointwise dualizing if /* is a dualizing complex of SpecO^.i for any x G X. A dualizing complex of a noetherian scheme is pointwise dualizing. Conversely, a pointwise dualizing complex of a finite dimensional noetherian scheme is dualizing. (2.10.5) Let (R, m) be a noetherian local ring, and / ' a dualizing complex of R, that is to say, the complex of Ox-modules associated to / ' is a dualizing complex of X, where X = Spec/?. Then ExtlR(R/m,I') is non-zero for one and only one i, and it is isomorphic to R/m. If the i such that Ext^(i?/m,7') ^ 0 is zero, then we say that /* is a normalized dualizing complex. Note that a normalized fundamental dualizing complex is unique up to isomorphisms of /^-complexes. For a fundamental dualizing complex I' of (R, m), any localization 7* at p £ Spec R is again a fundamental dualizing complex. If moreover / • is normalized, then /*[—dim/?/p] is normalized. If R is a finite dimensional Gorenstein ring, then a minimal injective resolution /* of the .R-module R is a fundamental dualizing complex of R. If R is local and d = dim R, then I'[d] is normalized. The following theorem due to T. Kawasaki is an affirmative answer to Sharp's conjecture for local rings. Theorem 2.10.6 ([94, Corollary 6.2]) Let R be a noetherian local ring. Then R has a dualizing complex if and only if R is a homomorphic image of a Gorenstein local ring. The following theorem is known as the local duality theorem. Theorem 2.10.7 ([69, Theorem 6.2]) Let (i?,m) be a noetherian local ring, I' a normalized dualizing complex of R, and M a finite R-module. Then there is an isomorphism Hlm(M) * Homfi(Ext^(M, / ' ) , ER(R/m)) which is natural with respect to M.
2. From commutative ring theory
61
Proof. As Uom'R(M, I') is bounded with .R-nnite homology groups, it has a free resolution F, with each term finite free. As /* is dualizing, there exist quasi-isomorphisms M -> H o m ^ H o m ^ M , /*), /*) ^ H o m ^ F . , /*). By Corollary 2.1.8, H o m ^ F . , /*) is an injective resolution of M. Hence, we have
H'm(M) * H i (r m (Hom^(F. 1 /'))) £
Hi{HomR{F.,rm(r))).
Note that we have F m (/*) is quasi-isomorphic to Eii(R/m), which is easily seen when we consider the case /* is fundamental. As En(R/m) is an injective module, we have quasi-isomorphisms F., ER(R/m))
* Hom R (Hom^(M, / ' ) ,
ER(R/m)).
Hence, we have the isomorphism in question, as desired.
2.11
•
The duality of proper morphisms and rational singularities
(2.11.1) Let A be an abelian category, and A' its thick subcategory. We denote the full subcategory of D1(A) consisting of objects X such that H^X) € A' for any i by D\,(A), where ? is either b, +, - or 0. Obviously, D^^A) is a triangulated subcategory of D1{A). For a locally noetherian scheme X, we denote DlohX(xM) (resp. Z ^ c o X U M ) ) by Dl(X) (resp. Dlc(X)). Note that the forgetful functor £>?(QcoX) -> D\C(X) is an equivalence for a quasi-compact scheme X for ? = +,0 [25, 6.7]. (2.11.2) Let X be a noetherian scheme. The following was proved by M. Nagata [118]. See also [103]. Theorem 2.11.3 Let f : Y -> X be a morphism of finite type between noetherian schemes. Then f is compactifiable in the sense that there exist some scheme Y, a proper morphism p : Y —> X, and an open immersion i: Y -¥ Y such that pi = / . A factorization pi = f as in the theorem is called a compactification of / . The following is known as the global duality theorem of proper morphisms, see [121] and [102]. See also [69] and its appendix by Deligne [41]. Theorem 2.11.4 Letp :Y -» X be a proper morphism between noetherian schemes. Then the derived functor Rpt : Dqc(Y) —>• Dqc(X) (the unbounded derived functor, see [137]) has a right adjoint p'
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(2.11.5) Let F(X) denote the category of X-schemes of finite type. Any morphism / : Y -> Y' of F(X) has a compactification pi = f. We define f := i'op1 : D+C(Y') -» D+C(Y), where p ! is the right adjoint of R+pt. Proposition 2.11.6 Under the notation as above, the following hold. 1 The definition of f' is independent of the compactification pi = f of f, up to isomorphisms of functors. 2 For two morphisms f and g in F(X), that go f is defined.
we have (g o / ) ' = / ! o g'-, provided
3 (Residue isomorphism) If h : Y —> Y' is a smooth X-morphism of relative dimension d, then h' is isomorphic to h* := h*? ®£>y wy//y[d]. 4 / / g : Y -» Y' is a finite X-morphism, g'RHom'Cr,(g*Qv,?), where g : (Y,OY) morphism of ringed spaces.
then g- is isomorphic to g^ := ->• (Y',gtOY) is the canonical
5 Let f : Y —> Y' be a morphism of F(X), and g : Z' —» Y' aflat morphism of noetherian schemes. If gf = fg' is a fiber square, then we have a canonical isomorphism (g')* o /• = (/')' o g*. 6 If X has a dualizing complex Ix, then Iy •= fix is a dualizing complex of Y for any morphism f :Y —> X of finite type. Corollary 2.11.7 (Duality for proper morphisms) Let p : Y —» X be a proper morphism between noetherian schemes, F e Dqc(Y) and G £ D*C(X). Then there is an isomorphism 9P : /Zp..RHomg v (F.p l G) = /ZHom^.(flp.F.G). which is functorial on F and G. Proof.
(Sketch) Consider the composite of the canonical maps RHom' ^
(Rp.F,e) >
where the first arrow is the natural map given in [102, (3.5.4)], and e : Rptp- -> Id is the counit of adjunction. It suffices to show that
RT(U,ep) : Rnom'0uy(W\UY, (p'G)\Uy) -» R is an isomorphism for any open set U of X, where Uy = p~l(U). As we may identify (P'G)\UY with (p|£/y)!(G|t/) by 5 of Proposition 2.11.6 and (Rp,F)\u by R(p\uy)*F\uY' w e m a y assume that U = X, after replacing X by U and p by p\uY • Then the assertion is clear, because p! is the restriction of p ! ' ub , and p ! ' ub is the right adjoint of RptD
2. From commutative ring theory
63
(2.11.8) Usually, if the base scheme X and its dualizing complex Ix are obvious from the context, then we call IY := f{Ix) 'the' dualizing complex of Y for any X-scheme / : Y —> X of finite type. If X = Spec /? is affine and R is a d-dimensional Gorenstein local ring, then we always consider that Ix = R[d}. (2.11.9) Let X be a noetherian scheme with a fundamental dualizing complex Ix, and Y a connected X-scheme of finite type. We set the minimal i £ Z such that H'(IY) ^ 0 to be r. We denote Hr{IY) by wy, and call it the canonical sheaf of Y. The coherent Oy-module wy is determined only by Y (not depending on X or Ix) up to the tensor product with an invertible sheaf. If Y is disconnected, then we define u)Y componentwise. Let (R, m) be a complete noetherian local ring. In this case, the fundamental dualizing complex IR of R is uniquely determined up to degree shifting. Hence, U/R is uniquely defined to be the non-zero cohomology group of IR. We call UIR the canonical module of R. For a not necessarily complete noetherian local ring (R, m), if there exists a finite ^-module K such that K = wR, then such a K is unique up to isomorphisms, where R is the m-adic completion of R, and K — R® K. We usually denote this K by KR, and call it the canonical module of R. If R has a dualizing complex, then we have GiR = KR. However, R may not have a dualizing complex even if there is a canonical module of R. Lemma 2.11.10 Let Y be a connected X-scheme of finite type. Then the following are equivalent. 1 For some d 6 Z, we have LJY = IY[—d] in D(Y). 2 wy has a finite injective dimension as an object of YM. 3 Y is Cohen-Macaulay. If the conditions are satisfied, then we have supp wy = Y, and in particular we have wy,y = u>oYjV for any y eY. Proof. 1=^2 is trivial. 2=>3 We set suppwy = Z. We define r to be the minimum i such that H'(Iy) j= 0. Let J be a fundamental dualizing complex which represents IY. Note that Z is the union of all irreducible components YJ of Y such that J(rji) is a direct summand of Jr, where T)t is the generic point of Yt. For y 6 Z, wy y is a non-zero, finitely generated Oy>!(-module of finite injective dimension by [69, Proposition II.7.20]. Hence, by Theorem 2.5.6, Y is Cohen-Macaulay at any point of Z. We denote by Z' the union of all irreducible components of Y not contained in Z. If Z' ^ 0, then as Y is connected, there is a point y G Z D Z'. As Jy is a fundamental dualizing complex of the
64
/. Background Materials
Cohen-Macaulay local ring Gy,y, we have that the positions s at which EoYn (K(rji)) appears as a direct summand of J* are equal for all generic points of irreducible components of Y which contain y. This contradicts y € Z fl Z', and we have that Z' = 0. Hence, we have suppwy = Z = Y, and Y is Cohen-Macaulay. We show 3=>1. An argument similar to above shows that suppwy = Y. For each y € Y, there is at most one i such that H^ (OY,y) ^ 0. By the local duality, there is at most one i such that Hl(IYiy) ^ 0. As we have ujYy 7^ 0, there exists some d G Z, which is independent of y, such that H~d(IY,y) = uy,y 7^ 0. Hence, we have Hl(IY) = 0 (i ^ -d), and we are done. • Corollary 2.11.11 Let f : Y -> Y' be a finite X-morphism between Xschemes of finite type. If Y is connected and Y' is Cohen-Macaulay, then we have where h = codimy Y. Proof. We may assume that Y' is also connected. The morphism of ringed spaces / : (Y, OY) -> (Y1, f,OY) is flat. There exists an integer d such that IY<[—d] is an injective resolution of uiyi. Hence, we have
W(IY) £ /T(f (Hamo^/.CV./y.))) s /•Ext££(/.CV,wK.)This cohomology group is equal to zero if and only if
for any y' G Y'. By the local duality theorem, this module is non-zero for i = — d + codim0v,, ,(/»Cy) y ', and is zero if i is smaller than this value, see [26, Corollary 3.5.11]. Hence, the minimum i such that H'(IY) ^ 0 is h - d, and we are done. Q Lemma 2.11.12 Let f : Y —> Y' be a smooth X-morphism between noetherian X-schemes of finite type. Then we have U)Y = LJyi ®
Proof. Obvious.
Wy/Yi.
Q
Proposition 2.11.13 Letk be afield, X an x-dimensional Cohen-Macaulay normal k-variety, Y a k-scheme of finite type. Let f : X -> Y be a proper k-morphism, and assume that Oy -> f,Ox is an isomorphism, RlftOx = 0 (i > 0), and there exists an r > 0 such that Rlf,u>x = 0 (i ^ r). Then Y is an (x — r)-dimensional Cohen-Macaulay normal variety, f is surjective, and Rrftux = wy-
2. From commutative ring theory
65
Proof. Note that Y is connected, as X is connected and the support of f,Ox is contained in exactly one connected component of Y. Set y := dimV. As Y is of finite type over Specfc, it is easy to see that u>y = H~V(IY) (we consider that the current base scheme is Specfc). By Corollary 2.11.7 and assumption, we have
H\IY) = Ex£,Y(f,Ox, h) = Ri+Xf^xBy assumption and Lemma 2.11.10, we have Y is Cohen-Macaulay, — y+x = r, and Rrftux = wy. It remains to show that Y is normal and / is surjective. As the question is local on Y, we may assume that Y = Speed, with A of finite type over k. As
A = H°(X,OX) =
f)Ox,x
xex is an intersection of normal domains, A is a normal domain. As / is domi• nating and is a closed map, it is surjective. (2.11.14)
A desingularization / : X —> Y which satisfies the conditions
Rif*Ox = 0 {i > 0), f.Ox = OY, and fff^x = 0 (i > 0) is called a rational resolution of Y. By Proposition 2.11.13, if Y has a rational resolution, then it is a Cohen-Macaulay normal variety. Assume that the characteristic of k is zero, and Y is integral. Then Y has a rational resolution if and only if any desingularization of Y is rational. If the equivalent conditions are satisfied, then we say that Y has (at most) rational singularities.
2.12
Summary of open loci results
Let R be a noetherian ring. Definition 2.12.1 We say that a finite .R-module M is of Gorenstein dimension 0 if M is reflexive (i.e., the canonical map M —> M** is an isomorphism), and Extjj(M, R) = 0 = Extjj(M*, R) for any i > 1. We set Q to be the full subcategory of RMJ consisting of modules of Gorenstein dimension 0. For N € RM^, we call Q -resol.dim N the Gorenstein dimension of N. L e m m a 2.12.2 Let R -> 5 be a homomorphism of commutative noetherian rings, N a finite R-module, M a finite S-module. 1 For each of the following conditions, the subset of Spec R consisting of p € Speci? such that the condition is satisfied is Zariski open: Mp — 0, Mf is Sp-free, Mp is of Gorenstein dimension 0 as an Sp-module.
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I. Background Materials
2 If S is of finite type over R, then the subset of Spec 5 consisting of P € Spec 5 such that Mp is RRnP-flat is open. 3 If S is essentially of finite type over a complete local ring or S is essentially of finite type over Z, then for each of the conditions CohenMacaulay, Gorenstein, l.c.i., and regular, the subset of Spec 5 consisting of P € Spec 5 such that the condition is satisfied for Sp is open. 4 If S has a dualizing complex, then for each of the conditions equidimensional, Cohen-Macaulay, and Gorenstein, the subset of Spec S consisting of P S Spec 5 such that the condition is satisfied for Sp is open. 5 If S is a homomorphic image of a Cohen-Macaulay ring, then the CohenMacaulay locus and the MCM locus of M are open. In particular, the Cohen-Macaulay locus of S is open. For the proof, see [110], [61], and [65]. Corollary 2.12.3 Let R be a noetherian ring, and M a finite R-module. For each i with 1 < i < oo and each of the following conditions, the subset of Spec R consisting of P G Spec R such that the condition is satisfied is open: proj.dim fip MP < i, Q -resol.dim MP < i and MP is zero or perfect of codimension i. If R is Cohen-Macaulay, then the subset of P such that dim Rp — depth R p Mp < i is also open. Corollary 2.12.4 Let tp : X —¥ Y be a morphism locally of finite type between locally noetherian schemes. Then for each P of the following properties, the subset [/(P,
2. From commutative ring theory
67
with / a perfect ideal of codimension h < oo, say, to prove the Gorenstein and l.c.i. assertions. As the Gorenstein locus of tp is nothing but the rankone free locus of the finite B-module Ext^(S, A), it is open in Y. The l.c.i. locus of (p agrees with the free locus of I/I2, see Lemma 2.13.2 below, hence it is open. To prove the assertion for a smooth locus, we may assume by the discussion above that
2.13
Normal flatness
(2.13.1) Let R be a commutative ring, / an ideal of R, and M an Rmodule. We set G := Gr/ R to be the graded ring associated to the ideal /. That is to say, G is the R//-module ®i>o P/P'+1 equipped with the Nograded ^//-algebra structure P/Ii+1 ®R/I P/Ij+1 -> Ii+J/Ii+i+l induced is a graded Gby the product of R. Note that Gr7 M := ©i>0 PM/Ii+1M module in a natural way. If Gr/ M is i?//-flat (in other words, if PM/P+1M is i?//-flat for any i £ No), then we say that M is normally flat along /. Let A" be a scheme, I a quasi-coherent ideal sheaf of Ox which defines a closed subscheme Y of X, and M £ Qco(X). Then we define Qx%M := ©i>0 TM/li+lM. Note that Gr z Ox is a sheaf of Oy-algebras, and Gr z M is a Grj Ox-module. If (Grj M)x is Ox.x/^i-flat for all x £ X, then we say that M is normally flat along I (or along V). We say that X is normally flat along Y if Ox is. Lemma 2.13.2 Let R be a noetherian local ring, and I a proper ideal of R. Then the following are equivalent. 1 / is a complete intersection ideal. 2 proj.dim R / < oo, and I/I2
is R/1-free.
2' proj.dimR I < oo, and R is normally flat along I. 3 I/I2 is R/1-free, and the canonical map Sym fl / 7 1/I 2 —• Gr/ R is an isomorphism. For the proof, see [110]. Theorem 2.13.3 ([110, Theorem 15.7]) Let R be a noetherian local ring, and I a proper ideal of R. IfwesetG = Gi[R, then we have dim G = dim R.
68
I. Background Materials Although we will not use it later, the following is also important.
Theorem 2.13.4 Let R be a noetherian local ring, and I a proper ideal of R. If G is Cohen-Macaulay (resp. Gorenstein, regular, normal), then so is R. For the proof, see [34, Theorem 3.9, Theorem 3.13]. This theorem is a corollary to Corollary II.2.4.3, see [16]. Lemma 2.13.5 Let (R, m) be a noetherian local ring, and M a finite Rmodule. Let P be a prime ideal of R, and assume M is normally flat along P. Then the following hold: 1 IfM^O,
then dim M = dim MP + dim R/P.
2 If R is normally flat along P, then M is R-free if and only if Mp is Rp-free. If, moreover, Ext'R(R/Pt M) is R/P-free for all i > 0, then the following hold: 3 We have depth M = depth Mp + depth R/P. In particular, M is CohenMacaulay if and only if both MP and R/P are Cohen-Macaulay. If this is the case, then we have type M = type MP • type R/P, where type denotes the Cohen-Macaulay type, see (2.8.6). In particular, M is Gorenstein if and only if both Mp and R/P are Gorenstein. 4 Assume that R/P i>0.
is Gorenstein.
Then /x£
Proof. 1 As M/PM ^ 0 and M/PM dim M/PM, and hence
lm
is R/P-hee,
(M) = ^lRp{Mp)
for
we have dim R/P =
dim M > dim MP + dim R/P. On the other hand, dim M - dim Gr P M < dim /t(m) <8> Gr F M + dim
M/PM.
As Grp M is /?/P-flat, we have dim K(ITI)
since the Hilbert function of K(m)
= dim K(P)^)GTP
MP,
M.
2. From commutative ring theory
69
We show 2. The 'only if part is trivial. We prove the 'if part. By assumption, the canonical map 7n
: (Pn/Pn+1)
®R/P M/PM
->
PnM/Pn+1M
is a surjective map of finite free R/P-iree modules for any n. The localized map (jn)p is an isomorphism by [110, Theorem 22.3], as we assume that Mp is .Rp-free. As 7 n is a surjective map between finite free modules of the same rank, it is an isomorphism. Using the local criterion [110, Theorem 22.3] again, M is fl-flat, and it is R-iree. We show 3. We set depth Mp = qQ and depth R/P = p0. There is a spectral sequence (2.13.6) E%« = ExtpR/P{R/m, As ExtqR(R/P, M) is R/P-tee, ExtR(R/P,M)P
ExtqR(R/P, M)) => ExtpR+q{R/m,
M).
ExtqR(R/P, M) = 0 if and only if = ExtRp(K(P),MP)
= 0.
Hence E%'q = 0 for q < q0. On the other hand, as ExtqR{R/P, M) is R/Pfree, we have E^ = 0 for p < p0. Hence Ext^(i?/m, M) = 0 for i < p0 + q0. On the other hand (2.13.7)
Ext%+qo(R/m, M) S E™M ^ E$°'qo
, Ext%(R/P, M)) ± 0. Hence depth M =po + q0 = depth MP + depth R/P. By 1, M is Cohen-Macaulay if and only if both Mp and R/P are CohenMacaulay. Assume that M is Cohen-Macaulay (and hence both Mp and R/P are Cohen-Macaulay), and set typeMp = r and type R/P = r'. Then as we have Extq£{R/P,M) ^ {R/P)Br, , M) S Ext£ / P (i?/m, {R/P))®r
£
by (2.13.7). This shows typeM = rr'. 4 As we have E$'q = 0 for p =£ p0 = dim R/P in (2.13.6), the spectral sequence (2.13.6) collapses, and the assertion is clear. • Remark 2.13.8 In [81] are listed some examples which show that even if R/P is regular, RP is Cohen-Macaulay and R is normally flat along P, R may not be Cohen-Macaulay. The freeness of Ext'R(R/P, R) is really necessary.
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I. Background Materials
Lemma 2.13.9 Let (R,m) be a noetherian local ring, and P G Spec/?. Assume that R is normally flat along P, and R/P is regular. Then we have: 1 emb.dim R — dim R = emb.dim Rp — dim Rp. In particular, R is regular if and only if Rp is regular. 2 Assume moreover that ExtfR(R/P, R/P) is R/P-free fori>2. Then R is a complete intersection if and only if RP is a complete intersection. Proof. We take bi,... ,bd G m so that the image of b\,... ,bd in R/P forms a regular system of parameters of R/P, where d = dim R/P. We set J := (&i,..., bd). As the image of 6 X ,..., bd in m f l / p / m ^ p = m/(m 2 + P ) is linearly independent, its image in m/m 2 is also linearly independent, and we have emb.dim R/J = emb.dim R — d. As &i,..., bd is an /?/P-sequence, the Koszul complex Kos(6i,..., bd; R/P) is acyclic. As P{/Pi+l is R/P-hee by assumption, Kos(6 1 ; ..., bd; P'/P'+1) is also acyclic. If j > 0 and a is a j-cycle of F := Kos(&i,... ,bd; R), then a G r\i>0(Bj(¥) + P'F 7 ) = Bj(¥), and hence F is also acyclic, where Bj(F) is the set of j-boundaries of F. We have that b\,..., bd is an it-sequence, and F is a free resolution of R/J. In particular, dim R/J = dim R- d. Next, take a free resolution G of R/P. As b\,... ,bd is .R/P-regular, we have Tox?{R/P,R/J) = 0 (i > 0). Hence, G ®R R/J is an R/J-iiee resolution of R/m = R/(J + P). As we have an isomorphism (G ® fi G) <8>K R/J s (G
R/J),
we obtain a spectral sequence Elq = Tor*(Torf (R/P, R/P), R/J) => Toi%.Jq(R/m, R/m) associated with the double complex (G ®R G) <8> F. As TOT*(R/P, R/P) = R/P and Torf {R/P, R/P) S P/P2 are R/P-hee by assumption, we have isomorphisms Torf/J (/?/m, R/m) s Torf (R/P, R/P) ®H R/J ^Tov«(R/P,R/P)®R/PR/m for i = 0,1,2. As Torf (R/P, R/P) is R/P-hee for i = 0,1, emb.dim R/J = (3*'J(R/m) = dimR/m Torf (R/P, R/P) ® H / P R/m = dim K ( P ) Torf p (/t(P),K(P)) = emb.dim RP. Combining this with Lemma 2.13.5, we have emb.dim R - dim R = emb.dim R/J + d - dim R = emb.dim RP — dim RP,
3. Hopf algebras over an arbitrary base
71
and we have proved 1. We show 2. As G, G) ®fi R/J ^ H o m ^ G ®fi R/J, G ®R R/J) and proj.dim^ R/J < oo, there is a spectral sequence Ep2'q = Tor%(ExtR(R/P, R/P), R/J) =» Ext^« (fl/m, R/m), see Lemma III.2.1.2. As Ext°R(R/P,R/P) S fl/P and ExtJ,(fl/P, R/P) S Hom fl/P (P/P 2 , fl/P), we have that Ext^(.R/P, i2/P) is R/P-hee for g > 0 by assumption. Hence E%q = 0 (p 7^ 0), and we have an isomorphism Ext^iJ/P, fl/P) (gifi/p fi/m ^ Extje/J(i?/m, i?/m). This shows
for i > 0. By Proposition 2.8.4, R/J is a complete intersection if and only if RP is. As J is a complete intersection ideal, R/J is a complete intersection if and only if R is. Hence, R is a complete intersection if and only if Rp is. D Notes and References. There is no new result in this section, except for some of the lemmas followed by proofs. Although some important topics such as Cohen-Macaulay rings and perfect modules are reviewed from the first definitions, this section is merely a glossary on commutative ring theory. For basic notation, terminology and results on commutative ring theory, see [110] and [26]. Undefined terminology on algebraic geometry should be found in [71]. We treat Cohen-Macaulay approximations and related topics in subsection 4.10.
3
Hopf algebras over an arbitrary base
This section is devoted to reviewing Hopf algebras over an arbitrary commutative ring R. All results in this section are basic, and some non-trivial results can be found in [90, 145].
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3.1
I. Background Materials
Coalgebras and bialgebras
(3.1.1) We say that A is an .R-algebra, if A is a ring, and a ring homomorphism u : R —> A such that u(R) C Z(A) is given, where Z(A) denotes the center of A. This is the same as to say that A is an .R-module, fi-linear maps u : R —> A and m : A ® A —> A are given, and the diagrams A® A® A
m
I
®lA-A®A I
I" A®A
I ^
»A
_ A®R—^—
_ A *—^—R®A
are commutative. In fact, if A is an .R-algebra, then A is an .R-module in a natural way, and if we define m : A ® A ->• A by m(a ® a') := aa\ then it is easy to see that the diagrams above are commutative. Conversely, if u and m are given so that the diagrams are commutative, then A is a ring with the product aa! := m(a®a'), and A is an .R-algebra by u : R —> A. We call m = rriA '• A ® A —> A the product map of A, and u = UA '• R —> A the unit map of A. The notion of R-coalgebra is the dual to that of .R-algebra. Namely, Definition 3.1.2 We say that a triple (C, A, e) is an R-coalgebra if C is an .R-module, and A : C -* C ® C and e : C -» .R are .R-linear maps such that the diagrams ) C < Aig>lc
tA .
®A ?,
g ( g ) g
^^
C
are commutative. We sometimes say that C is an .R-coalgebra if there is no confusion. The commutativity of the first (resp. second) diagram is called the coassociativity law (resp. counit law). We call A = Ac the coproduct of C, and e = £c the counit map of C. If, moreover, C is an .R-algebra and Ac and £c are .R-algebra maps, then we say that C is an R-bialgebra. An .R-coalgebra C is called cocommutative if r o Ac = Ac, where T : C ® C -> C ® C is the ii-linear map given by T(C ® d) = d ® c for c,d e C. Note that cocommutativity is the dual notion of the commutativity of an algebra. Let A and A' be .R-algebras. Then to say that
3. Hopf algebras over an arbitrary base
73
Definition 3.1.3 Let C and C" be i?-coalgebras. Then we say that ip : C -> C is an R-coalgebm map if ip is .R-linear, £ G ° V' = £ c , and Ac o ^ = (ip<S)tp) o A c Let S and 5 ' be i?-bialgebras. Then we say that / : B —> B' is an R-bialgebra map if / is both an .R-algebra map and an /2-coalgebra map. Example 3.1.4 Let B be an .R-bialgebra, and b € B. We say that b is a group-like element if es(b) = 1 and AB(b) = b ® b hold. Let G be a semigroup. Then the group algebra RG is an .R-algebra. It has a unique .R-bialgebra structure such that each g £ G is group-like. Any semigroup homomorphism G -¥ G' is uniquely extended to an .R-bialgebra map RG -> RG'.
3.2
Hopf algebras
(3.2.1) We say that G is an R-semigroup scheme if G is an R-scheme, endowed with R-morphisms e : SpecR —> G and /i : G XRG —t G subject to the semigroup-law Ato(/ix l G ) = / i o ( l G x / x ) ,
fio(lGxe)=pG,
no (e x 1G) = AG,
where pG • G x Spec R = G and AG : Spec i? x G = G are the canonical identifications. We call fi the product of G, and e the unit map of G. A homomorphism of .R-semigroup schemes is an .R-morphism which preserves (j, and e. The category of affine /{-schemes is contravariantly equivalent to the category of commutative /^-algebras. Hence, an affine ^-semigroup scheme G = Spec B, which is completely described in terms of objects and morphisms of the category of affine .R-schemes, can be descried in terms of commutative .R-algebras and .R-algebra maps between them. In fact, B is a commutative .R-bialgebra if and only if G = Spec B is an affine .R-semigroup scheme. A homomorphism of affine .R-semigroup schemes corresponds to an .R-bialgebra map. We say that G is an .R-group scheme if G is an .R-semigroup scheme such that there exists an .R-morphism t : G —> G such that fiG o (1 G x t) o A G = fiG o (t x 1G) o A G = e o uG holds, where uG : G -t Spec R is the structure morphism of G, and A G : G —> G Xft G is the diagonalization. For an .R-semigroup scheme G, such an i is unique, if it exists, and it is called the inverse of G. Considering the case that G is affine, translating the condition for t into the context of commutative bialgebras, and generalizing it to the noncommutative case, we get the following definition.
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I. Background Materials
Definition 3.2.2 Let B be an .R-bialgebra. We say that 5 : B -> B is an antipode map of B if 5 is .R-linear, and the equality rnB o ( 1 B ® 5 ) o A B = m f i o ( S ® 1B) o A B = uB holds. An .R-bialgebra is called an R-Hopf algebra if it has an antipode map. Remark 3.2.3 The following is known. i If an antipode 5 = 5 B of B exists, then it is unique. ii 5 is an anti-algebra anti-coalgebra map. Namely, the equalities S{bb') = Sb'• Sb,
5(1) = 1,
TO(5®5)OA
= Ao5,
eoS = e
hold, where T : B ® B -> B <8> B is the map given by b ® 6' i-» 6' ® b. iii If B is commutative or cocommutative, then we have 5 2 = id#. iv If B and B' are .R-Hopf algebras and
Ac
®Ac<> (c ®A c)
be the coproduct and ec ® £c be the counit map, C <S) C' is an A ® A'coalgebra. If moreover C and C are an .A-bialgebra and an >l'-bialgebra, respectively, then C
3. Hopf algebras over an arbitrary base
75
Exercise 3.2.5 Let 5 be an .R-algebra i?-coalgebra. Then B is an Rbialgebra (i.e., both eB and A s are .R-algebra maps) if and only if both UB and rag are fl-coalgebra maps, where the coalgebra structure of the tensor product B
3.3
Comodules
The notion of comodule is dual to that of module. Definition 3.3.1 Let C be an it-coalgebra. We say that M is a right Ccomodule if an .R-linear map w = % : M —> M ® C is given, and PM
°
(1M
® £c) ° uM = idM,
( % ® lc) ° w M = ( 1 M
hold, where p M : M
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I. Background Materials
3.4
Sweedler's notation
Let C be an i?-coalgebra, and M € M c . For n > 0, we define u/^ M®C®
n
: M ->
by an inductive definition; OJM : = idM and w^j := ( W M ® lc®<"->>)°
For m £ M , we express the element u>^ (m) <E M
(m)
where we consider that m(0) G M, and m^j G C (i = l,...,n). This notation is somewhat extraordinary, and we could write, for example,
However, the notation does not cause any confusion, and it is useful, because the summation itself tells us what the original element m is. This notation is called Sweedler's notation, and is accepted by many authors. If M = C, then we sometimes express ur^' also as A^ , and for c € C, we denote A £ + 1 ) ( C ) by
This notation is also called Sweedler's notation. For example, the counit law of C is equivalent to saying that the equality
(c)
(c)
holds for any c E C. Messy commutative diagrams are sometimes expressed as a family of equalities of elements in a convenient way, using Sweedler's notation. For an i?-coalgebra C, we define A' : C -> C <8> C by A'(c) := £(c) c(2) <8> C(!) for c G C. It is easy to see that (C, A', e) is an i?-coalgebra. We call this i?-coalgebra the opposite coalgebra of C, and denote it by C op . Obviously, we have (C o p ) o p = C as /?-coalgebras. A right (resp. left) C-comodule is a left (resp. right) Cop -comodule in a natural way, and we have equivalences £* M c ° p and M c S C ° P M.
3.5
Bicomodules, Horn and 0
We say that M is a ( C , C)-bicomodule if M is both a left C"-comodule and a right C-comodule, and ( l c <S>w) OLJ' = (w'(g) lc) °w holds in Hom(M, C ® M 0 C), where u>' (resp. w) denotes the structure map of M as a left C comodule (resp. right C-comodule). Note that a (C',C)-bicomodule and a
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C
provided the i?-module structures of A^ coming from the C-comodule structure of N and yl'-module structure of Af coincide (such a condition is not mentioned in the sequel, if there is no danger of misunderstandings). Lemma 3.5.1 Let R —>• R' be a homomorphism of commutative rings, C an R-coalgebra, M a C-comodule, and N a C
by cu{j){v)
: Hom(Vr, V) -> Hom(V, C ® V ) ^ C
:= £(„) vi ® f{v0). We also define
wC' : Hom(l/, V") -> Hom(K, V ® C ) S Hom(V, V") ® C by w C '(/) :=uv'°
f-
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I. Background Materials
Lemma 3.5.5 Let C,C',V and V be as above. Then Hom(V, V') is a (C,C')-bicomodule with the structure maps c& anduic- If, moreover, C" is an R-coalgebra, V is a (C",C)-bicomodule, and V is a (C",C')-bicomodule, then Homc"(V, V) is a (C, C')-subbicomodule of Hom(V, V). The canonical map Hom(V, R) ® V -> Hom(V, V) is a (C,C')-bicomodule map. Proof. We only prove the coassociativity with respect to cw. By definition, if we set c^f := 12(f) /i ® /o, then the equality
holds. As the isomorphism in Lemma 2.1.7 is natural on H, it suffices to prove (/)
(/)
(/o)
(/i)
Since we have
(/) (/o)
(/)
(v)
/u 0 = (A ® 1) ^ ui ® /u 0 = (A the assertion follows.
D
Lemma 3.5.6 ie< C and C be R-coalgebras, M a (C',C)-bicomodule, and V a left C-comodule. Then the map $ : Homc «(M, V) -»• Hom (c -, C) (M, V ® C) defined by $(/)(m) := E(m)/ m (o) ® "i(i) *5 well-defined, and an isomorphism which is natural with respect to M and V, where S(m) 7?io<E>mi is the image of m by the structure map of M as a C-comodule. Proof. It is easy to verify that $ ( / ) is a ( C , C)-bicomodule map, and hence $ is well-defined. We define * : Hom(c<,C )(M, V ® C) -> Hom C '(M, V) by *(p)(m) := Eg(m), where £ : V ® C -> V is denned by E(v ® c) := e(c)w. It is also straightforward to check that ty(g) is a C'-comodule map. When we set gm = Yli Vi ® Q, then we have
(m)
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This shows $\I> = id. On the other hand, we have
) = E{*(f)(m)) = e(ml)f(mQ) = fm, and hence * $ = id. Thus, $ is an isomorphism. Let C be an fl-coalgebra. Then the functor F right adjoint G =?
• c
: M -» RM has the M c , we call the cobar M, and denote it by M
t=0
By definition, we have: Lemma 3.5.7 Let V be an R-module, and M a C-comodule. Then we have V
Cobarc (M)' S Cobarc,(M') for any homomorphism of commutative rings R —t R!', where (?)' denotes the functor ? ® R'. We also have the following, by the definition of the C-comodule structure of Hom(V.M). L e m m a 3.5.8 Let C be an R-flat R-coalgebra, V a finitely presented Rmodule, and M a C-comodule. Then we have a canonical isomorphism Hom(V, Cobar c (M)) ^ Cobarc(Hom(Vr, M)). Lemma 3.5.9 Let C be an R-flat R-coalgebra. Then the set ofC-comodules of the form J®C with J an injective R-module, is an injective cogenerator of Mc. In particular, M.c has enough injectives. If, moreover, R is noetherian, then any C-injective comodule is R-injective. Proof. As G = ? ® C has an exact left adjoint F, it preserves injectives. Hence, G(J) = J®C is injective as a C-comodule for any injective .ft-module J. Let M e M c , and take an injective hull of M as an .R-module i: M ^> J. Then \I>i: M -» J ® C is an injective C-comodule map, and the first assertion follows, where * is the map in the proof of Lemma 3.5.6. The last assertion is obvious by Corollary 2.1.8. D
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Corollary 3.5.10 Let C be an R-flat R-coalgebra, V an R-finitely presented C-comodule, M a C-comodule, and R' an R-flat commutative Ralgebra. Then the canonical map Hom c (V, M) ® R! ->• Hom C 0 f f {V ®R',M®
R')
is an isomorphism. Proof. As both Hom c (V, ?)®R' and Homc®R>{V®R', ?®R') are left exact, we may assume that M is of the form J ®C with J an injective /?-module by Lemma 3.5.9 and the five lemma. In this case, as the canonical map Hom(V, M)®R'
->• Hom ff {V ®R',M®
R')
is an isomorphism, the assertion follows from Lemma 3.5.6.
D
L e m m a 3.5.11 Let C be an R-flat coalgebra. Then for any C-comodule M and any R-module V, we have Ext l c (M, V ® C) S ExVR(M, V). Proof. Obvious by Lemma 1.6.12.
D
Lemma 3.5.12 Let C be an R-flat R-coalgebra, V a finitely presented Rmodule, and M and N be C-comodules. Then the canonical map Hom(V, N) ® V ->• N
{f®v^fv)
is a C-comodule map. The isomorphism Hom(M
•
If R is noetherian, then for an /?-flat .R-coalgebra C, any C-injective comodule is /?-injective (Lemma 3.5.9). Hence, for any finite .R-module V, Ext'R(V, ?) is the derived functor of Hom(Vr, ?) in the category M c . Hence, Ext'R(V, M) has a canonical C-comodule structure for a C-comodule M. Proposition 3.5.13 Let R be noetherian, C an R-flat R-coalgebra, V an R-finitely presented module, and M and N be C-comodules. If M orV is R-flat, then there is a spectral sequence
Ep2'q = Extpc{M, Ext9fi(y, N)) =* Ext£ +9 (M ® V, N).
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Proof. By Lemma 3.5.12, we have an isomorphism of left exact functors Hom c (M
3.6
The restriction and the induction
(3.6.1) Let B and C be .R-coalgebras, and
be the structure map, we have M e M c . It is also easy to see that this defines a functor res B : M B -> M c . Note that res B M is nothing but M itself, as an i?-module. For a left C-comodule L and a right C-comodule N, we define the cotensor product of N and L, denoted by N K c L, by the exact sequence 0 —> N K c L —» N <S> L
PN L
' ) N
where pN]L := wN ® 1L - 1N ® LJL. If C" and C" are i?-flat i?-coalgebras, N is a ( C , C)-bicomodule, and L is a (C, C")-bicomodule, then N ® L and N®C®L are ( C , C")-bicomodules in a natural way, and pniL is a ( C , C")bicomodule map. Hence, N^FL has a (C',C")-bicomodule structure. Note that the definition of cotensor product is dual to that of tensor product. Almost by definition, the next lemma holds. Lemma 3.6.2 Let C be an R-coalgebra, L a left C-comodule, and N a right C-comodule. Then we have the following. 1 ? S c L and Nfflc? preserve any filtered inductive limits. 2 LetF:Mc^> 3IfC
RM
be the forgetful functor. Then ? E c L is F-left exact.
and L are R-flat, then 1MC L is a left exact functor from Mc to RM.
L e m m a 3.6.3 Let V be an R-module, N a right C-comodule, and L a left C-comodule. Then there are isomorphisms of R-modules N S c (C <S> V) = N
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I. Background Materials
Proof. The cobar resolution Cobar c (iV) : 0 -t N - ^ N ®C —> N
is 0 for M 6 M and V € M c . As the sequence 0 -> Hom B (M, indg(V)) - ^ Hom B (M, V ® B) - ^ Hom B (M, V ® C ® 5 ) is exact, we get a map q : Hom c (resg(M), V) -> Hom B (M,indg(V)), which is natural with respect to M and V. To prove that q is an isomorphism, we may assume that V is of the form V = VQ ® C with Vo an .R-module by the five lemma, as the sequence is exact. In this case, we may identify ind B (V) = Vo ® B, while u : ind B (^) -> V ® £ is identified with
Hence, as we have J2(b) £c(fh)b2
= b for b G B, q is given by
q(
and hence q is identified with the composite map Hom c (resg(M), Vo ® C) S Hom(M, Vo) S Hom B (M, Ko ® B) (see the proof of Lemma 3.5.6), and is an isomorphism.
•
3. Hopf algebras over an arbitrary base Example 3.6.6 Let B the i?-coalgebra map e an /^-module V, where UJ(V ®b) = v ® AB(b). but WJU, and hence w^
83
be an i?-flat /J-coalgebra. Set C := R, and consider : B -> C = R. Then we have ind5. V = V ® B for V ® B is endowed with a B-comodule structure by The counit of adjunction M -4 M ® 5 is nothing is a B-comodule map.
Definition 3.6.7 Let C be an .R-coalgebra. We say that B is an Rsubcoalgebm of C if B is a pure i?-submodule of C, and Ac(B) C B ® B holds. A subalgebra subcoalgebra of a bialgebra is called a subbialgebra. (3.6.8) Assume that C is i?-flat. Note that B C C is an it-subcoalgebra of C if and only if it is a (C, C)-subbicomodule of C, which is also a pure i?-submodule. If this is the case, B is also R-R&t. If B is an .R-subcoalgebra of C, then ind^(iV) is identified with the C-subcomodule {n e N | wjv(n) G N ® 5 ( c W ® C)} of TV for iV € M c , via the injective map indc(A0 = N®cB
C N®CC
^ AT.
Remark 3.6.9 In particular, if C is an .R-flat .R-coalgebra and 5 C C is an i?-subcoalgebra of C, then the counit of adjunction (ind^ ores^)(M) -> M is an isomorphism. Hence, res§ is fully faithful (and is obviously exact, as res^(M) = M). Thus, M B is identified with the full subcategory of M c consisting of C-comodules Af such that wN{N) C N ® B. Note that M B is closed under subobjects, quotients, and direct sums. The converse holds in the following weak form. Lemma 3.6.10 Let k be a field, C a k-coalgebra, and B a full subcategory of Mc closed under subobjects, quotients, and direct sums. We denote the inclusion B t-> M c by i, and denote its right adjoint (it does exist, see Lemma 1.10.2) by j . Then there is a unique k-subcoalgebra B of C such that any B-comodule is in B, and res5. : M B —> B is an equivalence. In fact, B is given as j(C). Proof. Let V be a /c-vector space, and M S M c . Then the canonical map V ® jM —> j(V ® M) is an isomorphism. This is trivial when V = k, and hence also for the case dim V < oo, as j is additive. As j is also compatible with filtered inductive limits by Lemma 1.10.3, the general case follows. Now let N £ B. Then the coaction ui^ : N —> N <8> C is a C-comodule map, and hence it factors through j(N ®C)=N<E> j(C). Considering the case Af = j(C), we have that j(C) is a fc-subcoalgebra of C. This also shows that any object of B is a j(C)-comodule. Conversely, as j(C) € B, any j(C) comodule N is in B, as N is a subobject of N ®j(C). Hence, the existence of B is proved.
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I. Background Materials
We show the uniqueness. of j . We show j(C) C B. we have that any object of 5-comodule. For c € j(C), uniqueness is proved.
As B G B, we have B C j{C) by the definition As M B is closed under isomorphisms in Mc, B is a B-comodule. In particular, j(C) is a we have c = E(c) £c(c(i))c(2) 6 B, and the D
Lemma 3.6.11 Let C be an R-coalgebra, C and C" be R-flat coalgebras, P an R-finite R-projective (C,C')-bicomodule, N a (C", C)-bicomodule. Then there is a (C"', C')-bicomodule isomorphism
which is natural with respect to P and N. Proof. It is easy to see that the kernel of the map N ®P -> N ®C
a Corollary 3.6.12 Let B —» C be an R-coalgebra map. If B is R-finite projective, then there is an isomorphism of functors ind^ = Hom c (S*,?). (3.6.13) From now on, until the end of this subsection, we assume that C is an R-flat .R-coalgebra. Note that M c is abelian, and the forgetful functor F : MP —> ijM is faithful exact, and its right adjoint G = ? <S> C is also exact. Definition 3.6.14 For M 6 M c and a left C-comodule N, we denote R'F(? S c N)(M) by Cotor^M.TV), and call it the ith cotorsion module of M and N. By definition, we have Cotor'c(M, TV) = /T(Cobarc(M) ®c N). As Cobarc(M) Mc N is the complex of the form
it is symmetric with respect to M and N, up to sign change. We denote it by Cobar c (M, N). We have Cobarc(M,AT) ^ CobarCop(7V,M). Hence, we have Cotor'c(M, TV) S /T(Cobar c (M,A0) ^ ifi(CobarCoP(iV, M)) ^ Cotor'cop(N, M).
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Lemma 3.6.15 For a homomorphism of commutative rings R —• R', we have
Cobarc(M, N)' Si CobarC'(M;, N') in a natural way, where (?)' = ? <8> R' •
Proof. Obvious. By Lemma 3.6.11 and Lemma 1.6.12, we have the following. Lemma 3.6.16 Let M be an R-finite R-projective C-comodule, and N a C-comodule. Then we have natural isomorphisms
Ext'c(M,JV) S ^ i ( £* H<(Cobarc(JV, AT)) S Cotor^AT, M*). The isomorphisms are natural with respect to M and N. Lemma 3.6.17 Let N be a left C-comodule, and (MA) a filtered inductive system of C-comodules. Then lim Cotor' c (M A , N) £ Cotorlc(lim Mx, N) fori > 0. Proof. Easy.
D
Lemma 3.6.18 Let I be an F-injective C-comodule, and N an R-flat left C-comodule. Then I is? E c N-acyclic as an object ofMF. In particular, if N is R-flat, then we have
Cotor^M, N) £ R\l E c N){M) for any C-comodule M. Proof. We may assume that / is F-cofree, that is to say, / = V (8) C for some fl-module V. We take an i?-injective resolution J* of V. Then we have
JP(? E c N)(V) = ^((J'
ig> C) S c N) Si H\r
® N).
As N is .R-flat, we have that J' <S> N is quasi-isomorphic to V <8> N, and the assertion follows. • Corollary 3.6.19 Let B be an R-flat coalgebra, and B -> C an R-coalgebra map. Then we have
R{ ind£ S // i (Cobar c (?, B)) = Cotor^?, B).
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I. Background Materials We have the following.
Proposition 3.6.20 Let R-> R' be a flat morphism of commutative rings. We denote the functor ? ® R! by (?)'. For i > 0, we have 1 For M e Mc and N e CM, we have Cotor^(M, N)' £ Cotorjy(M#, N'). 2 Let M be a C-comodule, and B -> C an R-coalgebra map with B R-flat. Then we have 3 Assume that R is noetherian. Then for any R-finite C-comodule V and any C-comodule M, we have Ext^V, M)®R' = Extj^V, M'). 4 Assume that R is noetherian. For any R-finite C-comodule V and any C-comodule N, we have
The proof is straightforward. We give a remark on 4. If 7* is a Cinjective resolution of M, then /*
3.7
Locally noetherian property
Throughout this subsection, we assume that C is an i?-flat /i-coalgebra. Lemma 3.7.1 Let C be an R-flat R-coalgebra. Let M € M , and X be an R-finite submodule of M. Then there exist some C-subcomodule L of M and an R-finite R-submodule N of M such that X C L C N C M. Proof. Take a generator x\,...,xn oi X as an i?-module. We express w(xi) = Zj rriij <8> (kj, and define N to be the .R-nnite .R-submodule of M generated by all m^'s. The inverse image of N <8>C by the C-comodule map u>M : M -> M <2> C = ind£ res£(M), say L := v]rf(N<8>C), is a C-subcomodule of M, and it contains X. On the other hand, as we have L = id M (L) = (idM
®C)cN, D
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Corollary 3.7.2 In the lemma, if moreover R is noetherian, then there exists an R-finite C-subcomodule of M which contains X. In particular, M is the filtered inductive limit of its R-finite C-subcomodules. We have already seen that M c has enough injectives. By the following corollary, we see that M c has injective envelopes by Theorem 1.7.6. Corollary 3.7.3 Let C be an R-flat R-coalgebra. Then the abelian category M° is Grothendieck. If moreover R is noetherian, then M c is locally noetherian, and Mj consists of all R-finite C-comodules. Proof. We already know by Lemma 3.3.3 that M c is an abelian category which satisfies (AB5). We show that M c has a small set of G-generators. The isomorphism classes of C-comodules which are /?-submodules of infinite modules are parameterized by a small set / . Taking representatives from these isomorphism classes, and collecting them, we have a small set of G-generators {Ui)i€i of M c by the lemma. This shows M c is Grothendieck. Now we assume that R is noetherian. Obviously, any infinite C-comodule is noetherian. Hence, M c is locally noetherian, as each Ui in the first paragraph is .R-finite. As any object in M^ is a homomorphic image of a finite direct sum of copies of U^s, it is infinite. • Proposition 3.7.4 Let R be noetherian, C a flat R-coalgebra, and V an R-finite C-comodule. Then the canonical map lim Ext l c (V, MA) -» Ext^(V, lim Mx) is an isomorphism for any filtered inductive system (M\) of C-comodules and any i>0. Proof. Follows immediately from Corollary 3.7.3 and Lemma 1.9.5.
3.8
•
The dual algebra of a coalgebra
(3.8.1) For ^-modules V and W, we define p : V*
We call the i?-algebra C* the dual algebra of C.
(b*,c'£C',ceC).
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I. Background Materials
Definition 3.8.2 Let R be a field. Let V be an .R-space, and W a subspace of V*. We say that W is a dense subspace of V*, if V —> W (v >-> (w >-> (w,v))) is injective. Let R be a general commutative ring, and V and W be iZ-modules. We say that an .R-linear map / : W —> V" is universally dense, if the .ft-linear map 9u : 1/ <8> V -> Hom(H/, £/)
u ® u H> (w H (to, t>)u)
is injective for any .R-module f/. In the definition above, the pairing W ® V -> i? which corresponds to / by the isomorphism Hom(W, K*) = Hom(VK ® V, R) is denoted by (—,—)• For a homomorphism R —)• R' of commutative rings, we still denote the pairing W ®RI V —> R' obtained by the base change by the same symbol (—,—). From the pairing (—,-),
f':RomRI(W',HomK(V',R')) is induced. Note that the associated map V -> Homfl'(Vy', R') (v' t-^ (w' H* {w',v'))) is nothing but the composite map
V = R' ® V ?£> Rom{W, R') S Note also that 6V in the definition above is natural with respect to U. Lemma 3.8.3 Let R be a field, V an R-vector space, and f : W —¥ V* an R-linear map. Then the following are equivalent. 1 f is universally dense. 2 OR : V -¥ W* is injective. 3 Imf
is a dense subspace of V*.
Proof. 1=>2 is obvious. As OR : V —> W* is the composite of V -4 (Im/)* and the injective map (Im/)* —• W*, 2-O3 is obvious. We show 2=>l. It is obvious that Qu is injective for finite dimensional U. Now consider the case dim fl U = oo, and assume that 0[/ is not injective. As there is a finite dimensional .R-subspace Uo of U such that Ker 6y (~\ (Uo ® V) ^ 0 and 0 is D natural, 6u0 is not injective, and this is a contradiction. Lemma 3.8.4 Let f : W —> V be an R-linear map. Consider the following two conditions.
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1 f is universally dense. 2 V is R-flat, and the induced map
is universally dense for p £ Spec R, where ?(p) denotes the functor K(P)<8? : ftM ->
K ( P ) M.
In general, we have 1=>2. //, moreover, R is noetherian, then we have 2=>1. Proof. We prove 1=>2. For any injective ^-linear map g : U -> U', we prove that 5®lyr : t/®V -> t/'®V is injective. As g, : Hom(VK, £/) -> Hom(W, [/') is injective, 0[/ is injective, and 0 is natural so that 6u> ° (g <8) 1) = g, ° 6u holds, we have that g ® 1 is injective. This shows that V is /?-flat. As we have is injective, we have that f(p) is universally dense by Lemma 3.8.3. Next, assuming that R is noetherian, we prove 2=>1. Assume the contrary. Then we have an i?-module U such that 8u is not injective. Take a non-zero element X^Ui (8)i»i in Kerfly, and set UQ to be the i?-submodule of U generated by all ttj's. Then 8u0 is not injective, and we may assume that U is fi-finite, replacing U by Uo. As V is i?-flat, if 0 -> f/j ->• t/2 -> I/3 is an exact sequence of fl-modules and 6Vl and ^u3 are injective, then 6y2 is also injective by the five lemma. As any finitely generated module over R has a finite filtration whose successive subquotients are of the form R/p with p £ Spec./?, there exists some p € SpecR such that 6R/V is not injective. So we may assume that U = R/p. Replacing R/p by R, we may assume that R is an integral domain, and OR : V —• W* is not injective. We set K = K(0) to be the quotient field of R. As r : R <-> K is injective and V is fi-flat, r i g i l v ^ - ^ / ^ i E i V i s injective. On the other hand, 6K • K <8>V -> rlomK(K (8) W, K) is injective. As the injective map 6K ° {r <S) lv) agrees with the composite map V -> W* -> rlomK{K ® W, K), we have that 0 fi is injective, and this is a contradiction. •
Lemma 3.8.5 Let W -> V* be a universally dense R-linear map. Then for any R-module U, we have that 6V : U ® V —> Hom(W, U) is R-pure. In particular, 6R:V -¥ W* is R-pure.
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I. Background Materials
Proof. Let X be an /^-module. Then the composite map
X&UQV
±^% X ® Hom(Wl U) -»• Hom{W,X ® U)
agrees with $x®u, which is injective. Hence, lx <8> 9u is also injective for any X, and this shows that 9y is i?-pure. D Exercise 3.8.6 Prove that if V and W are fl-projective modules, then V* ® W* —» (V <E> VF)* is universally dense. Considering the case W = R, i d y : V* —> V* is universally dense, if V is /?-projective. Utilizing this, prove that if C is an .R-projective i?-coalgebra, then C is cocommutative if and only if C* is commutative. Let M be a right C-comodule. Then M is a (left) C*-module with the action c'm := 5Z( c * m (i)) m (o) (c* £C*, m€ M). (m)
Any C-comodule map is a C*-linear map. Exercise 3.8.7 Prove the assertions above. Thus, for an i?-algebra map A —» C*, the exact functor $ : M c -> C'M. —» ^M is denned in an obvious way.
3.9
The dual coalgebra of an algebra
Let A be an i?-algebra. In general, A* does not have a canonical .R-coalgebra structure even if R is a field. This is because even if / 6 A*, iri*A(f) £ (A <8> A)* does not belong to A*
A°:={fE
A'\mrA(f)eA*®A*}.
Lemma 3.9.1 For f £ A*, the following conditions are equivalent. i / € A°, that is, mA(f) £ A* ® A' ii mA(f) G A0
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3.10
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Rational modules
In this subsection, R denotes a general commutative ring again. Let C be an i?-coalgebra, and A -> C* a universally dense i?-algebra map (hence, C is R-ftat). We denote the canonical functor M c —» ^M by $. As an .R-module, we have $(M) = M. Let V be an A-module. We denote the .A-action A ® V —» V by ay- The canonical isomorphism Hom(.4
Uom(A, Vrat) -»• Hom(A, V),
the coaction U)VTM : Vrat -> Vrat ® C is canonically defined, and Via% has a C-comodule structure. The A-module structure of $(Kat) agrees with that o/Vrat 05 an A-submodule ofV. Proof. For v € Vrat> we may write {pva,v){v) = £< ^KC^t^Ci)- For o, a' £ /I, as we have
({pvav){av))(a!) = a'(av) = (a'a)t;
= 5^(a'a, Ci)vi = 53 53< i
t
( Ci )
we have
(pvav)(av) = 5ZS
()
and it follows that at) € Kat- Hence, Kat is an .A-submodule of V. This shows that (pvav){v) (v S Vrat) is contained in the image of Hom(v4, V^-at) <-> Hom(>l, V), and hence it is zero in Hom( J 4,y/V r rat ). As J4 is universally dense, Ov,vM • V/Vxat ® C -
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I. Background Materials
is injective, and hence £ 4 vt ® Q is zero in V/Kat ® C by naturality of 0. As C is i?-flat, we have £ { «<
the counit law is satisfied. Next, the map P: is injective. In fact, P is the composite of the injective maps V
Hom(A,
V)®C , V)) S
For any a, a' € A we have wi) = a^a!,
Ci)vi = a(a'v) = {aa')v
t
a'). As P is injective, the coassociativity law also holds. By the definition of wy,,,, the .A-module structure of $(VJat) agrees with that of Kat as an .A-submodule of V. O Proposition 3.10.2 Let V e Mc, W € AM, and f G Then we have f(V) C Wvat, and an isomorphism
EomA($V,W).
HomA(9V, W) a Hom c (^, W rat) is induced. In particular, if W e AM and g G HomA(W, W), then we have 0(W/at) C W"*> and letting g rat = g\w^t e Homc (M / r' at, W rBt), (?) rat is a functor. Moreover, (?)rat is right adjoint to <£>, id^ : V = V = ($V) ra t is the unit, and $(Wrat) = W rat t-*W is the counit of adjunction. Proof.
Easy.
Corollary 3.10.3 The functor (?)ra t is left exact, and preserves injective objects. $ is fully faithful.
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Proof. As (?)rat has an exact left adjoint $, it is left exact and preserves injective objects. The unit of adjunction ($V) r a t -> V is clearly an isomorphism, hence $ is fully faithful. • If V is an .4-module such that Vrat = V, then we say that V is rational. For a C-comodule M, we have that M = $ M is rational. Conversely, if V is rational, then V = Viat is a C-comodule. So we identify rational ^-modules with C-comodules. Exercise 3.10.4 Rational .4-modules are closed under submodules, factor modules, and inductive limits in ^M. Exercise 3.10.5 If M is a flat /t-module and V is an i4-module, then the canonical map Vrat
3.11
FPCP coalgebras and IFP coalgebras
Throughout this subsection, R denotes a noetherian commutative ring. Let C be an i?-coalgebra. Definition 3.11.1 We say that C is ind-finite projective (IFP, for short) if for any i?-finite i?-submodule M of C, there exists some /2-finite Rprojective /?-subcoalgebra of C containing M. By definition, we have L e m m a 3.11.2 If C is an IFP R-coalgebra, then a base change R' (g> C is an IFP R'-coalgebra for any commutative noetherian R-algebra R! of R. Lemma 3.11.3 If C is IFP, then C is R-Mittag-Leffler. In particular, C is R-fiat. If R is noetherian and C is R-countable, then C is R-projective. Proof. Follows immediately from Lemma 2.2.11 and Proposition 2.2.6. D Now we assume that C is /?-flat. Definition 3.11.4 We say that C satisfies the finite projective cover property (resp. projective cover property), FPCP (resp. PCP) for short, if for any i?-finite C-comodule (resp. any C-comodule) M, there exists some surjective C-comodule map P —> M with P /?-finite projective (resp. i?-projective). Lemma 3.11.5 If C satisfies FPCP, then C satisfies PCP. Conversely, if C satisfies PCP and gl.dim R<2, then C satisfies FPCP.
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Proof. Assume that C satisfies FPCP and M is a C-comodule. By Corollary 3.7.2, M is the inductive limit limMA of the inductive system (MA) of it-finite C-subcomodules of M. As C satisfies FPCP, we can take a surjective C-comodule map f\.P\-> MA with PA infinite projective for each A. Then the composite map
®P A ^0M A ->lunM A = M \
A
is a surjective C-comodule map, and ® A PA is fi-projective. This shows that C satisfies PCP. Assume that C satisfies PCP, gl.dimi? < 2, and M is an it-finite Ccomodule. As C satisfies PCP, there exists some surjective C-comodule map / : P -> M with P it-projective. We can take an it-finite C-subcomodule N of P such that f(N) = M, by Lemma 3.7.1. We can also take an it-free module F with a basis B which contains P as its direct summand. Then we may consider N C P C F, and there exists some finite subset Bo of B such that Fo := R • Bo contains N. Now we define Q to be the kernel of the composite of the C-comodule maps p:P^P®C^>F®C^
F/Fo ® C.
For q € Q, we have q = £( 9 ) £c(c(i))c(o) € FQ, and hence Q C Fo is ii-finite. As both P and F / F o <8> C are i?-flat and gl.dim R < 2, we have that Q is .R-fiat, and hence is .R-finite projective. As we have p(N) = 0, it follows that f(Q) D f{N) = M, and hence the restriction / | Q : Q —> M is a surjective C-comodule map. Hence, C satisfies FPCP. Lemma 3.11.6 If C is IFP, then C satisfies FPCP. Proof. Let M be an .R-finite C-comodule. As the image of w^ : M —> M ® C is .R-finite, there exists an .R-finite i?-projective .R-subcoalgebra D of C such that Im u M C M ® £>, by assumption. This shows that M is a Dcomodule, and hence is a D*-module. As D* is fl-finite .R-projective, there exists some surjective D'-linear map / : P —» M such that P is .R-finite .R-projective. As a D*-module is always a Z)-comodule and a DMinear map is a £>-comodule map, we have that C satisfies FPCP. • Lemma 3.11.7 Let R be a noetherian ring, R-t K an injective homomorphism of commutative rings, P an R-projective module, and MK a finite K-submodule of PK := K
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Corollary 3.11.8 Let R be a reduced noetherian ring, P an R-projective module, and M an R-submodule of P. If Mp is a finite dimensional Rpvector space for any p G Min(/Z), then M is R-finite. Proof. Set K to be the total quotient ring npgMin(fl) ^p O I R- By the lemma, MK n P is fi-finite, where MK := M ®K. As M C MK n P, M is also fi-finite. D Lemma 3.11.9 Let R be a hereditary {i.e., gl.dim/? < 1) noetherian ring, and C an R-projective R-coalgebra. Then C is IFP. That is to say, for any R-finite R-submodule M of C, there exists some R-finite protective Rsubcoalgebra D of C which contains M. Proof. First, we consider the case gl.dim R = 0. As the image of M by A'2) : C —> C®C®C (c i-> Y.(c) c(i)®c(2)®C(3)) is .ft-finite, there exists some .R-finite .R-submodule N of C such that A (2) (M) is contained in C®N®C. then we have M C D. It is easy to If we set D := ( A ^ ) " 1 ^ ®N®C), check that A(D) CD®D. As gl.dimR = 0, D is an fi-subcoalgebra of C. If d € D, then d = E(d)e(^i)e(^3)^2 £ N, and hence D C N. This shows that D is .R-finite. It is .R-projective, as gl.dim R = 0. Next, we consider the case gl.dim R = 1. Let K be the total quotient ring of R. As gl.dim K = 0, there is a A'-finite /C-subcoalgebra D ^ of K<%>C which contains K®M. Set L> := DKC\C. By Lemma 3.11.7, D is fi-finite. As C/D is torsion-free, it is i?-flat, since gl.dim R = 1. This shows that D is a pure submodule of C, and D is .R-nnite projective. Moreover, the composite map D ^ c A c ® C 4 i C ® {C/D ®C®C® C/D) is zero by the choice of DK. subcoalgebra of C.
3.12
Hence, A(£>) C D ® D, and Z) is an RD
In this subsection, R denotes an arbitrary commutative ring again. (3.12.1) Let U be an fl-Hopf algebra. For V,W G y M , we define the (/-module structure of V ® W by u£U,veV,w€
W),
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and the [/-module structure of Hom(V, W) by
(«/)(«) = Euw(f((Su(2))v))
( u € U , f € KomR(V,W),
v e V),
where S = Su denotes the antipode of U. Exercise 3.12.2 Check that the definitions above do give [/-modules. (3.12.3) An J?-module M endowed with the [/-module structure given by urn := e(u)m (u 6 U, m 6 M) is called a trivial [/-module. If we want to emphasize the trivial [/-structure, we denote it by M t n v . However, by the [/-module R we always mean the trivial module Rtnv, unless otherwise specified. Lemma 3.12.4 Let V, W, and X be U-modules. maps (3.12.5) (3.12.6) (3.12.7) (3.12.8) (3.12.9)
The standard R-linear
Hom(V, W) ® Hom(X, V) -> Hom(X, W) Hom(V ® X, W) S Hom(V, Hom(X, V<S)R = V^ R®V (v®l >-> v (V®W)®X &V®(W®X) ({v®w)®x>-*v®{w®x)) X
are U-linear, and natural with respect to V, W, and X, where the map in (3.12.6) is given by /4(()H(li-j/(lJ® X))).
Proof. Straightforward.
D
(3.12.10) See Lemma 2.1.7 for sufficient conditions for the map (3.12.9) to be an isomorphism. If W = R, then (3.12.9) is nothing but the map X ® V -)• Hom(K,X)
(x®ip^{v^(ip,v)x)).
If U is cocommutative, then r :V ®W = W ®V (r{v ® w) = w ® v) is also [/-linear and natural with respect to V and W. (3.12.11) We denote the functor Ext\j{R, ?) by H{(U, ?). For [/-modules V and W, we have V, W) s Hom c/ (fl, HomR(V, W)) = H°(U, Hence, taking H°(U, ?) of both sides of (3.12.6), we have (3.12.12)
Homu(V ®X,W)^
Homt/(Vr,Homfl(X, W)).
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This shows that ? ® X is left adjoint to Hom^A", ?). In particular, considering the case that V is [/-projective and X is i?-projective, as the left-hand side is an exact functor on W, we have that V <8> X is [/-projective. Considering the case V = Hom/e(A", W) in (3.12.12), the element of the left-hand side corresponding to idy is nothing but the evaluation map ev: KomR(X,W)®X->W
(f®x^fx).
Hence, ev is [/-linear. If U is cocommutative moreover, then the [/-linear map which corresponds to ev or by the isomorphism Homy (A" ® Homfi(A:, W), W) S Homy (A, Hom /J (Hom fi (X, W), W)) is nothing but the duality map n - » ( / 4 fx). Hence, the duality map is [/-linear, if U is cocommutative. (3.12.13) Let H be an /2-Hopf algebra. For //-comodules M and TV, we define the //-comodule structure of M ® N by M ® N -> M ® JV
(
(3.12.14) We have seen that in the following cases, Hom(M, ?)
) := 5Z S (/ (o))(o) (m)
for / £ Hom(M, A^). It is left to interested readers to check that these definitions do give //-comodules. (3.12.18) For an fl-module M, the //-comodule M with the structure map u>(m) = m ® 1 is again denoted by M. If necessary, it is denoted by M t n v . The functor (?) triv is nothing but the restriction via the /?-coalgebra map UH '• R -»• H, and its right adjoint is the induction ? ® H. The Hcomodule R means Rtm, unless otherwise specified. We denote the functor Ext^H (R, ?) by / / ' ( M " , ?). Note that the functor ^ ( M " , ? ) = Hom M «(/?,?) is nothing but the induction via UH, and it is called the H-invariance.
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3.13
The dual Hopf algebra
(3.13.1) Let R be a commutative ring, H an i?-bialgebra, and U an Rbialgebra, and (—,—) : U <S> H —> R an i?-linear map. We assume the following conditions. 1 ( - , - ) is a pairing of i?-bialgebras. Namely, the induced map U -t H' is an i?-algebra map, and H -> U* is also an it-algebra map. 2 U -t H* is a universally dense injective map. We call such a pair U and (—, —) a generalized hyperalgebra of H. (3.13.2) When is there a generalized hyperalgebra of HI As a necessary condition, H must be it-flat by Lemma 3.8.4. In the following two cases, a generalized hyperalgebra of H exists. Example 3.13.3 Let R be a field, and H an it-Hopf algebra. As H is an it-algebra, U := H° is an it-coalgebra. It is easy to verify that H° is an /Z-subalgebra of H*, and is an i?-Hopf algebra. We call H° the dual Hop} algebra of H. The canonical map H -4 U* is an algebra map. If H is commutative, then H° is cocommutative [1, Corollary 2.3.17]. If H is commutative and of finite type over R, then the inclusion H° —>• H* is universally dense, and hence H° is a generalized hyperalgebra of H. We prove the last assertion. By Lemma 3.8.3, it suffices to show that the canonical map 0 : H -> (H°)* (6(h)(h*) = (h*, h)) is injective. Assume that h e Ker#. For any maximal ideal m of H and n > 1, as H/mn is a finite dimensional /t-space by the Hilbert Nullstellensatz (see [110, Theorem 5.3]), the image of h in H/mn = (H/mn)** is 0 by Lemma 3.9.1. Hence, we have • m ^ supp Hh. As m is arbitrary, we have Hh = 0. Example 3.13.4 Assume that H is i?-finite projective. Then as we have = H' (and idy) is a generalized hyperalgebra of H' ®H* ^ (H®H)\U H. If H is commutative, then U is cocommutative. (3.13.5) Let U be a generalized hyperalgebra of H. If M is an Hcomodule, then M is an i/*-module, hence is a [/-module, and an exact functor $ : MH -» yM is induced. As U -» H* is a universally dense algebra map, there is a right adjoint (?) rat : (/M -> MH of $, and $ is fully faithful (Corollary 3.10.3). In this situation, more is true. The functor $ also preserves tensor products. That is, for M,N e MH, the identity map $M®$N = M®N = $ ( M ® N) is a [/-isomorphism. Moreover, $ preserves Horn. That is, if
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M,N e MH and one of (3.12.15-3.12.17) is satisfied, then the identity map Hom($(M),$(TV)) = $(Hom(M, TV)) is a [/-isomorphism. Moreover, the functor $ preserves trivial representations. That is to say, $(M t r i v ) = M t r i v for any fl-module M. Hence, rational [/-modules are closed under tensor products, and if moreover one of the conditions (3.12.15-3.12.17) is satisfied, then they are also closed under Horn.
3.14
Module algebras and comodule algebras
The reference for this subsection is [114]. (3.14.1) Let R be a commutative ring, and U an i?-Hopf algebra. We say that A is a U-module R-algebra if A is an .R-algebra and a [/-module, and the product TUA '• A
(a, b € A,
u,v€U).
It is easy to see that A#U is an .R-algebra. Note that both A —> A#U (a i-> a(8)l) and U -* A#U (u i-> l®u) are i?-algebra maps. So any >l#[/-module is in a natural way a [/-module >l-module, which is also a (U, .4)-module. Conversely, if M is a (U, .4)-module, then defining (a <S) u)(m) = a{wm) for a®u £ A#U and m £ M, M is an j4#[/-module. These correspondences are quasi-inverse to each other, and an A#U-module and a ([/, /l)-module are one and the same thing. Thus, we have that ^ y M and y ^ M are equivalent. We always identify an A#U-modu\e and a ([/, i4)-module. As i?#[/ ^ [/, we have UiRM = VM. Note that H°(U, A) is an fl-subalgebra of A, and H°(U, ?) is a left exact functor from y ^ M to 7/°(y,/i)ML (3.14.4) Let /f be an fi-flat Hopf algebra. We say that B is an Hcomodule R-algebra if B is an .R-algebra //-comodule, and the product
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TUB : B <S> B —> B is an //-comodule map. By an //-comodule /?-algebra map we mean an //-comodule map which is also an /?-algebra map. We say that M is an (H, B)-Hopf module if M is an //-comodule B-module, and the action B
3.15
Coalgebras and comodules over a scheme
Let A" be a scheme. We say that C is an Ox-coalgebra if C is a quasicoherent Ox-module, Ox-module maps e : C —> Ox and u> : C —> C ®Ox C are given, and the coassociativity and the counit laws are satisfied. Similarly, Ox-algebra, Ox-bialgebra, and Ox-Hopf algebra are defined, replacing an /^-module by a quasi-coherent Ox-module. In [71], an Ox-algebra is called a quasi-coherent Ox-algebra.
Notes and References. For basics on Hopf algebra theory, see [140], [1], [93], and [114]. As the base ring R in our text is not restricted to a field, we have discussed some difficulties arising from this point. In particular, the notion of universal density and related results on rational modules and generalized hyperalgebras, and the notion of IFP, FPCP, and PCP are new here. Some of the important properties of a flat coalgebra and its comodules in this section are proved in [145].
4. From representation theory
4 4.1
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From representation theory Group schemes as faisceaux
(4.1.1) Let X be a scheme. The category of X-schemes Sch/X with the fppf topology is a site, and a sheaf with respect to the Grothendieck topology is called an X-faisceau, see Example 1.8.14. For an X-scheme Y, y(Y) = Homsch/x('i Y) ' s a set-valued X-faisceau. An X-faisceau is called representable if it is isomorphic to y(Y) for some Y £ Sch/X, see (1.1.7). (4.1.2) We denote the full subcategory of the category of X-schemes Sch/X consisting of affine X-schemes by X-aff. Note that X-aff is also a site with the fppf topology. If F is an X-faisceau, then F is completely determined by its restriction to X-aff. Hence, F can be viewed as a faisceau over X-aff, and is a covariant functor from the category of X-algebras X-alg to Set, where an X-algebra is a commutative ring A together with a morphism Spec A -> X (note that X-alg is contravariantly equivalent to X-aff). We call a covariant functor on X-alg an X-functor. The functor which maps an X-faisceau F to the X-functor F has a left adjoint (?) by (1.8.8). For an X-functor P, the sheafification P is called the associated faisceau of P. For more, see [43]. We remark the following. Lemma 4.1.3 Let F be a subfunctor of an X-faisceau G. Then F is the subfunctor of G given by F(A) = {x£ G{A) | 3 fppf A-algebra B such that x £ F{B)} for A € X-alg, where fppf means faithfully flat of finite presentation. Definition 4.1.4 A group (semigroup)-valued X-functor G is called an Xgroup scheme if G, viewed as a set-valued functor (with composing the forgetful functor), is a representable X-faisceau. By Yoneda's lemma (Lemma 1.1.6), to say that an X-scheme G is a semigroup-valued functor is the same as to say that X-morphisms HG • G xx G -> G and e : X -¥ G are given, and the semigroup laws HG ° (1 G x no) = ficiVG x 1G), MG ° (e x 1G) o AG' = 1 G = /xG o (1 G x e) o p^1 are satisfied, where AG : X x * G -> G and pG : G xx X -> G are the canonical identifications. Further, G is group-valued if and only if there is an X-morphism t G : G -> G such that He o (1 G X ( C ) o i G = e o u G = | i C o (t G x 1G) o A G is satisfied, where uG : G —• X is the structure map, and A G : G —• G xxG is the diagonalization. Thus, we see that the definition above agrees with
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that in (3.2.1), when X is affine. In particular, if both X = Spec/? and G = Spec H are affine, then to give an X-group scheme (resp. X-semigroup scheme) structure to G is the same as to give an fl-Hopf algebra (resp. i?-bialgebra) structure to the commutative .R-algebra H. (4.1.5) We say that a semigroup X-scheme G acts on an X-scheme Y (from the right) if the X-functor G acts on Y from the right. Translating this situation in terms of Yoneda's lemma, an action of G on Y is nothing but an X-morphism a : YxxG —> Y such that ao(ax 1G) = ao(l y x^iG) and the unit element acts as the identity morphism. The left action is defined similarly. Unless otherwise specified, an action of a semigroup scheme on a scheme is a right action. However, if G is a group scheme, then a right action a :Y xxG —>Y is sometimes viewed as the left action G xxY ->Y given by g - y : = y g ~ l . The quotient of Y by G, denoted by Y/G, is the associated faisceau of the X-functor F defined by F(A) := X(A)/G(A). We say that a subscheme Z of Y is G-stable if Z(A) is a G-stable subset of Y(A) for any X-algebra A.
4.2 (4.2.1)
Rational representations of an algebraic group Let R be a commutative ring.
Definition 4.2.2 Let G be an affine i?-semigroup scheme with H = R[G]. We call an //-comodule a G-module or a rational G-module. There is an alternative definition, which is more natural. Let X be a scheme, and G an X-semigroup scheme. For a quasi-coherent C?x-module M, we define an X-semigroup functor End(A^) (resp. X-group functor
GL(M)) by End(7W)(y) := EndOy(/*7W)
(resp. GL(M)(Y) := End Oy (/*7W) x )
for each X-scheme / : Y —> X. We say that M is a G-module, if A^ is a quasi-coherent Ox-module, equipped with a morphism G —> End(M) of X-semigroup functors. This definition looks more like that of group representation. If G is an X-group scheme, then the representation G —> End(.M) factors through GL(M). If, moreover, M is locally free coherent, then we have End(A4) = Spec(SymHornOY(.M, M)v), and both End(M) and GL(M) are representable. In this case, the representation G —> End(A^) or G-> GL(M) is a morphism of X-schemes, by Yoneda's lemma.
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(4.2.3) We briefly review the correspondence between the two definitions provided in the last paragraph in the case where both X = Spec R and G = Spec H are affine [901. If M is an H-comodule, then we have a morphism G ~ GL(M) given by 9 f-t (a ® m
f-t
L ag(m{l)) ® m(O) (m)
for each A and 9 E G(A) = HOmR-alg(H, A). Conversely, assume that a morphism h : G ~ GL(M) of R-group functors is given. Then as idH E G(H) = HOmR-alg(H, H), we have hH(id H) E EndH(M ® H). It is easy to see that M is an H-comodule, letting the composite map
be its coaction. A little more generally, if G = Spec 1£ is affine over X, then 1£ is an Ox-Hopf algebra, and a G-module and an 1{-comodule are the same. In the sequel, we only consider group schemes G = Spec 1£ affine over the base scheme X. --
(4.2.4) Let X be a scheme, and G an X-group scheme affine over X. Let M and M' be G-modules. We say that If : M ~ M' is a G-linear map if cp is an Ox-module map, and for any morphism f : Y ~ X, r(Y,/*cp): r(Y,/*M) ~ r(Y,/*M') is G(Y)-linear. We denote the category of G-modules and G-linear maps by eM!. Note that eM! is equivalent to the category M!1i.
(4.2.5) For G-modules M and M', we define the G-module structure of M ®ox M'. For f : Y ~ X with Y affine and 9 E G(Y), 9 acts on /*(M ®ox M') ~ f*M ®Oy /*M' so that the action on the right hand side is given by 9 ® g. This definition agrees with the tensor product of 1£-comodules.
Lemma 4.2.6 Let A be both a G-module and an Ox-algebra. Then the following are equivalent. 1 The coaction WA : A
~
A ®ox 1£ is an Ox-algebra map.
2 The product map A ®ox A
~
A is G-linear.
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(4.2.7) If the conditions above are satisfied, then we say that A is an Kcomodule algebra or a G-algebra. Applying the functor Spec to the coaction w,4, we get a right action az : Z xx G -> Z, where Z = Spec A. Conversely, if an affine morphism / : Z —> X and a right action az : Z xx G -> Z are given, then f,Oz is a G-algebra in a natural way. Thus, a G-algebra and a right G-action affine over X are one and the same thing. Lemma 4.2.8 Let R be a noetherian commutative ring, and G = Specif an affine R-group scheme of finite type. Assume that H is IFP. Then the coordinate ring H of G is R-projective. Moreover, there exists some n such that G is a closed subgroup of GLn(R). Proof. Note that H is countably generated as an it-module. The first assertion is obvious by Lemma 3.11.3. As H = k[G] is of finite type over R, there exists some it-finite projective it-subcoalgebra D of H, which generates if as an it-algebra, by the definition of IFP group. Note that the dual algebra D* is an impure subalgebra of (EndD*) op = EndD, via the right multiplication. This is trivial when R is a field, and the general case follows easily from Lemma 2.1.4. This shows that there is a surjective coalgebra map (End D)* -4 D. The composite coalgebra map (EndD)* -> D '-* H is uniquely extended to a fc-algebra map Sym(EndD)* —>• H, which is obviously an it-bialgebra map. This map is surjective, because the image of this map contains D, which generates if as an it-algebra. Taking the corresponding geometric morphism, we have a closed immersion it-semigroup homomorphism G -¥ End D. As D is a direct summand of Rn for some n, there is a closed immersion itsemigroup homomorphism G -> End it". Because G is a group, this map factors through G t-> GLn, which is also a closed immersion. • (4.2.9) Let tp : H -> G be a homomorphism of it-flat affine semigroup schemes. Then we have a bialgebra map R[G] -> R[H]. The restriction res fl//i an<^ induction indmS are respectively denoted by res% and ind^, and called the restriction functor and the induction functor. However, note that this induction is not the same as induction in the context of representations of finite groups or representations of (the enveloping algebras of) Lie algebras, defined via the tensor product. The terms 'induction' and 'coinduction' are sometimes interchanged depending on the context. Inductions via tensor products are right exact, while our induction via cotensor products is left exact but not necessarily right exact. (4.2.10) Let G be an affine flat it-group scheme of finite type. Then G acts on G itself via the adjoint action ((x,y) H-> y~1xy). Thus, H = R[G] is a G-module. As the unit element e is fixed by the action, the defining ideal I := Ker£ W of e is a G-submodule of if. As the product of H is
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G-linear, I/I2 is also a G-module. The Zariski tangent space (I/I2)* of the unit element is denoted by Lie(G), and called the Lie algebra of G (it is an i?-Lie algebra, in fact). The G-module Lie(G), as (I/I2)*, is called the adjoint representation of G.
4.3
Algebraic tori
(4.3.1) For a positive integer n, the X-group scheme GLO®" is denoted by GL{n,X) or GLn(X). We denote O$ = GL(1, X) by Gm,x or G m . The direct product G£, of G m is called the n-fold split torus. An X-group scheme which is isomorphic to the n-fold split torus for some n is also called a split torus. An X-group scheme T is called an n-torus if T is affine flat of finite type over X, with its all geometric fibers n-fold split tori. An n-fold split torus is an n-torus, but the converse is not true. (4.3.2) Consider the case X = Spec R is affine. Then we can express /?[Gm] = Jty.f"1 ]) with t group-like. Hence, if G is an affine i?-group scheme, then to give a rank-one i?-free representation of G is the same as to give a homomorphism of R-group schemes G —> G m , and it is the same as to give a bialgebra map R[t, t'1] —> R[G), which is given by a group-like element of R[G]. Thus, the set of isomorphism classes of rank-one i?-free representations X(G) of G, and the set of group-like elements X(/?[G]) of R[G] are identified. Moreover, the canonical bijection X(G) —y X(R[G]) is an isomorphism of abelian groups, where the product of X(G) is given by tensor products, and the product of X(R[G]) is the product of R[G). However, it is common to view X(G) as an additive group, and express its product by '+'. The group X(G) or X(R[G\) is called the character group oiG. (4.3.3)
As the coordinate ring of T := G£, R is expressed as
with each tt group-like, it is easy to see that X(H) = {tx \ A G Z"}, where as usual tx := tXl ••• tx" for A = (A x ,..., An) e Z n . By the map given by tx y-¥ A, we have an isomorphism of additive groups X(H) = Z n . When T is an i?-split torus, we call n = rankzX(T) the rank of T. Note that X(H) above is an i?-basis of H, and H is a direct sum of rank-one i?-free i?-subcoalgebras. (4.3.4) Let T = Spec H be as in (4.3.3). If V is a T-module, then we have a direct sum decomposition V = ©Agx(//) Vx> where (4.3.5)
Vx =
{veV\wv{v)=v®\).
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I. Background Materials
If Vx 7^ 0, then A is called a weight of V. If / : V —> V is a T-homomorphism, then obviously we have f(V\) C {Vx'). Thus, we have a canonical functor from j-M to the category of X(T)-graded i?-modules. Conversely, letting (4.3.5) be the definition, we have its quasi-inverse, and we see that a Tmodule is nothing but an X(T)-graded module. Even if the base scheme X is not affine, for a split torus T = G£, we have X(T) = Z" by the same reasoning, and X(T)-graded quasi-coherent Ox-modules and T-modules are the same thing.
4.4
Maximal tori, Borel subgroups, and reductive groups
(4.4.1) Let k be an algebraically closed field, and G a reduced affine algebraic fc-group scheme. In this situation, G and the group G(fc) are sometimes identified. Any Zariski-closed subset of G(k) is considered as a reduced closed subscheme of G. G has a maximum connected normal solvable subgroup, which is a closed subgroup of G, called the radical of G. We say that G is reductive if G is connected and non-trivial, and the radical of G is a torus. If G is reductive, then the connected component Z(G)° of the center Z(G) of G containing the unit element agrees with the radical of G. For example, a torus, GL(n, k), SL(n, k), SO(n, k), Sp(n, fc), and their direct products are reductive. If G is reductive, then the derived subgroup [G, G] and G/Z(G) are also reductive. (4.4.2) Let k and G be as in (4.4.1). A maximal connected solvable subgroup of G is called a Borel subgroup of G. A Borel subgroup is Zariski closed. It is not unique, but is unique up to conjugacy. For a closed subgroup P of G, P contains some Borel subgroup of G if and only if G/P is a fc-projective variety. If the equivalent conditions are satisfied, then P is called a parabolic subgroup of G. If, moreover, there is no closed subgroup Q of G such that P C Q C G , then P is called maximal. A subgroup which is maximal among closed subgroups which are tori is called a maximal torus. Note that maximal tori are conjugate to one another, hence their ranks are equal. We call the rank of a maximal torus of G the rank of G. By Lemma 4.2.8, G is a closed subgroup of some GL(n, k). We say that x £ GL(n, k) is unipotent if its eigenvalues are 1 only. As this is equivalent to (x - 1)" = 0, the set of unipotent matrices in GL(n, k) is Zariski closed. Hence, the set of unipotent elements Gu in G C GL(n, k) is a closed subset of G. Note that G u is independent of the embedding G <-* GL(n,k). If G = GU, then we say that G is unipotent. Let B be a Borel subgroup of G. Then Bu is a normal subgroup of B, B contains a maximal torus T of G, and B is a semidirect product of Bu and T.
4, From representation theory
107
4.5 Split reductive groups (4.5.1) Let R be a noetherian commutative ring. We say that G is a reductive .R-group scheme if G is an affine flat .R-group scheme of finite type, and all geometric fibers of G are reductive in the sense of (4.4.1). By definition, a reductive .R-group scheme is /^-smooth with connected geometric fibers. Any base change of a reductive group scheme is again reductive. Let G be an /J-smooth group scheme with connected geometric fibers. A closed subgroup H of G is called a maximal torus of G if H is a torus, and all geometric fibers H
108
I. Background Materials
1 For any commutative /?-algebra A, any t € T(y4) and any a € A = Ga(A), txa(a)t~l = xa(a(t)a). 2 The tangent map dxa is an isomorphism LieGa = (LieG)Q. are satisfied. Note that xa is unique up to isomorphisms of Go by the action of R* by multiplications. We always assume that xa is the base change of an xa defined over Z (and hence is uniquely determined up to sign change). Note that xa is a closed immersion. We denote the scheme-theoretic image xa(Ga) by Ua, and call it the root subgroup of G with respect to a. The group functor Ua represents A i-> Im(a;Q(yl)). (4.5.4)
Let R, G, T and E be as in (4.5.3). The set y(T):=HomR. groU p(G m ,T)
is an abelian group. If T ^ G^, then Y(T) * Z n . For f,g e Y{T), the sum / + g is nothing but the composite
AG
"^ in ~ ir (^'g)> T v T 'fH T >
For A € ^ ( T ) and / £ ^(T), the integer which corresponds to A o / by the isomorphism ((t H4 t r ) ^ r)
Homfl. group(Gm, G m ) = Z
is denoted by (A,/). Note that (?, - ) : X(T) x Y(T) -^ Z is bilinear, and induces an isomorphism Y(T) = Homz(X(T),Z). (4.5.5) Let R, G, T and E be as in (4.5.3). For a £ E, there exists a unique homomorphism of .R-group schemes <pa : SLQ —* G such that for any commutative R-algebra A and a € A, n
?
= Xa(a)
and
y>Q
hold. For any A and a € A*, we have
and thus we have a definition of a v G V(T). Note that a v is independent of the choice of xa, and is determined only by a. We set X(T) R := X(T)® Z R and r ( T ) R := r(T)<8)zK. Then the pairing (-,?):
X(T)®ZY(T)->Z
4. From representation theory
109
is extended to the pairing (-,?) : X(T)R ®R Y(T)u -> R, and we may identify Y(T)R ^ HomR(X(71),R) in a natural way. For a e E, we define sQ : X(T)M -> *(T) R by saA:=A-(A,av)a
(A G X(T) R ).
Note that sQ is an R-linear involution of X(T) R . We call the subgroup of GL(X(T)R) generated by {sa \ a £ E} the M^ey/ group of G, and denote it by W = W(G). Obviously, W maps -Y(T) to X(T). (4.5.6) Let fl, G, T and E be as in (4.5.3). The set of roots S of G is an abstract root system [87, p. 229] of the R-span R • E of E in X(T)R. Note that the restriction of sa to R • E is the reflection which corresponds to a. For any /?-algebra A which is an integral domain, we have W{G) <* NG(T)(A)/T(A)
£* (NG(T)/T)(A),
where in the first isomorphism, for any a 6 E, sa € W(G) corresponds to the image of
[
0
in NQ{T){A)/T(A). This map is always defined regardless of what A is, and it is injective unless A is the null ring. Hence, we have a canonical map W -»• NG(T)/T, and W acts on T via the adjoint action of NG(T)/T on T. Hence, W acts on both X(T) and Y(T), and the pairing (-,?) is W-invariant (note that the action of W on X(T) so obtained agrees with the action of W on X(T) given in (4.5.5)). As E is a root system, we can take a base A = AG- We say that A is a base of E, if A = {aj,..., at} is a basis of R• E, and for any a £ E, when we uniquely express a as the linear combination a = £ i Qa^ of this basis, Q are non-zero integers of like sign. If the Ci's are all positive (resp. negative), we say that a € E is a positive root (resp. negative root). The set of all positive roots (resp. negative roots) is denoted by E + = E j (resp. E~ = Eg). Note that the positivity/negativity of roots is a notion which depends on the choice of A. By the definition of a base, we have E = E + IJ E~. In the rest of these notes, if a split reductive i?-group scheme G is given, then we take it for granted that a split maximal torus T defined over Z and a base A = {on,..., a;} of EG are fixed. Note that W is a finite group generated by {sa | a G A}. For t u e l V , the minimum h such that there exists an expression w = sQj(1) • • • sa (1 < i(l),... ,i(h) < I) is called the length of w, and is denoted by l(w). There is a unique element of maximum length in W. We call it the longest element of W, and denote it by WQ. An element w in W equals u>o if and onlyifu)(E+) = E-.
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I. Background Materials
(4.5.7) Let R be a noetherian commutative ring, and G a split reductive /?-group scheme. We denote by U the closed subgroup of G generated by all Ua with a € E~. By the product map (in any order), we have an isomorphism of /J-schemes
n va*u. Hence, as an fl-scheme, we have U = A * s . Note that the maximal torus T normalizes U. The semidirect product B = TU = T K U is called the negative Borel subgroup of G. If R = k is an algebraically closed field, then B is a Borel subgroup of G in the sense of (4.4.2). (4.5.8) Let / C A. We set E 7 := E fi II, and we define L 7 to be the closed subgroup of G generated by T and all Ua with a £ E/. We have that Li is again a split reductive fl-group scheme, and its Weyl group is Wj := (sa | a S / ) . We call a closed subgroup of the form Lj for some / a (standard) Levi subgroup of G. (4.5.9) For A,/i 6 X(T), we say that A < \i if we have an expression /x - A = £i=i CiCti with Cj € No. This defines an ordering of X(T). We call this ordering the dominant order of X(T). We say that A e ^ ( T ) is dominant if for any a € A, we have (A, a v ) > 0. We denote the set of dominant weights by XQ. Lemma 4.5.10 We have that XQ, as a subset of X(T), is a countable ordered set. For any A 6 XQ, there are only finitely many /z € XQ such that fj, < A. For the proof, see [88, 13.2, Lemma B]. For A € X(T), A e X^ if and only if -wo\ € X%. For A G X{T), we denote — u/oA by A*. We have A** = A, and A > \i if and only if A* > fi*.
4.6
General linear groups
Taking G = GL(V) = GL(n, R) as an example, we review the basics on reductive groups explained in (4.5) more explicitly. (4.6.1) Let R be a noetherian commutative ring, and G = GL(V) = GL(n, R). We choose T as the subgroup of invertible diagonal matrices. (4.6.2)
In this case, any rank-one R-hee representation of T is of the form h
«n
4. From representation theory
111
(A = (Ai, A2l • • •, An) G IT). Thus, we have X(T) S Z n in a natural way. (4.6.3) We set £i = ( 0 , 0 , . . . , 0 , 1 , 0 , . . . , 0) G Z = X(T) (the j t h entry is 8ij, where 5{j denotes Kronecker's delta). Then E = { £ j - £ j | l
i^j)
is the set of roots of G (with respect to our choice of T). When we set a* := €i - £j + i, then A = { « i , . . . , a n _i} is a base of the root system S. Now wefixthe choice of the base A of E as above. Then the set of positive roots is E + = {ei - e}; 11 < i < j < n}. (4.6.4) Let T and A be as above. Then the negative Borel subgroup B is nothing but the subgroup of invertible lower triangular matrices. Note that U consists of all elements of B whose diagonal entries are all 1. (4.6.5) In the situation above, we have W(G) = S n , and W acts on T as the adjoint action of the group of permutation matrices. The action of W on X(T) = Z" is given by the permutation of entries. The longest element w0 is the permutation given by wo(i) = n + 1 - i. (4.6.6) We have (A, ( e i - e J ) v ) = ^ - A , for A = (A x ,..., An) G Z n = X(T) and 1 < i ^ j < n. Thus, A is dominant if and only if Aj > A2 > • • • > An.
4.7
Representations of reductive groups over an algebraically closed field
Let R be a noetherian commutative ring, G an .ft-split reductive group, T a split maximal torus of G defined over Z, and let A be a base of the root system E of G. We recall that U is a normal subgroup of the negative Borel subgroup B of G, and B is a semidirect product of U and T, with U normal. Let A G X(T), namely, A is a rank-one i?-free T-module. Then letting U act on A trivially, A is a rank-one R-iree B-module, whose restriction to T is the original A. We denote this rank-one i?-free J5-module by RxDefinition 4.7.1 For A G X+, we denote ind£(i?A) by V(A) = V G (A), and call it the induced module of highest weight A. The G-module V(A*)* is denoted by A(A) = Ac(A), and called the Weyl module of highest weight A. (4.7.2) From now on, R = k denotes an algebraically closed field. We denote R\ by k\. The following is well-known. (4.7.3) The set {k\ \ A G X(T)} is a complete set of representatives of the isomorphism classes of simple 5-modules. Any simple [/-module is trivial.
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I. Background Materials
(4.7.4) If M is a finite dimensional B-module, then •R'indg(M) is also finite dimensional. If i > dimG/B, then we have R'md^(M) = 0 (this vanishing holds also for infinite dimensional B-modules, see Lemma 3.6.17). In particular, for A e X+, we have that VG(A) and AG(A) are finite dimensional. (4.7.5)
For A e X(T), we have ind^(kx) ^ 0 4=> A G X%.
In the representation theory of the reductive group G, induced modules and Weyl modules play important roles. Theorem 4.7.6 ( K e m p f s vanishing) IfX £ X+, then for i > 0 we have #ind|(A: A ) = 0. (4.7.7) For A £ X+, we have soc(V(A)) = top(A(A)), and they are simple. We denote this simple G-module by L(X) = LG(X), and call it the simple G-module of highest weight A. As a result, we have that {L(X) | A £ X+} is a complete set of representatives of isomorphism classes of simple Gmodules. Moreover, if A, fi E X+ and L(/x) is a subquotient of rad A(A) © V(A)/socL(A), then fi< X. Theorem 4.7.8 (Cline-Parshall-Scott-van der Kallen, [39]) For any dominant weights X,fx£ X Q , we have
(4.7.9) Let V be a finite dimensional G-module. Then it is a T-module by restriction, hence we have a decomposition V = 0* e x(r) ^A- The element
ch(V):=
Y,
is called the formal character of V. It is easy to see that ch(V) € (ZX(T))W, where W is the Weyl group of G. As a consequence of Weyl's character formula [90, p. 250], we have that for any A 6 X£, ch(Ac(A)) = ch(Vc(A)), and they are determined only by A, and independent of A;. In particular, we have dim/t AG(A) = dim^ VG(A) are independent of k. (4.7.10) Another important property of weights of induced (or Weyl) modules is: if VG(A) M ^ 0, then we have u;0A < n < X- Moreover, VG(A)A = 1. Similarly for AG(A) and LQ{X). The name 'highest weight' comes from this fact. It follows that, if V is a G-module, A € XQ, dim^ VA = 1, and
dim* VM = 0 for n € X£ \ {A}, then V £ LG(A).
4. From representation theory
113
(4.7.11) If Jfc is of characteristic 0, then we have A(A) =* V(A) for For the results above, we refer the reader to [90].
4.8
XeX+.
Universal module functors
(4.8.1) Let X be a scheme. We say that U — (UA) is a universal family over X if for any X-algebra A (i.e., a morphism Spec>l -» X), a full subcategory UA of ^M closed under isomorphisms corresponds, and for any X-algebra map A -> B (i.e., a morphism Spec B —> Spec .4 of X-schemes), M € UA implies M ®A B £ UB- For example, the family Vx = (PA), where PA is the full subcategory of ^M consisting of finite projective /1-modules, is a universal family. Definition 4.8.2 Let s, t > 0, andZ^i,... ,Us+t, and V be universal families. We say that M = ((MA), (pf)) : Wi x • • • x U3 x U°^ x • • • x U%t -> V is a universal functor of type (r, s), if for each commutative X-algebra A, MA : (U,)A X • • • X ([/,)* x ( ^ + 1 ) 7 x • • • x (f/s+f^p -» V^ is a functor, for each X-algebra map / : A —> 5 , p/ : (B
X ((5®?) 0 p )')
is a natural isomorphism, and for any composable X-algebra maps
the diagram
C®BB®A
MA
C
®Bpf
A
. C®B MB((B®A?y, (B®A?)S) MC((C ®B (B®A?)Y, (C ®B \Mc(ar,as)
C ®A MA
^Z
. M
is commutative, where a : C®B(B®A7) -> C®^? is the usual identification. If Id = • • • = Us+t = Vx and V = (AM), then we say that M is a universal module functor of type (r,s). If Wi = • • • = Us+t = V = Vx, then we say that M is a universally projective functor of type (r, s). If it happens that X = Spec/? with R a PID, then a universally projective functor is sometimes referred as a universally free functor.
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I. Background Materials
This definition could be made as a special case of part of the theory of fibered categories and pseudo-functors [66, VI]. In the sequel, we only treat universal module functors, for simplicity. Definition 4.8.3 Let M = ((MA), (/>,)) and M = ((NA), (p'f)) be universal module functors of type (r,s) over X. We say that
o pf
holds. We have a category of universal module functors of type (r, s) over X. We denote this category by UMF(r,s;X). Note that UMF(r, s\X) is abelian. Exercise 4.8.4 The functor (Hom,j(?, —)) is a universally free functor of type (1,1) over SpecZ. (pf)(M,N) : B ® RomA(M,N) ->• HomB(B ®A M,BAN) is given by 6®/i >-> ( b ' ^ m H bb'®h(m)). The tensor product (Vi,..., Vr) i-> V\ ®A • • • ®A Vr is a universally free functor of type (r, 0). Pf(Vu • • •, K ) : B ®A (Vi ®A • • • ®A Vr) -> (B ®A Vi) ®B • • • ®B {B ® Vr)
is given by b ® fa ® • • • ® vr) i-> b{vx
®\)®---®{vr®\).
1
Similarly, the exterior power V i-> A V, the symmetric power V i-> SiV, and the divided power V ^ D{V = {SiV*)* are universally free of type (1,0) forz >0. Exercise 4.8.5 Note that a universal module functor (resp. universally projective functor) of type (0,0) is nothing but a quasi-coherent (D^-module (resp. locally free coherent Ox-module). be universally projective Lemma 4.8.6 Let r,s > 0, and M\,...,Mr+S functors of type (ri,si),..., (r r + s ,s r + 5 ) over X, respectively. If Af is a universal module functor (resp. universally projective functor) of type (r,s), then the 'composite' C:=Afo(Mu...,Mr+.) defined by is a universal module functor (resp. universally projective functor) of type r
s+\
r
s+1
i=r+l
i=l
i=r+l
4. From representation theory
115
(4.8.7) Let X be an i?-scheme. We denote the category of locally free coherent Ox-modules by Px- Let M = ({MA), (pf)) be a universal module functor of type (r,s), Vi,..., Vr+S € Px, and r > 1. As an extreme case of Lemma 4.8.6, we have that M(V\,..., Vr+S) is a universal module functor of type (0,0), i.e., a quasi-coherent Ox-module. Lemma 4.8.8 Let M = ((MA), (p/)) be a universal module functor of type (r, s) over X, and Vi,..., Vr+S € PX- Then the quasi-coherent sheaf M := M(Vi,..., Vr+S) is a module over G:=GL(V 1)x-.-xGL(V r+s) with the action
4.9
• • • ,9s,9s+i,-
• -,9s+r)
= MA(gi,.
• • ,9s,97+u-
• • >57+r)-
Tilting modules
Let A be a ring. Definition 4.9.1 We say that T 6 yiM is a tilting A-module if the following conditions are satisfied. (4.9.2) T e (4.9.3) v l e ( a d d T ) v (4.9.4) Ext^(r,r) = 0 ( i > 0 ) . (4.9.5) Let A be a ring, and T a tilting ,4-module. Set B := Endi4(T)°P. Then T is an (A, 5)-bimodule in a natural way. As a right 5-module, T is a tilting module, and we have EndB(T) = A. In this sense, the axioms for tilting modules are symmetric with respect to B and A. Hence, we sometimes say that T is a tilting (A, 5)-bimodule. (4.9.6) Let A, T and B be as above. For e > 0, we define KEe(T) to be the full subcategory of ^M consisting of ^4-modules M such that Extjj(T,M) = 0 for i ^ e. We define KTe(T) to be the full subcategory of flM consisting of 5-modules N such that Torf (T, N) = 0 for i ^ e. Then KEe(T) and KTe(T) are equivalent. In fact, ExteA(T, ?) : KEe(T) -4 KTe(T) and Torf (T, ?) : KTe(T) -> KEe(T) are the quasi-inverse of each other. (4.9.7)
We have left.gl.dim A < oo <=> left.gl.dim B < oo.
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I. Background Materials
(4.9.8) Let A and B be artinian algebras (i.e., module finite algebras over artinian commutative centers). Let us denote by G(A) and G(B) the Grothendieck groups of (^M)/ and ( B M)/, respectively. Then the maps
Ext: G{A) -> G(B), [X] -> i>0
and
Tor : G(B) -> G(A), [Y] -> B - l f l T o r f (T,y)] are inverse to each other. (4.9.9) Let A and B be as in (4.9.8). Then the number of isomorphism classes of indecomposable direct summands of T, the number of isomorphism classes of simple v4-modules, and the number of isomorphism classes of simple B-modules are all equal. For the details of the results summarized above, see [113]. See also [67], [68] and [35]. Let T be a tilting ,4-module. When we set F := Hom A (T, ?), then F : KE 0 (T) -> KT 0 (T) is an equivalence by (4.9.6), and add(T) corresponds to add(B) by F.
4.10
Cotilting modules
Let A be a left noetherian ring, and B a right noetherian ring. Definition 4.10.1 ([112, p. 586]) Let U be an (A, B)-bimodule. We say that U is a cotilting (A, B)-bimodule if the following conditions are satisfied. (4.10.2) AU e AMf, UB € M B / (4.10.3) inj.dim,i[/ < oo, inj.dimC/B < oo (4.10.4) Ext^f/, U) = 0 (i > 0), Ext B (£/, U) =
0(i>0)
(4.10.5) The canonical maps Bop -> End^ U and A -> End B U are isomorphisms. By definition, the condition for being a cotilting module is left-right symmetric. In other words, an (.4, S)-bimodule U is cotilting if and only if it is cotilting as a ( S o p , i4op )-bimodule. By condition (4.10.5), a usage such as '^f/ is a cotilting .4-module' or '[/ B is a cotilting right B-module' makes sense. Assuming that A is both left and right noetherian, a cotilting (,4,.4)-bimodule is called a dualizing bimodule of A.
4. From representation theory
117
(4.10.6) Let k be a field, and A and B be finite dimensional algebras over k. Then T is a tilting (A, B)-bimodule if and only if T* is a cotilting (B, j4)-bimodule. (4.10.7) Cotilting modules are deeply related to Auslander-Buchweitz contexts (1.12) in ^M/. Theorem 4.10.8 ([112]) Let A be a left noetherian ring, B a right noetherian ring, and U a cotilting (A, B)-bimodule. When we set A := /iM/, X := XU, u := addU, and y := u>, then (X,y,w) is an AuslanderBuchweitz context in A. The functor
is a contravariant equivalence of exact categories, with Homfl(?, AUB) its quasi-inverse. By Lemma 2.11.10, we have the following. L e m m a 4.10.9 If A is a noetherian commutative ring with a dualizing Amodule UA, then A is Cohen-Macaulay, and the minimal injective resolution of UA is a dualizing complex of A. Conversely, if A is a Cohen-Macaulay ring with a dualizing complex I', then the A-module r(Spec<4,wgpec/i) is dualizing, where wspecA is the canonical sheaf (2.11.9) of Spec A. In this case, A is of finite Krull dimension. (4.10.10) If the conditions in the lemma are satisfied, then the assumption of Theorem 4.10.8 is satisfied, and we have an Auslander-Buchweitz context {XA^A, W/i). More generally, even if A is not finite dimensional, we can construct an Auslander-Buchweitz context. Let A be a commutative noetherian ring. An .4-module M is called pointwise dualizing if M is ^4-finite, and for any p £ Speed, the j4p-module Mp is dualizing. By Theorem 2.5.6, a commutative noetherian ring with a pointwise dualizing module is Cohen-Macaulay. An .4-module M is called a maximal Cohen-Macaulay module if M is Afinite, and for any p € Spec A, Mp is a maximal Cohen-Macaulay /l p -module in the sense of (2.5.1). Proposition 4.10.11 Let A be a commutative noetherian ring, and KA a pointwise dualizing module of A. Set AA := yiM/, UA '•= add KA, XA t° be the full subcategory of AA consisting of maximal Cohen-Macaulay Amodules, and yA to be the full subcategory of AA consisting of M E AA such that for any p € Speed, inj.dim^ Mp < oo. We define VA '•= add A. Then the following hold.
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I. Background Materials
(4.10.12) XA = LKA (4.10.13) yA = LJA (4.10.14) (XA,yA,ujA)
is an Auslander-Buchweitz
context.
(4.10.15) YlomA(?,KA) is a contravariant equivalence of exact categories from XA to itself, with KomA(?,KA) itself its quasi-inverse. (4.10.16) EomA(KA, ?) is an equivalence of exact categories from yA to VA, with KA®A1 its quasi-inverse. In fact,
given by q® f >-> fq is an isomorphism for N G yA, and M
-+HomA{KA,KA®M)
given by m *-* (q >-> q ® m) is an isomorphism for M G VA. Proof. Let
For M G AA and p 6 Spec,4, we set cM(p) := dimylp — depthM p . F
: • • . - > F2 % Ft i > Fo ^ M -> 0
be a free resolution of M with each Ft .A-finite. We set fi*M := lmd{. Then it is easy to see that (fijM)p is a maximal Cohen-Macaulay .Ap-module if and only if CM(P) < i. By Corollary 2.12.3, CM is an upper semicontinuous function over Spec A A similar argument applied to PM(P) := proj.dim^ M p instead of CM yields that PM is also upper semicontinuous. As Spec>l is quasi-compact, we have d := maxcM(p) < °°) a n d by definition, Sl^M is maximal Cohen-Macaulay. Hence, we have M e XA, and AA = XA. Similarly, an object M of AA belongs to VA if and only if proj.dim^ Mp < oo for any p G Spec A Next, we show that XA = ±KA. To prove this, we may assume that A is local. In this case, as KA is a dualizing complex of A, the assertion follows easily from the local duality (Theorem 2.10.7). Also, for any M e XA, we have that RomA(M, KA) G XA and M^EomA(}iomA(M,KA),KA) is an isomorphism. This is also checked after localization. Hence, (4.10.15) holds. Next, we show that uiA is an injective cogenerator of XA. We already know that UJA = &ddKA is #-injective, and u>A C XA. When we take an exact sequence 0 -> N -> F -> EomA{M, KA) -> 0
4. From representation theory
119
with F .A-finite free, then the sequence is an exact sequence in XA. Applying the functor Hom^?, KA) to the exact sequence, we have an exact sequence 0 -> M -> EomA(F, KA) -» EomA(N, KA) -+ 0 in XA again. As B.omA(F, KA) G u>A, we have that wA is an injective cogenerator of XA. XAnyA, Next, we show that XAf\yA C CJA- To verify this, we take M e and it suffices to show that HomA(M, KA) e VA. Hence, we may assume that (>l,m) is local. As KomA(M,KA) is maximal Cohen-Macaulay, it suffices to show that pTO).dimAUomA(M,KA)
< co
by Theorem 2.5.3. Let F be the minimal free resolution of A/m, and /* the minimal injective resolution of KA. Then as we know that M £ ±KA, we have
REom'A(F,M) £ REomA{¥,REom'A{REomA(M,I'),I')) £ REomA{¥ ®LA Hom^(M, I'),I') s RHom^(F ®J Hom j 4 (M,^),/'). Note that /2Hom^(F, M) has bounded homology groups, since M € Hence, F ®A EomA{M, KA) S i? EomA(R Hom^(F ®$ Hom yl (M, ^ ) , /*), / ' also has bounded homologies. This shows proj.dim^ Hom^(M, KA) < CXD. Hence, we have that (4.10.14) holds. The assertions (4.10.12) and (4.10.13) are consequences of Theorem 1.12.10. The assertion (4.10.16) is well-known. It was first proved by Sharp [136], and is generalized by Avramov and Foxby [18, Corollary 3.6]. We prove the first assertion of (4.10.16). As N G y& = &A, there is a finite o^-resolution W of iV. Since KA ®A EomA{KA, KA) —> KA is an isomorphism, we have that KA®A EomA(KA, W) -> W is an isomorphism of complexes. As the augmented complex W —> N —> 0 is a bounded exact complex consisting of objects of yA C KA, we have that EomA(KA, W) is a resolution of EomA(KA, N). By the five lemma, KA®A EomA(KA, N) -> N is also an isomorphism. The second assertion is proved similarly, utilizing Theorem 4.10.19 below. D As in the proof of the theorem, the following is easy to prove. Lemma 4.10.17 Let R be a regular ring, and V E RMJ. Then we have proj.dim fi V < oo.
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I. Background Materials
(4.10.18) Let R be a noetherian commutative ring. We say that an Rmodule N is locally of finite flat dimension if flat.dimfip Np < oo for p G Spec R. Note that if N is of finite flat dimension, then it is locally of finite flat dimension. The converse is true if the Krull dimension of R is finite, see [55, Corollary 3.4]. Related to the proof of (4.10.16), the following holds [18, Corollary 3.6]. Theorem 4.10.19 Let R be a Cohen-Macaulay ring, and M a maximal Cohen-Macaulay R-module. If N is an R-module locally of finite flat dimension, then we have Torf (M, N) = 0 for i > 0. Note that the proof is easily reduced to the case that (R, m) is a complete local ring, and we may also assume that M = QR by (4.10.14). Corollary 4.10.20 Let R and M be as in the theorem, and let N be a perfect R-module of codimension h. Then we have Ext'R(N, M) = 0 (i ^ h). By Lemma 2.9.1, we have proj.dim R Ext^(iV, R) < oo, and Ext^N,
M) S T o r j ^ E x t * (JV, R), M).
By the theorem, the assertion follows.
•
Proposition 4.10.11 is well-known as the Cohen-Macaulay approximation. The name 'approximation' comes from the fact that XA is contravariantly finite (see Theorem 1.12.10). In Theorem 4.10.8, we have that any object of y is of finite injective dimension. However, in the situation of Proposition 4.10.11, this is not true any more if dim .4 = oo. (4.10.21) For a finite dimensional algebra A over a field, AuslanderBuchweitz contexts in (/iM)/ and basic cotilting modules of A are in one-toone correspondence. This beautiful result was proved by Auslander-Reiten [11]. Let A be a ring, and M an yl-module. We say that M is basic if M does not have any direct summand of the form N © N, where N is a non-zero i4-module. Theorem 4.10.22 Let k be a field, and A a finite dimensional k-algebra. We set AA •= CiM)/. Consider the following. a A full subcategory u in AA such that u/ = adda> C u>L and Xu = AA, where Xu is the full subcategory of AA consisting of X G x u such that there exists an exact sequence
such that T ' e w and Im/' G x w.
4. From representation theory
121
b A full subcategory X of AA which is closed under extensions, epikemels, and direct summands, and has an injective cogenerator, such that X = AA. c A covariantly finite full subcategory y of AA closed under extensions, monocokernels, and direct summands such that AA* G y and any object in y is of finite injective dimension. d An Auslander-Buchweitz context (X,y, w) in AAe The isomorphism class of a basic cotilting A-module T. The objects a—e above are in one-to-one correspondence by the following correspondences. a=>b u) to Xu. b=>c X to Xx. c=>a y
to-Lyny.
a,b,c=>d Obvious correspondence. a,b,c,d=>-e As we have AA* G X1 = u, there is an exact sequence 0 - » u)n - » • • • - > wi - > w 0 -> A-A* - > 0 withuji G u. Starting with the Krull-Schmidt decomposition ofu0®---(B u)n, we get a basic module T, removing N whenever we find N © TV in the decomposition. The isomorphism class ofT is uniquely determined by (X,y,ui), and T is the corresponding cotilting module.
e=>a T to addT. Corollary 4.10.23 Let A be a finite dimensional algebra over a field k, and (X,y,u) an Auslander-Buchweitz context in ,jM/. Then the number of isomorphism classes of indecomposable objects in u is equal to the number of isomorphism classes of simple A-modules. Proof. By the theorem, the number of isomorphism classes of indecomposable objects in w agrees with the number of indecomposable direct summands of T in the theorem, which agrees with the number of indecomposable direct summands of T". As T* is a tilting j4op-module, these numbers agree with the number of simples of Aop, which is equal to the number of simples of A by (4.9.9). D Notes and References. There is no new result at all in this section. In this section, we listed basic results in the representation theory of algebraic groups and algebras. For more, we refer the reader to [87], [90], [42] for algebraic groups, and to [12], [11], [112], [113] for algebras. Although we assume that a reductive group (over an algebraically closed field) is connected, note that it is not always assumed in the literature.
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I. Background Materials
Basics on equivariant modules
5.1 Cocommutative Hopf algebra actions (5.1.1) Let R be a commutative ring, U a cocommutative .R-Hopf algebra, and A a commutative [/-module /^-algebra. Let M be an A#U-module, and N a [/-module. The /^-module M
d: M ®(A®N)
-¥ M ®N
by d(m <8) a ® n) = am ®n — m® an, then it is an ^4#[/-linear map, and M ®A N = Cokerd also has an A^U-module structure. (5.1.2) Let M be an A#U-modu\e, and N a [/-module. Then Hom(M, N) is an i4#[/-module in a natural way. The action of U is given as in (3.12.1), and the action of A is that on M. If M is a [/-module and iV is an j4#[/-module, then letting A act on N, Hom(M,N) is an A#U-module. If both M and N are /l#[/-modules, then there are two different ways to see Hom(M, N) as an i4#f/-module. We give priority to the definition in which A acts on N. If both M and ./V are .4#[/-modules, then RomA(M,N) is an A#Usubmodule of Hom(M, N). Note that we have EomA#u{M, N) = Honu(M, N) (1 Eomv{M, N) = H°(U, HomA(M, N)). (5.1.3) Let R and U be as in (5.1.1). Let A and B be commutative Umodule algebras, and ip : A —t B a [/-module algebra map. Then we define : A#U = A®U->B®U
= B#U
to be (p®\du- It is easy to see that y # [ / is an .R-algebra map. In particular, any 5#[/-module is an >l#[/-module by restriction. Let M be a S#[/-module, and V an A#U-modu\e. Then M ®A V, V ®A M, UomA(M,V) and UomA(V,M) are B#U-modules in a natural way, and as yl#[/-modules they agree with the ones defined in (5.1.1) and (5.1.2).
5. Basics on equivaiiant modules
123
(5.1.4) Let R, U, and
M —> EomA(A,M) ( m 4 ( a H am)) UomA(V, W) ®A Hom / i(M, K) -> Hom i4 (M, IV) Hom^lV, M) <S)A Hom/i(Vr, W) -^ Hom A (y, M) Hom/i(Vr ®yi W, M) ^ Hom/i(K! Ho m / i(iy, M)) M O.4 V, W) S Hom j4(M, Homj4(Vr, V (8.4 M, W) s Homii(Vp, Ho my i(M, Hom B (V ^ M, TV) ^ Hom^V, Hom B (M, N)) EomB(M ®A V, N) S Hom B (M, Hom,i(Vr) N)) HomA{M ® B -?V, V) ^ Hom B (M, Hom^(A^, V)) (M ®^ V) (8)4 IV «* M ®>i (V ®^ IV) M ®,i V = V ®A M M -4 Hom/i(Hom/ 4(M, V), V) (the duality map) r M ®A Homj4(V, IV) -> Hom/i(V , M ®^ IV) M ®B Hom/i(Vr, AT) -> ftomA{V, M ®B N)
are B#t/-linear maps, which are natural with respect to M, N, V, and W. As a special case, we will use the case A = R or A = B frequently. Taking the invariances H°{U,?) of both sides of (5.1.11), (5.1.12), and (5.1.13), we get natural isomorphisms (5.1.20) * : Homfl#C/(Vr ®A M, N) (5.1.21) V:1iomB#u(M<8>AV,N) (5.1.22) * : HomA#u(M ® B N, V)
HomA#u(V, Hom B (M, N)) HomB#u(M,HomA(V,N)) Hom B # u (M, Homyl(A^, V)),
respectively. In particular, we have Lemma 5.1.23 Let R, U, f : A-> B, M, N, and V be as in (5.1.4). Then the following hold. 1 If N is B#U-injective injective. 2 If N is B#U-injective
and M is A-flat, then UomB(M,N) and V is A-flat, then Hom/i(Vr, N) is
3 IfN is B-flat and V is A#U-injective,
then YiomA(N, V) is
4 If M is B-projective and V is A#U-projective, projective.
is
A#U-
B#U-injective. B#U-injective.
then V ®A M is
B#U-
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I. Background Materials
5 If M is B#U -projective and V is A-projective, then M <8u K is # # [ / projective. 6 If N is A-projective and M is B#U-projective, projective. In particular, if V is an A#V-projective projective.
then M ® B N is A#U-
module, then V
<S>A
B is
B#U-
Moreover, since B#[/ =* B®A{A#U) 2* (A#U)®AB as 4#[/-modules, we have Corollary 5.1.24 / / B is A-projective, then any B#U -projective module is A#U-projective. When we consider the case M = B in Lemma 5.1.23, 1, we have the following. Corollary 5.1.25 If B is A-flat, then any B^U-injective injective.
5.2
module is A#U-
Tor/4 and Ext^ as A#C/-modules
(5.2.1) Let R be a commutative ring, U a cocommutative /?-Hopf algebra, A and B commutative [/-module algebras, and ip : A -> B a [/-module algebra map. We assume that U is ii-projective. Lemma 5.2.2 Any A#U-projective injective module is A-injective.
module is A-projective.
Any
A#U-
Proof. We prove the first assertion. It suffices to show that A#XJ is Aprojective. As the action of A on A#U is given by a(b ® u) = ab
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125
(5.2.3) Let R, U, and
since any B#t/-projective module is .A-flat in this case. Similarly, we define
Torf(V,M)
:= L4(? ®
and these are B#f/-modules. We have Li(V<8)-4?)(M) ^ Torf(V.M) in i B # [ / M , if 5 is 4-flat. We have /? (Hom i4 (?, V))(M) = E x t ^ M , V) in B #i/M, if 5 is 4-projective. We have i?i(Hom^(Vr, ?))(M) = Ext^V, M) in B # l / M , if 5 is 4-flat, by Corollary 5.1.25. Proposition 5.2.4 Let M and N be B#U-modules, If M is A-flat, then we have spectral sequences (5.2.5) (5.2.6)
£ 2 M = Ext^(V,Ext B (M,./V)) EZ" = ExtpA#u(V,ExtqB(M,N))
and V an A#U-module.
=> ExtpB+q(V ®A M,N) =• Ext££v(V ®A M,N)
in the categories B #uM and H°{U,B)^I respectively. If, moreover, M is Bprojective, then we have isomorphisms (5.2.7) (5.2.8)
Extj4(Vr,HomB(M,iV)) ExtA#u(V,HomB(M,N)) and
H°(U,B)^,
£ S
Ext l B(V ®A M,N) ExVB#u(V ®A M,N)
respectively.
Proof. We prove the existence of a spectral sequence (5.2.5). Let F be an .4#{/-projective resolution of V, and I a B#[/-injective resolution of N. Then we have quasi-isomorphisms of complexes Hom^(F,Hom B (M,I)) =* Hom B (F®,t M,I) <- HomB(K
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I. Background Materials
(5.2.9) For later use, we need to study unbounded derived functors. Let R and U be as in (5.2.1), and A a commutative [/-module algebra. For F,G £ C(A#UM), the complex Hom^(F,G) is defined as in (1.4.2), and it is again a complex of j4#[/-modules. It is easy to see that the construction induces a bifunctor Horn;, : K(A#uM)op
x K(A#,jM)
->
K(A#UM).
Lemma 5.2.10 / / F is an exact A-complex and G is a K-injective complex, then Hom^(F, G) is an exact A-complex.
A#U-
Proof. We have a standard isomorphism
Hom^(F,G) S HomA#u{{A#U)
®AW,G).
By the proof of Lemma 5.2.2, the j4-module A#U induced by the right regular action (A#U)A#u is i4-projective. Hence, {A#U)®A¥ is still exact. As any i4#[/-linear chain map (A#U) <8>A W[—n] -> G is null homotopic by the /C-injectivity of G, the complex Hom^ #[/ ((A#U) ®AF,G) is exact, and hence so is Hom^(F,G). • By the lemma and Theorem 1.4.12, there is a derived functor REomA
: D(A#uM)op
x D(A#UM)
-»
For F,G £ K(A#UM), 5Hom^(F,G) is Hom^(F,I), where I is the Kinjective resolution of G. By the lemma, we also have that the construction of i?Hom^ is compatible with the forgetful functor D(A#u^-) —> D(AM). (5.2.11) For A G C(A#u^), there is a quasi-isomorphism F —¥ A in C(-4#c/M) such that F is the inductive limit of an inductive system indexed by N, of >l#C/-projective complexes bounded above [137], [25]. This shows that the bifunctor ®^ induces a bifunctor ®A • D(A#UM)
x D(A#UM)
->
D(A#uM),
and the construction of ®A is compatible with the forgetful functor D(A#UM)
-> D(AM).
For F e D(A#M), the composition induces R Hom^ (F, G) ®^ R Hom^ (E, F) -> R Hom^ (E, G), which is natural with respect to E € D(A#uM)op is calculated as the composite y ••
and G € D(A#UM).
This
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127
where G —> G' and F -> F' are /f-injective resolutions, and
is a quasi-isomorphism, where Q is the inductive limit of an inductive system indexed by N, of .4#[/-projective complexes bounded above. Taking the cohomology, we have the Yoneda product of hyper-Ext groups E,F) -> Ext i+i (E,G), which is i4#(/-linear. Concretely, it is given as the composite map Ext^F.G) ®A Ext£(E,F) = IP(Q) ®A ^ j (Hom^(E,F')) -> Hi+j(Q®\
Hom^(E,r))
H + (v)
' ' ) iT+^Hom^E.G')) = Ext i+ '(E,G).
Clearly, this is nothing but the usual Yoneda product as an .4-linear map, and hence it is associative. In particular, for an >l#f/-module M, the Yoneda algebra ©i>0 Ext\(M, M) admits an ^4#(/-action.
5.3
(G, >l)-modules
(5.3.1) Let R be a commutative ring, G = Spec.// an affine flat il-group scheme, and A a commutative G-algebra (or an i/-comodule algebra). Set X := Spec A If / : A —> B is an i/-comodule algebra map, then we say that / is a G-algebra map, or sometimes that B is a (G, A)-algebra. We call an (H,A)-Uopi module a (G, A)-module, and instead of using AMH, we use G ^ M . Note that MH = G,RM is nothing but G M. Instead of writing Extj, M , ff(M", ?), or Ext'G AM, we use the notation Ext G , H^G, ?), and Ext G X , respectively. The functor H°(G,7) is also written (?) G , and is called the G-invariance functor. A (G, >l)-submodule of the (G, .<4)-module A is called a G-ideal. If / is a G-ideal of A, then A/1 is a (G, ,4)-algebra in a natural way. The following is checked easily. L e m m a 5.3.2 Let R, A and G be as above. Then c,/iM is an abelian category which satisfies the (AB5) condition. The forgetful functors G,y»M -» G M and G,/IM -> AM are faithful exact and preserve inductive limits. (5.3.3) Let R, A and G be as above. Let M,N € G,AM and V, W e G M. Then M ® W and V ® N are G-modules. They are yl-modules with the actions given by a(m ® w) := am ® w and a(v ® n) := v ® an, respectively, and are easily checked to be (G, ,4)-modules. Then we have two different (G, j4)-module structures of M®N. As in the case of y4#C/-modules, we give priority to the one in which A acts on M, unless otherwise specified. The j4-module G-module M ®A N is a (G, .A)-module, which is a quotient object
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I. Background Materials
of M
* : Hom G , B (V ®A M, N) S HomG,,i(V, Hom B (M, N))
(5.3.7)
* : Hom G , B (M ®A V, N) S Hom G , B (M, Hom^jV, N))
natural with respect to V, M and N. Here we require that M is of finite presentation as a B-module in (5.3.6), and V is of finite presentation as an j4-module in (5.3.7). In particular, we have L e m m a 5.3.8 Let R, G, and tp : A —» B be as in (5.3.5). Let M and N be (G, B)-modules, and V a (G, A)-module. Then the following hold:
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129
1 If M is a (G,B)-module which is A-flat and of finite presentation as a B-module, and N is an injective (G, B)-module, then HOITIB(M, iV) is an injective (G,A)-module. 2 Let V be an A-finite protective (G, A)-module, and N an injective (G, B)module. Then Hornby, N) is an injective (G,B)-module. Corollary 5.3.9 If B is A-flat, then any (G, B)-injective module is (G, A)injective. Lemma 5.3.10 For any M € G,AM and any A-finite A-submodule V of M, there exists some A-finite A-submodule N of M and a (G, A)-submodule L of M such that V C L C N. If, moreover, R or A is noetherian, then we can take L to be A-finite. Proof. Let x\,...,xn be a set of generators of V as an /l-module. By Lemma 3.7.1, there exists some .R-finite /t-submodule iV0 of M and some Gsubmodule LQ of M such that {x\,..., xn} C LQ C NQ. We set L\ := A®LQ, and iVi := A®NQ. When we define L and N to be the image of the canonical maps L\ —>• M and N\ —> M, respectively, then the required conditions are satisfied. • Corollary 5.3.11 The category G,/IM is Grothendieck. In particular, C,A^ has injective hulls. If A is noetherian, then G,A^. is locally noetherian, and hence G,/IM satisfies the (AB3*) condition. In this case, G,A^/ consists of A-finite (G, A)-modules. Proof. This is similar to Corollary 3.7.3, and we omit the proof.
•
Corollary 5.3.12 / / either of R and A is noetherian, then any (G, A)module is the filtered inductive limit of its A-finite (G, A)-submodules. Corollary 5.3.13 Assume that A is noetherian. Let M G G,A^J and let (N\) be a filtered inductive system in G,A^- Then the canonical map
lim ExtjjiA(M, Nx) -> E x t ^ ( M , ljm Nx) is an isomorphism for i > 0. Remark 5.3.14 Even if R = k is a field and A = k, the forgetful functor QM = G.fcM -> jtM does not preserve direct products in general. Notes and References. All the results stated here are straightforward, and there is no reference.
Chapter II Homological Algebra of Equivariant Modules and Matijevic-Roberts Type Theorem 1
Homological aspects of (G, A)-modules
1.1
Construction of Ext ,4
(1.1.1) Let R be a commutative ring, G an affine i?-flat group scheme, A and B be G-algebras, and
Let R, G, and A be as in (1.1.1). We define a : A -> A ® H by
a(a) := a® 1, 0 : A-¥ A® H by 0 := uAt and 7:.4
(a® h H-> ^ a
0
® (Scn)h).
(a)
For an i?-algebra C, we consider the trivial G-action on C, and C is a G-algebra. However, if C has another G-algebra structure, we denote the trivial G-algebra C by C, to avoid confusion. For a C-module M, when we consider the trivial G-action, then M is a (G, C')-module. We
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II. Equivariant Modules
denote this by M', to avoid confusion. Note that a : A —> A ® H and 7 : A ® H -> A' ® H are G-algebra maps, whence so is 0 : A —> A' ® H. Note also that a : A' -> A' ® H is a G-algebra map, which we denote by a', to avoid confusion. Lemma 1.1.3 Let C and D be G-algebras, and V a (G,C)-module. Let a : C -» C ® D be the G-algebra map given by a(c) — c 0 1. Then a#V is isomorphic to the (G, C <8> D)-module V <%> D, where the C <8> D-action is given by (c ® d)(v
ip : a#V = V®C{C®D)-*V®D by v <8> (c <S> d) i-> cu ® d, then it is a G-isomorphism. It is C ® ZMinear as well, by the definition of C ® D-action on V (8) Z). D By the lemma, a#V = V ®H.
Lemma 1.1.4 Let V be a (G,A)-module. Then (3#V is isomorphic to the (G, A'®H)-module V®H with the A'®H-action (a®h)(v®h') := £( o) aov® Proof. We have /3# = 7 # o a # ^ ( 7 ~ 1 ) # ° a # . Hence, @#V is the (G, /l' module
(j-lf(V
®A (A ® H)) ^ ( 7 " 1 ) # (^ ® H)
by Lemma 1.1.3. Now the assertion is trivial.
D
L e m m a 1.1.5 Let A be a G-algebra, and M a G-module A-module. Then M is a (G, A)-module if and only ifuiM '• M —> M'®H is a G-linear A-linear map from M to M ® H = /3 # a' # M'. Proof. As M is a G-module, w M is G-linear. For a £ A and m € M, we havew M (am) = H{am)(arn)o® (am)u and aw M (m) = E(o),(m)aomo®onmi, D and we have u)M is .A-linear if and only if M is a (G, i4)-module. L e m m a 1.1.6 We have the following. 1 Let M be a' (G,A)-module. When we define UM : M ® H -> M' ® H by OM{m ® h) := Y,(m)mo ® fn\h, then we have that D : (3# —> a# is a natural isomorphism between functors from G,A^ to /t®//M. Moreover, the composite map KP ® 1H) O 0\#M
= [(lA ® A H ) O (3}#M
(U8AH)#
° M ) [{lA ® AH)
1. Homological aspects of (G,A)-modules
133
agrees with the composite map [(/?
{m )
" *°M)
= [o12 o f3]#M
[(/3 ® iH) o a]#M (ai2)# M
° ) [a12 o a ] # M = [(U ® A w ) o a)#M,
where a ] 2 : A® H -* A® H ® H is given by an(a® h) := a® h® 1. 2 Conversely, if an A-module M and an isomorphism D M : (3#M -> a # M suc/i t/io<
M is a G-module, and hence is a (G,A)-module. Proof. The map DM in 1 is nothing but the A ® 7/-linear map which corresponds to U)M '• M —> (ffia#M by the adjunction between 0# and /?#. This map is an isomorphism, as the inverse is given by Q^/(m <E> h) — £(m) Tna®S{m\)h. The naturality is obvious. The map U>M in 2 corresponds to OM (given in 2) by the adjunction. To say that the diagram in question is commutative is the same as to say that the corresponding maps M —> M ® H ® H to the two composite maps (by adjunction) (lM ® AH) o UJM and (WM <8> 1//) ° % agree. Hence, this is equivalent to the coassociativity of OJM- Hence, 1 is obvious. We show 2. The base change of OM by A ® H A®£> A ® R = A is an ^-isomorphism, and it agrees with the composite map
When we denote it by p, we have p o p = p by the coassociativity. As p is an isomorphism, we have p — \M, and the counit law follows. Now we know that M is a G-module. As % is >l-linear, we have that M is a {G,A)module by Lemma 1.1.5. • (1.1.7) Let R, G, and A be as in (1.1.1). As a' : A' -> A' ® H and 0 : A -> A' ® H are G-algebra maps, we have /? # (a' # (V")) is a (G,A)module for any >l-module V. More explicitly, /? # (a^(V")) is the G-module V'®H, equipped with the ,4-module structure by a(v®h) := £(„) aov®aih. If M is a (G, ,4)-module, then uM:M-*M'®H = /3*{a'#(M')) is 4-linear (Lemma 1.1.5) and G-linear (Example 1.3.6.6), and hence it is (G, .4)-linear. Lemma 1.1.8 Let M 6 G ^ M , and V G ^M. Then the map $ : HomA(M,V)
->
EomG>A{M,0*(a'#(V')))
defined by $(/)(m) := £( m ) frriQ ® m\ is an isomorphism which is natural with respect to M and V. In particular, the forgetful functor C,A^ —> >iM has SA '•— /8*a^(?') as its right adjoint.
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II. Equivariant Modules
Proof. As we have $ ( / ) = /? # (a' # /') o u, $ ( / ) is certainly a [G,A)linear map by the remark above. We define \& : Homc^M,/? # (a' # (V'))) —> H o m ^ M , V) by $(g)(m) := E(gm), where E : V ® 77 -> V is given by E(v
D
Corollary 1.1.10 Let M e G,A^> V e AM, and i > 0. Then there is an isomorphism $ : ExeA(M,V) S Extj, i>l (M,S(V)) natural with respect to M and V. Proof. Let I be an ^4-injective resolution of V. Then as H is exact and preserves injectives, H(I) is an injective resolution of H(V). Hence, we have a sequence of isomorphisms E x t ^ M , V) S i7 i (Hom x (M,I))
which is natural with respect to M and V. L e m m a 1.1.11 Let A be noetherian. If J is an injective then Homyi(?, J) is an exact functor on G,A^-
D (G,A)-module,
Proof. First, we prove that Hom^?, J) is an exact functor on G,A^/By Corollary 1.1.9, we may assume that J = H(7), with 7 an j4-injective module. Hence, we have isomorphisms of functors Hom,i(?, J) = Honu(?,/? # a # 7) ^ Hom y l ( / 9 # ?,a # 7) £ H 0 m / 1 ( a # ? , a # 7 ) ,
1. Homological aspects of (G, A)-modules
135
where the last isomorphism is KomA{0,a#I), see Lemma 1.1.6. Note that the G-structures of modules are irrelevant here, because the question is the exactness of Hom^(?, J), so the symbol (?)' is omitted here. As ? in question is of finite presentation as an >l-module, we have HomA(a#?,a#I) =* a # Hom,t(?, /) by Lemma 1.2.1.7. As a# and HomA(?,I) are exact, the exactness of Hom/i(?) J) as a functor over G,/IM/ is proved. Hence, we have that HoniG,/i(?, H( J)) is an exact functor over G,A^/ by Lemma 1.1.8. By Lemma 1.1.9.4, E(J) is an injective object of GjAM, and D hence Hom^?, J) is exact over G,/»M, by Lemma 1.1.8 again. Now we can prove the first theorem in this chapter. Theorem 1.1.12 Let R be a commutative ring, G an affine flat R-group scheme, and A a noetherian commutative G-algebra. Then for any M 6 l GlAM and an injective (G,A)-module J, we have Ext A(M, J) = 0 (i > 0). Proof. We proceed by induction on i > 1. First, consider the case M = V
in G,>IM, where K := Ker(p'
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Proposition 1.1.14 Let R be a commutative ring, G an affine flat Rgroup scheme, A —>• B a G-algebra map between commutative noetherian G-algebras, V £ G,/M, and M,N 6 G ] B M. Then the following hold. 1 If M is A-flat and B-finite, then there is a spectral sequence of R-modules Ep2-q = ExtpGiA(V, Ext«B(M, N)) => E x t £ | ( V ®A M, N). If, moreover, V is A-finite, then there is a spectral sequence of (G,A)modules EP'q = Ext&(V, Ext e (M, N)) => Ext B+*(V ®A M, N). 2 IfVis
A-finite protective, then there are isomorphisms
Ext^B{M
® V, N) S Extjj ]B (M, Hom^(Vr, N)) S E x t ^ B ( M , V* ® N),
where V* := HomA{V,A). Proof. This follows easily from (1.5.3.6), (1.5.3.7), the (G, yl)-module version of (1.5.1.11), and Lemma 1.5.3.8. • If V is ^-finite and we have either B is yl-flat or there is a resolution F —¥ V of V in G,AM such that each term of F is .4-finite projective (e.g., the case G satisfies FPCP, see the next subsection), then the second spectral sequence in 1 can be taken to be that of (G, B)-modules. This is proved similarly to Proposition 1.5.2.4. (1.1.15) Let R, G and A be as in Theorem 1.1.12. Let / be a G-ideal of A. Then Tj = limHom,i(.A// n , ?) is a left exact functor from C,A^ to itself. By Theorem 1.1.12, F o ( i ? T / ) agrees with the local cohomology functor Hj, where F : G,/IM —> A^ is the forgetful functor. Hence, for M € G,/IM, the local cohomology H}(M) is equipped with a (G, ^4)-module structure. The canonical map limExt' /1 (j4//",M) -¥ H)(M) is an isomorphism of (G, A)modules.
1.2
Equivariant modules of a split torus
(1.2.1) Let R be a commutative ring, n > 1, and T the n-fold split torus GU, over R. A T-module and an X(T)-graded fl-module (i.e., a Z"-graded .R-module) are the same thing (1.4.3). For two T-modules V and W, the tensor product V <8> W is again a T-module, and it is graded. It is easy to see that the grading is given by (V®W)X=
0
VM®WV
1. Homological aspects of (G, A)-modules
137
That is, the grading of the tensor product V ® W is given by the total grading. Let A be a T-module, and an it-algebra. To say that A is a T-algebra is the same as to say that A ® A —> A preserves the grading. Hence, A is a T-algebra if and only if A is an X(T)-graded i?-algebra. Similarly, a (T, yl)-module is nothing but a graded j4-module.
1.3
FPCP groups and IFP groups
(1.3.1) Let R be a noetherian commutative ring. Let G = Spec H be a flat affine .R-group scheme, and A a commutative G-algebra. Definition 1.3.2 We say that G satisfies PCP (resp. FPCP) if the coordinate Hopf algebra H satisfies PCP (resp. FPCP) (see Definition 1.3.11.4), as an it-coalgebra. By Lemma 1.3.11.5, we have that if G satisfies FPCP, then G satisfies PCP, and the converse is true if gl.dim R < 2. Lemma 1.3.3 // the R-group scheme G satisfies FPCP (resp. PCP), then for any A-finite (G,A)-module (resp. any (G, A)-module) M, there exists some R-projective P € cM/ (resp. GM) such that there is a surjective (G, A)-linear map A®P^M. Proof. Let Mo be an it-finite G-submodule (resp. G-submodule) of M which generates M as an yl-module. As G satisfies FPCP (resp. PCP), there exists some surjective G-linear map P —> MQ, with P a G-module which is it-finite projective (resp. it-projective). Now, the composite map
is surjective.
•
Proposition 1.3.4 Let G be an affine flat R-group scheme which satisfies FPCP. Assume that R and A are noetherian, and A is R-flat. Then for a (G,A)-module I, the following are equivalent. 1 I is G-injective, and for any R-finite G-module V, any (G,A)-submodule N of A®V, and any (G,A)-linear map f : N —> I, f is extended to a (G, A)-Unear map A
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II. Equivariant Modules
Proof. 1=>2 Let M be an .4-finite (G, A)-modu\e. Then there exists an exact sequence of (G, ^4)-modules 0 -> N -> A ® V -> M -* 0 such that V is an ^-finite G-module.
Ext^iA
By Proposition 1.1.14, we have
®V,I) = ExtG (V,/) = 0. As the map EomG,A{A ®V,I) ->
HomG,A(N, /) is surjective by assumption, it follows that ExtxG A(M, I) = 0. By Lemma 1.1.9.4 and Corollary 1.5.3.11, we have that / is injective. We show 2=>1. By Corollary 1.5.3.9, we have that / is G-injective. When we set M to be the cokernel of JV -» A®V, then as we have E x t ^ ^ M , /) = 0, the map Homc,A{A ®V,I)-> Homc,/i(A^, /) is surjective. O (1.3.5) Let R be a noetherian commutative ring, and G an affine flat Rgroup scheme. If G satisfies PCP, then by Theorem 1.1.4.9, a left derived functor L(? ®A N) : D-(G,AM)
->
D-(G,AM)
exists. For M,N € G,AM, we have L4(? ®A N){M) = Li(M®A?)(N), and they coincide with Torf (M, AT), simply as ^-modules. Hence, if G satisfies PCP, then Torf(M, N) is equipped with a (G, .4)-module structure in a natural way. If G satisfies FPCP, A is noetherian, and N € G,XM, then there is a derived functor RHomA{?,N)
: D-{G,AMf)
-> D+(C
and for M e G ^ M / , we have /?i Hom(?,Ar )(M) S Ext^M.iV), as (G,X)modules. Definition 1.3.6 Let i? be a commutative ring, and G an affine .R-group scheme. We say that G is IFP if the coordinate ring R[G] of G is IFP as an i?-coalgebra (Definition 1.3.11.1). By definition and Lemma 1.3.11.6, an IFP .R-group scheme satisfies FPCP. Lemma 1.3.7 Let R be a noetherian commutative ring, and G = Spec// an affine IFP R-group scheme of finite type. Then H is R-projective. Proof. Follows immediately from Lemma 1.3.11.3.
•
2. Matijevic-Roberts type theorem
2 2.1
139
Matijevic—Roberts type theorem Stability of various loci
(2.1.1) Throughout this subsection, let it be a noetherian commutative ring, G a flat .R-group scheme of finite type, and X a locally noetherian right G-action. Lemma 2.1.2 Assume that G has integral geometric fibers. If Y is an irreducible (resp. reduced, integral) locally noetherian R-scheme, then so is Y xG. Proof. As Y x G is flat over Y and any fiber is reduced by assumption, Y x G is reduced if Y is, by [110, Theorem 23.9, Corollary]. Now it suffices to consider only the irreducibility. If Y — Spec k with k a field, then the assertion is obvious, as Spec k x G —» Spec k x G is faithfully flat, where k denotes the algebraic closure of k. Now consider the general case. Let Y be irreducible, with its generic point r). As the fiber G,, is irreducible by the previous paragraph, we have its generic point, say, £. It suffices to show that {£} =Y x G. Assume the contrary. Then there is an affine open set U = Spec B of Y x G which does not intersect Gv. We take an irreducible component of U, and let ( be its generic point. As U is flat over Y, we have that £ must be mapped to rj by O the going-down theorem, which contradicts £ ^ Gv. (2.1.3) Let the notation be as in (2.1.1). We denote the action of G on X (resp. the first projection) X x G -> X by a = ax (resp. p = px = Pi)- We define hx '• X x G —> X x G by hx(x, g) := (xg, g). As hx is an isomorphism with hxl(x,g) = (z^"1,5) and px ° hx — ax, we have that ax is also flat of finite type. For a subscheme Y of X, we say that Y is G-stable if the action Y xG —» X ((y,g) ^ yg) factors through Y <-• X, see (1.4.1.5). If Y is a G-stable subscheme of X, then we have a unique action YxG -+Y such that Y ^ X is a G-morphism. Lemma 2.1.4 The union of a family of G-stable open subsets is G-stable. The intersection of finitely many G-stable subschemes is G-stable. Thus, X is a topological space, taking G-stable open sets as its open sets. However, we will not use this (usually too coarse) topology later. For a subscheme Y of X, the image closure of Y x G —> X is denoted
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II. Equivariant Modules
Lemma 2.1.5 If Y is a subscheme of X, then Y* is the smallest G-stable closed subscheme of X which contains Y. In particular, Y is a G-stable closed subscheme of X if and only ifY = Y*. IfY' is also a subscheme of X and Y C Y', then we have Y* C (Y1)*. If G has integral geometric fibers and Y is irreducible (resp. reduced, integral), then so is Y*. Proof. By definition, Y* is a closed subscheme of X. As the composition Y = Yx{e}<-*YxG-iX agrees with the inclusion Y <-> X, Y* contains Y. As G is /?-flat, the image closure of a x 1 : Y x G x G -> X x G is Y* x G. On the other hand, a s o o ( a x l ) = a o ( l y x /xG) : Y x G x G —> X factors through Y*, we have that a : Y* x G -> X also factors through V*, which shows that Y* is G-stable. If Z is a G-stable closed subscheme of X which contains Y, then, as the action Y x G -> X factors through Z, we have Y* C Z. Hence, Y* is the smallest among such Z. Assume Y C Y'. Then Y C {Y')*, and hence Y* C (Y1)*. Finally, assume that G has integral geometric fibers. If Y is irreducible (resp. reduced), then so is Y x G by Lemma 2.1.2. By the definition of Y*, we have that Y* is irreducible (resp. reduced). D (2.1.6) An argument similar to the proof above shows that if U is an open subscheme of X, then the image of U x G —>• X is the smallest G-stable open subscheme of X which contains U (as ax is flat of finite type, it is an open map). Lemma 2.1.7 X has a covering consisting of quasi-compact G-stable open subschemes. Proof. Let (Ui) be a covering consisting of quasi-compact open subschemes of X. Then (a* (£/;)) ' s a covering of the desired type. D If (Vj) is a covering of X consisting of quasi-compact G-stable open subschemes, then for any subscheme Y of X, we have that Y is G-stable if and only if Y n Vt is a G-stable subscheme of V* for any i. (2.1.8) Let the notation be as in (2.1.1). Assume that G has integral geometric fibers. For x € X, we denote the generic point of the G-stable integral closed subscheme {x}* of X by x*. If we have x = x*, then we say that x is a G-stable point of X. Lemma 2.1.9 Let X = XQxG be a principal G-bundle, where Xo is locally noetherian. Then any G-stable open subset of X is of the form V x G, with V an open subset of Xo-
2. Matijevic-Roberts
type theorem
141
Proof. Let U be a G-stable open subset of X. Then V := Pxo(U) is an open subset of X o , as pXo is fiat of finite type. It suffices to show that U = p'1^) = V x G. Assume the contrary. Then there exists some geometric point £ = Specif —> Specii of Specft, and some (x,g) € V(£) x G(£) \ U((). As U -» V is faithfully flat of finite type and /f is algebraically closed, we have that £/(£) —> V(£) is surjective by Hilbert's theorem. Hence, there exists some g' G G(£) such that (x,g') € £/(£)• This shows (x,g) = (x,g')((g')~1g) € £/(£)> which is a contradiction. D Lemma 2.1.10 Lei X and X' be locally noetherian G-actions, and
¥>xlG| X' x G
a
* . X
,P*
\
ax>
X x G
|yxlG X'xG
is a fiber square. As we have ((p x 1G) o hx = hx> ° (
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II. Equivariant Modules
By [43, III.3.2.5-6], the quotient map TT : P -> P/G is faithfully flat. By Theorem 1.2.7.10, 2', we have that n is l.c.i. Hence, the fiber T T " 1 ^ ^ ) ) = G is l.c.i., as desired. Assume moreover that R D Q. Then G is smooth. To verify this, we may assume that R is a field of characteristic zero. In this case, both P = GLn(R) and P/G are smooth varieties, and the field extension k(P)/k(P/G) is separable. This shows that the smooth locus of n is non-empty. As vr is a P-morphism and the only P-stable non-empty open subset of P is P, we have that TT is smooth. Hence, the fiber G is geometrically regular. Lemma 2.1.13 Let G be an R-group scheme flat of finite type. Then G has reduced geometric fibers if and only if G is R-smooth. Proof. Assume that G has reduced geometric fibers. As G is flat of finite type over R, it suffices to show that G has regular geometric fibers. Hence, it suffices to show that G is regular, assuming that R is an algebraically closed field and G is reduced. The regular locus of G is G-stable open, and it is empty or G itself by Lemma 2.1.9. As G is reduced, G satisfies Serre's (RQ) condition, and hence the regular locus of G is non-empty. The opposite direction is trivial. D Corollary 2.1.14 Let G be an R-flat group scheme of finite type. Then G has integral geometric fibers if and only if G is R-smooth with connected geometric fibers. Lemma 2.1.15 Let k be afield, and G a group scheme over k of finite type. Then G is a disjoint union of its irreducible components. In particular, if G is connected, then it is irreducible. Proof. Let K be the perfect field consisting of p r th roots of elements of k for some r. As (G ®k K)Te& is a reduced group scheme over K, it is /^-smooth by an argument similar to the proof of Lemma 2.1.13. Let / be the ideal of H
2. Matijevic-Roberts
type theorem
143
theorem, T T " 1 ^ ) has points in two different irreducible components of Gk>In particular, •n~1(g) has at least two points. This contradicts the fact TT is radical [65, IV.6.15.3.1]. • Lemma 2.1.16 / / the Cohen-Macaulay locus U = U(CM,X) of X is an open subset of X, then U is G-stable. Similarly for the Gorenstein and l.c.i. properties. If, moreover, G —>• Spec R has regular fibers, then we also have a similar result for the regular locus. Proof. If / : X' -» X is a locally Cohen-Macaulay flat morphism, x' 6 X', and x = f(x'), then Gx,x is Cohen-Macaulay if and only if Ox',x' 1S CohenMacaulay. Hence, we have f~x{U) = U(CM,X'). Applying this to the Cohen-Macaulay morphisms ax : X x G —> X and px : X x G —» X, we have a-^U) = U(CU,X x G) = p~x{U). This shows that U is G-stable. Similarly for the other properties. • Lemma 2.1.17 Let Y be a reduced closed subscheme of X. IfG has integral geometric fibers, then the following are equivalent. 1 Y is G-stable. 2 Any irreducible component ofY (i.e., maximal integral closed subscheme of Y) is G-stable. 3 The open set X -Y
is G-stable.
Proof. 1=>2 Let Z be an irreducible component of Y. By Lemma 2.1.5, Z* is integral, and we have Z C Z* C Y* = Y. As Z is maximal among integral closed subschemes of Y, we have Z = Z*. 2=>3 Let (Ui) be a quasi-compact G-stable open covering of X. It suffices to show that Ui \ Y is G-stable. As an irreducible component of Ui n Y is nothing but a non-empty intersection of an irreducible component Z of Y with Ui, we may assume that X is quasi-compact, replacing X by C/j. As Y is noetherian this case, we can express Y as a finite union Y\ U • • • U YT of its irreducible components. As we have X — Y = f\(X - Yi), it suffices to show that each X — Yi is G-stable, and we may and shall assume that Y is integral. We set U := X - Y. What we want to prove is U' := UG = U. Assume the contrary. Then there exists some geometric point 77: Spec K -» Spec R such that U(rj) -4 U'{rj) is not surjective. On the other hand, the action U x G —> U' is surjective of finite type. By Hilbert's theorem, we have that U(rj) x G(rj) -> U'(n) is surjective. Hence, there exist some u G U(i]) and g e G(v) such that ug e U'{n) \ U{rj). As we have Y(r)) = X{rj) \ U(TJ), we have ug £ Y(rj). As Y is G-stable, this shows u — (ug)g~1 £ Y{q), and we have u € Y(rj) n U(rj) = 0. This is a contradiction.
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II. Equivariant Modules
3=>1 An argument similar to above shows that the image of Y x G —> X does not intersect U :— X\Y. Hence, Y* is contained in Y set-theoretically. D As Y* is reduced, we have Y = Y*. Lemma 2.1.18 Assume that G has integral geometricfibers,and the CohenMacaulay {resp. Gorenstein, Lei., regular) locus U of X is an open subset of X. If any G-stable point of X is contained in U, then we have U = X. Proof. Assume the contrary. Consider Y := X — U with the reduced structure. Then Y is a non-empty G-stable closed subscheme of X by Lemma 2.1.17. Let r? be the generic point of an irreducible component of Y, which does exist. Then 77 is a G-stable point of Y by Lemma 2.1.17, D but it is not contained U, which contradicts the assumption. The assumption that U is open in the lemma is not necessary, in fact. This will be proved in (2.4).
2.2 Universal density of hyperalgebras (2.2.1) Let R be a noetherian commutative ring, and G = Spec H be an affine flat .R-group scheme of finite type over R. Definition 2.2.2 Let / := Kere# be the kernel of the counit map. We define the hyperalgebra of G, denoted by Hy G, to be \im(H/In)*. Note that U = Hy G is an i?-subalgebra of the dual algebra H* of H. In fact, we have 1H. = e <E (H/I)' C U, and (H/In)*{H/Im)* C {H/In+m-l)\ Definition 2.2.3 We say that G is infinitesimally fiat if H/In is infinite i?-projective for any n > 1. This is equivalent to saying that 7 " / / n + 1 is .R-finite .R-projective for n > 1. As we are assuming that G is of finite type over R, this is also equivalent to saying that H is normally flat along / . (2.2.4) Let R and G = Spec// be be as in (2.2.1). Set / := Kere H and U := HyG. We assume that G is infinitesimally flat. Then as we have (////")*(/" • H + H -In) = 0 and {H/In ® # / / " ) * S* ( # / / " ) '
•
2. Matijevic-Roberts (2.2.6)
type theorem
145
Let R, G, H, I and U be as in (2.2.4). As we have U = lim(J7/J n )*
by definition, U* = lim H/In.
In other words, U* is the 7-adic completion
oiH. Lemma 2.2.7 Let the notation be as above. Assume that R = k is a field, and G is connected. Then the canonical inclusion U —^ H* is universally dense. Proof. It suffices to show that the 7-adic completion H —> U* is injective, in other words, Di>o7l = 0. By assumption and Lemma 2.1.15, G is irreducible. By Corollary 2.1.11, G is Cohen-Macaulay, and hence H has a unique associated prime by [110, Theorem 17.3]. This shows that any zero-divisor of H is nilpotent. If b is a non-zero element of fli>o I\ then by Krull's intersection theorem [110, Theorem 8.9], there exists some a € H such that a - 1 e / and ab = 0. As a is a zero-divisor, a is nilpotent. As / is a prime ideal, a G /. As a - 1 £ / , it follows that 1 = a - (a — 1) G / ^ H, which is a contradiction. • Theorem 2.2.8 Let G = SpecH be an R-flat, infinitesimally fiat affine R-group scheme of finite type with connected fibers. Then U = Hy(G) is a generalized hyperalgebra of H. In particular, U —> H* is universally dense. Moreover, the coordinate ring H is R-projective. Proof. As the fiber G(p) is connected, we have that the map H(p) -> Hom K(p) (Hy(G(p)),«(p)) = hm(////")(p) is injective by Lemma 2.2.7. As this map factors through
we have that U —> H* is universally dense by Lemma 1.3.8.4. Now it is easy to see that U is a generalized hyperalgebra of H. Next, we show that H is i?-projective. By assumption, H/In is Rprojective for any n. Since the projective system (H/In) consists only of surjective maps, it satisfies the Mittag-Leffler condition. Hence, by Proposition 1.2.2.15, U* = lim H/In is also Mittag-Leffler. As U -> H* is universally dense, the canonical map H —> U* is .R-pure by Lemma 1.3.8.5. Hence, by Corollary 1.2.2.8, H is also a Mittag-Leffler /?-module. As H is of finite type over R, it is countably generated as an .R-module. By Proposition 1.2.2.6, H is .R-projective. • Corollary 2.2.9 If G is R-smooth with connected fibers, then U -> H* is universally dense, and H is R-projective.
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2.3 A generalization to equivariant sheaves (2.3.1) Let R be a noetherian commutative ring, and G = Spec// an affine flat /?-group scheme of finite type. We denote the category of G-schemes (i.e., an /^-scheme with a right Gaction) and G-morphisms by G-sch. For U £ G-sch, we define cov U to be the set of families of G-morphisms {ipi :{/;—»[/} with the same codomain U, such that each fi is flat locally of finite presentation, and that Ui
2. Matijevic-Roberts type theorem
147
(2.3.4) Let R, G and X be as in (2.3.3). Let A be a G-faisceau of Galgebras over X. We say that M is a (G, ,4)-module (resp. (G,A)-module presheaf) if M is both a G-faisceau (resp. G-presheaf) of G-modules and a G-faisceau (resp. G-presheaf) of ,4-modules (.A is viewed as a G-faisceau of ii-algebras), and for any (/ : Y -> X) € TG/X, the G-module A{Y)module M(Y) is a (G,.4(y))-module. By definition, if M is a (G,A)module presheaf, then for any Y' —> Y, the canonical map M{Y) —> JA(Y') is(G,A(Y))-lmea.r. The notion of aff-(G, fi)-module (presheaf) is denned similarly, for an aff-G-faisceau of G-algebras B, replacing TG/X by (TG/X)-afi. (2.3.5) We define Ox by OX(Y) = T(Y,OY) for Y € TG/X. Note that Ox is a G-faisceau of G-algebras. By restriction, Ox is also an aff-G-faisceau of G-algebras. Using the same symbol Ox is an abuse of notation. We say that Ai is a quasi-coherent Ox-module if M. is an Ox-module (where Ox is viewed as a G-faisceau of commutative rings), and for any morphism / : Y -> Z in (TG/X)a, the canonical map OX(Y) <8>ox(z) M(Z) -» M(Y) is an isomorphism. Assume that X is locally noetherian. M is a coherent Ox-module if M is quasi-coherent, and if Y € (TG/X)a then M(Y) is a finitely generated Ox(l^)-module. In the sequel, when we mention something on coherent Ox-modules, we always assume that X is locally noetherian. A (G, Ox)-module is called quasi-coherent (resp. coherent), if it is quasicoherent as an Ox-module. Similarly, quasi-coherent and coherent (for X locally noetherian) aff-GOx-modules and aff-(G, Ox)-modules are defined. (2.3.6) Let R, G and X be as in (2.3.3), and let A be a G-faisceau of G-algebras on X. Let M be a (G, .4)-module presheaf. Then the sheafification a(M) of M is a (G, .A)-module (faisceau) in a natural way. For (G, ,4)-modules M and TV, the tensor product M <S>A -A/" is defined to be the sheafification of the (G, Ox)-module presheaf given by (M <8u For a G-morphism / : Y —»• X and a (G, Oy)-module M, f*M has a (G, Ox)-module structure in a natural way. If A/" is a (G, Ox)-module, then f*M := OY ® / - I O X / ~ W i s a ( G . Oy)-module. Similar definitions are also made for Ox-modules, aff-(G, Ox)-modules, and aff-G-Ox-modules. (2.3.7) Let R and G be as in (2.3.3). Let us denote by \i : G x G -> G the product of G. For X S G-sch, we denote by ax,Px • X x G -> X the action of G and the first projection, respectively. The following definition is due to Mumford [115].
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Definition 2.3.8 A G-linearized Ox-module is a pair (M,<j>), where M is an Ox-module (with respect to the Zariski topology), and cj>: a*xM —» pJM is an OxxG-isomorphism such that the following condition is satisfied: The isomorphism [a x o (ax x 1G)]*M = [ax ° (lx x /xG)]*M
(1
* x " c ) ^ fa ° (lx x
MG)]*M
agrees with the composite [ax o (ax x 1G)]*M
(axXlc) 0
' ) fa o (a* x 1G)]*M = [ax op12]'M
^
fa o pn]*M = fa o (lx x /iG)]*M. By Lemma 1.1.6, if X = Spec .A is affine, then a quasi-coherent Glinearized Ox-module is the same as a (G, >l)-module. It is easy to see that the category of G-linearized Ox-modules is abelian, and the forgetful functor to the category xM is exact. Moreover, direct image functors and inverse image functors are enriched to the G-linearized versions, which are compatible with forgetful functors. (2.3.9) Let R, G and X be as in (2.3.3). If M = {M,<j>) is a quasicoherent G-linearized O x -module and (/ : Y -» X) £ (TG/X)a, then f*M = (f*M,(f x IG)*4>) is a quasi-coherent G-linearized Oy-module. Hence, letting W(M){f) := T{Y,fM) for (/ : Y -t X) € TG/X, we have that W{M) is a (G, Ox)-module. By (1.2.3), it is easy to see that W(M) is quasi-coherent. If X is locally noetherian, then W preserves the coherence. (2.3.10) Let JR, G and X be as in (2.3.3). A (G,Ox)-module is an aff-(G, Ox)-module by restriction. Note that this restriction preserves the quasi-coherence, and coherence. (2.3.11) Let R, G and X be as in (2.3.3). Let us assume that M is a quasi-coherent aff-(G, 0x)-module. We want to construct a G-linearized quasi-coherent Ox-module Q{M), so that Q is a quasi-inverse of the functor W in (2.3.9). First, we construct Q(M), assuming that X is quasi-compact. Take a finite afiine open covering (Vj,..., Vr) oiX, and set Y := (Lit Vi) x G. Note that Y € (TG/X)-aft. We denote the canonical map Y -> X by TT. Note that n is faithfully flat, and Y xx Y is also affine. We denote the ith projection Y xx Y —> Y by TTJ. Note that 7T* is a morphism of (TG/X)-aff for i = 1,2. Let r : Y xx Y --* Y xx Y be the isomorphism of (TG/X)-aff given by (2/1,2/2) >-> (2/2,2/1)- We denote the composite map (*i)*{M(Y)) s A*(y xx V) ^ >
A^(r xx 7) s (*2Y(M(Y))
by 1/). Then it is easy to verify that the cocycle condition TT^V' = '^Z2^)O'K7.\{I) holds, where n^ : Y xx Y xx Y -> Y xx Y is given by (2/1,2/2,2/3) ^
2. Matijevic-Roberts
type theorem
149
(yi, 2/j). By Proposition 1.2.3.8, we have a quasi-coherent Ox-module Q(M) corresponding to the descent datum (M(Y),ip), and we have M(Y) = n*(Q(M))- (as Y is affine, we identify M(Y) with its associated Zariski sheaf). Now Q(M), as an Ox-module (in the usual Zariski topology), has been defined. Next, we give a G-linearization to Q(M). By definition, Q(M) is Ker(hi - h2), where hi and h2 are as in (1.2.3.7). As xp is G-linear, both h\ and h2 are compatible with G-linearizations. Hence, Q(M) has a natural G-linearization (J>M such that -K*4>M = 4>i where <j> is the G-linearization of M(Y). Thus, we have a G-linearized quasi-coherent Ox-module (Q(M), <J>M) associated to the aff-(G, Ox)-module M, if X is quasi-compact. If / : X' —> X is a morphism of TQ/X, both X and X' are quasi-compact, and / is flat, then (f'(Q(M)),(f x 1G)*
Ox-modules.
The equivalence 1=>2 is given by the restriction. The equivalence 2=>3 is given by Q. The equivalence 3=>1 is given by W. If X is locally noetherian, then the coherence is preserved by these equivalences. The argument above is still valid (and much easier) for quasi-coherent Ox-modules without G-action. Hence, Proposition 2.3.13 Let X e G-sch. Then the following categories are abelian, and equivalent to one another. 1 The category of quasi-coherent Ox-module G-faisceaux. 2 The category of quasi-coherent Ox-module aK-G-faisceaux. 3 The category of quasi-coherent Ox-modules in the Zariski topology. The equivalence 1=^2 is given by the restriction. The equivalence 2=>3 is given by Q. The equivalence 3=3-1 is given by W. If X is locally noetherian, then the coherence is preserved by these equivalences.
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(2.3.14) If A is a commutative noetherian ring, B an ^4-flat commutative algebra, M an >l-finite module, and N an .4-module, then the canonical map Ext'^M, N)®AB-> Ext'B(M ®B,N®B) is an isomorphism. Let R, G and X be as in (2.3.3). Assume that X is locally noetherian. If M is a coherent (G, Ox)- module, and Af a quasi-coherent (G, C?x)-module, then we define a quasi-coherent aff-(G, Ox)-module ExtpA. (Ai. AO by Ext*,x(M,JNOOO := Ext' Ox(y) (A4(r), for Y G (TG/X)-aff. Hence, Ext^ x (X > 7V) has a quasi-coherent {G,OX)module structure. If G satisfies PCP and M and Af are quasi-coherent (G, C?x)-modules, then we can also define a quasi-coherent (G,Ox)-module structure of the sheaf Tgr?x(M,Af) in a similar way. Proposition 2.3.15 Let R be a noetherian ring, G an affine flat R-group scheme of finite type, X a locally noetherian right G-action, M. a coherent {G, Ox)-module, and r > 0. Then the set Ur(M) defined by UT{M) :={x£X\Mx^
OTXx (as 0 x,x-modules)}
is a G-stable open subset of X. If X is Cohen-Macaulay, then the maximal Cohen-Macaulay locus of M is also G-stable. Proof. It is well-known that UT(M) is an open set of X. Let A —» B be a flat local homomorphism of noetherian local rings, and M a finite .4-module. Then M = AT if and only if M ®A B = BT (easy). Hence, as we have a G-linearization ax{M) = px{M), we have that
px\Ur{M)\
and Ur(M) is G-stable. The last assertion is proved similarly, as we know that G is CohenMacaulay over R. • By the proposition, we have that the complement of the support of a coherent (G, Ox)-module is G-stable open. More precisely, we have: Lemma 2.3.16 Let R, G, X and M be as in Proposition 2.3.15. Then the annihilator axtnM of M is a G-ideal sheaf of X, namely, a (G,OX)submodule of Ox. Proof. As aim M is nothing but the kernel of the (G, C?x)-linear map
the assertion is obvious.
•
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151
(2.3.17) Let R be noetherian, G be i?-smooth with connected geometric fibers, and A be a commutative noetherian G-algebra. We say that a Gideal / is a G-maximal ideal if it is maximal among G-ideals with respect to the incidence relation. It is easy to see that a G-maximal ideal is a prime ideal. The intersection of all G-maximal ideals of A is called the G-radical of A. Note that the G-radical of A is a G-ideal by Corollary 2.2.9 and Exercise 1.3.10.6. Lemma 2.3.18 Let G and A be as above, M an A-finite (G, A)-module, and J an ideal of A contained in the G-radical of A. If M = JM, then we have M — 0. Proof. We may assume that J is the G-radical of A. Then any Gmaximal ideal is not contained in supp M by Nakayama's lemma. If M ^ 0, then suppM is a non-empty G-stable closed subset of Spec .4, and by Lemma 2.1.17 there is some G-prime ideal which supports M. So there is a maximal one among such, and it is a G-maximal ideal, which is a contradiction. D (2.3.19) Let X be a locally noetherian G-scheme. We denote the category of aff-(G, 0x)-modules by G]xM, and the full subcategory of quasi-coherent (G, Ox)-modules by Qco(G, X). Note that Qco(G, X) is an abelian category in a natural way, so that T(Y, ?) is exact for each Y € {Tc/X)-aS. If Y € (TG/X)-aS is faithfully flat over X, then T{Y, ?) is also faithful. The category Qco(G,A") satisfies (AB5). For each Y € (TG/A>aff, r(V, ?) preserves arbitrary inductive limits. Hence, for each quasi-compact Y € TQ/X, the functor F(Y, ?) preserves filtered inductive limits. Obviously, we have a forgetful functor F% : Qco(G, X) -» Qco(X), which is faithful exact and preserves inductive limits. Let X' denote the jR-scheme X with the trivial G-action, and M £ Qco(X). As M. together with the trivial G-linearization lies in Qco(G, X'), and px • X' x G —> X' is a G-morphism, we have that p*xM. is a quasicoherent (G, (9x'xG)-module. Hence, (ax),pxM G Qco(G,X), as ax : X' x G -*• X is an affine G-morphism. Taking uM • M -> (ax),axM
M
'*M)
{ax),VxM
as the unit of adjunction and (ax).PxAf
-> (ax),e,e*pxM = N
as the counit of adjunction (where e : Spec R -> G is the unit element of G), we have that (ax).p*x is right adjoint to the forgetful functor Fx. It follows
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II. Equivasiant Modules
that (ax)*Px preserves injectives. As u ^ is a monomorphism, it follows that the set of all objects of the form (ax)*pxZ with I an O^-irijective quasi-coherent Ox-module is an injective cogenerator of Qco(G, X), see [69, Theorem II.7.18]. Hence, we have Lemma 2.3.20 The category Qco(G, X) has enough injectives. We also have Lemma 2.3.21 Let J be an injective object in Qco(G, X). If U is a Gstable open subset of X, then I\u is injective in Qco(G, U). Proof. We may assume that J = {ax)*pxT, with J an C?*-injective quasicoherent Ox-module. As I\v is injective in yM, the assertion follows easily.
• (2.3.22) Let / : Y —> X be a quasi-compact G-morphism of locally noetherian G-schemes. By (2.3.6), we have canonical functors
and and these functors are obviously compatible with the forgetful functors:
F§°f? = / . o ^ and F$orc = /• o F§. Proposition 2.3.23 Let f : Y —> X be a quasi-compact G-morphism of locally noetherian G-schemes. If J is an injective object ofQco(G,Y), then we have R{f.J = 0 fori>0. Proof. As the question is local on X and X is covered by quasi-compact G-stable open subschemes, we may assume that both X and Y are quasicompact. We may and shall assume that J = (ay).pyl, with I an OYinjective quasi-coherent Oy-module. Then we have
fff,{aY).rM £ (<*).#(/ x 1G).PYI = {ax).Px&fa = 0. The first isomorphism is by the Leray spectral sequence (note that ay and ax are affine and / o aY = ax ° (/ x 1Q)). The second one comes from the flat base change of higher direct images, see [64, (1.4.15)]. Thus, we have Kf.J = 0 for i > 0, as desired. • By the proposition, we have
= R\f. for any M € Qco(G, V). Hence, R'ftM structure of Rlf?M in a natural way.
is endowed with a (G, Ox)-module
Corollary 2.3.24 Let f be as in the proposition. Then R*f^ preserves filtered inductive limits.
2. Matijevic-Roberts type theorem
2.4
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Matijevic—Roberts type theorem
(2.4.1) Throughout this subsection, let R be a noetherian commutative ring, and G an affine smooth /?-group scheme with connected geometric fibers. Let X be a locally noetherian G-action. In this subsection, we prove the following, which is the main theorem in this chapter. Theorem 2.4.2 Let x e X. Then the following hold: 1 / / Ox,x' is o, regular local ring {resp. complete intersection), then so is Ox,x- In particular, if X is regular (resp. l.c.i.) at its all stable points, then X is regular (resp. l.c.i.). 2 LetM be a coherent (G, Ox)-module. IfMx> is Gorenstein (resp. CohenMacaulay, free), then so is Mx. In particular, if for any stable point x, Mx is Gorenstein (resp. Cohen-Macaulay, free), then M. is Gorenstein (resp. Cohen-Macaulay, locally free). Let us consider the case that X = Spec A in the theorem is affine. For an ideal I of A, the sum of all G-ideals contained in / is again a G-ideal, and is the largest G-ideal contained in /, which we denote by /*. Clearly, we have Spec A/1* = (Spec A/1)*. Hence, if / is a prime ideal, then so is /*. If / corresponds to x E X, then /* corresponds to x*. As a split torus G = GJJ, satisfies the assumption of the theorem (one can consider that R = Z), we have the following by (1.2). Corollary 2.4.3 Let A be a Zn-graded noetherian commutative ring, and P £ Spec A. Then the following hold: 1 If Ap' is regular (resp. l.c.i.), then so is Ap. In particular, if the localization of A at any graded prime ideal is regular (resp. l.c.i.), then A is regular (resp. l.c.i.). 2 Let M be an A-finite graded A-module. If Mp- is Gorenstein (resp. Cohen-Macaulay, free), then so is M. In particular, if the localization of M at any graded prime is Gorenstein (resp. Cohen-Macaulay, free), then M is Gorenstein (resp. Cohen-Macaulay, projective). Note that Corollary 2.4.3 was originally conjectured by Nagata [119] (the case n — 1 for the Cohen-Macaulay property), and proved in [85, 109, 7, 108, 60, 34, 16], except for the l.c.i. property in 1 for the case n > 2. So we could say that this has been well-known. The theorem generalizes Corollary 2.4.3 to more general G. Now we start proving the theorem. Lemma 2.4.4 Let Y be a subvariety of X, and r\ its generic point. the following hold:
Then
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1 Oy,r) is a regular local ring. 2 Let M be a coherent (G,Oy)-module. Then Mn is Oy^-free. If M is a coherent (G,OX)-module, then Ext'Ox (MV,NV) is also Oy,v-free for i > 0. Proof. We show 1. Let tp : Y x G -» Y* be the action (y,g) >-» yg. Then we have Flat() ^ 0, and hence we have F =£ 0 and I J E F . It follows that the composite morphism i>: SpecK(r?) x G -¥ Y x G ^ Y* is flat, because the first morphism is flat, and the second morphism is flat on the image of the first. As G is /?-smooth, we have that Spec ^(77) x G is /t(?7)-smooth, and in particular, it is a regular scheme. As the unit element Spec/c(77) x {e} is mapped to r\ by tp, we have that Oy,v is also regular local. We show 2. As Y* is integral, the free locus U of M is non-empty, and is a G-stable open subset of Y* (Proposition 2.3.15). Assume that UC\Y = 0. Then Y* \ U with the reduced structure is a G-stable closed subscheme of X containing Y (Lemma 2.1.17). As U ^ 0, this contradicts the minimality of Y*. Hence, U D Y ± 0. As r) E U D Y, Mn is Oy^-free. We prove the last statement. As Ext'Cx(M,Af) is a coherent (G,Oy-)-module for i > 0,
is Cfy.|r;-free, by the first part.
D
Let X, x and M be as in the theorem, and set Y := {x}, B := Ox,x, B/P = OY',X, and M := Mx. Lemma 2.4.5 The following hold: 1 B/P is a regular local ring. 2 B is normally flat along the prime ideal P. 3 Extj,(.B/P, B/P) is B/P-free fori>0. 4 Bp is regular local (resp. a complete intersection), by assumption of the theorem. 5 M is normally flat along P. 6 Ext^(B/P, M) is B/P-free for i > 0.
2. Matijevic-Roberts type theorem
155
7 Mp is Cohen-Macaulay (resp. Gorenstein, free) by assumption of the theorem. Proof. Note that 4,7 are assumptions of the theorem. Note that Mp = AiX' and Bp = Ox,x'- The assertion 1 is obvious by Lemma 2.4.4. We denote the denning ideal sheaf of Y* by V. Note that P i s a (G, Ox)-
submodule of Ox, and VnM/Vn+1M is a coherent (G, Oy)-moA\x\e, for n > 0. Hence, {VnM/Vn+1M)x = PnM/Pn+1M is B/P-free, by Lemma 2.4.4. That is, 5 holds. In particular, considering the case M = Ox, we have 2. Similarly, 3,6 follow from Lemma 2.4.4 easily. D Now statement 2 of the theorem is obvious by Lemma 1.2.13.5. On the other hand, statement 1 in the theorem is obvious by Lemma 1.2.13.9. So the proof of the theorem is now complete. •
Chapter III Highest Weight Theory over an Arbitrary Base So-called highest weight theory plays an important role not only in the theory of algebraic groups, but also in various areas of representation theory. This chapter is devoted to reviewing and studying the theory of good filtrations, Schur algebras and the theory of quasi-hereditary algebras over an arbitrary base ring, from the viewpoint of comodules. As the central purpose is to reconstruct S. Donkin's Schur algebra over an arbitrary base, we will not go into the theory of highest weight category originated by ClineParshall-Scott [37], but give a more concrete treatment using coalgebras. For highest weight theory over an arbitrary base from a different approach, see [48, 51, 145].
1
Highest weight theory over a field
This section is devoted to reviewing the highest weight theory over a field. In view of the later characteristic-free treatment, simple comodules, which depend on characteristic, do not appear in the first definition.
1.1
Weak split highest weight coalgebras
(1.1.1) For an ordered set P and x,y € P, we use the interval notation such as [x,y] := {z £ P \ x < z < y} and (—oo, x) := {z G P \ z < x}. A subset Q of an ordered set P is called a poset ideal of P if q G Q, p G P and p < q together imply p G Q. For p G P , the intervals (—oo,p] and (—oo,p) are poset ideals of P. For a finite totally ordered set C, we define rank C := #C— 1. In general, we define rank P :— sup rank C CCP
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III. Highest Weight Theory
for any ordered set P, where C runs through all finite totally ordered subsets of P. Thus, rankP is an integer greater than or equal to — 1, or oo. For p € P, we define htp := rank(—oo,p],
and
cohtp := rank[p, oo).
Lemma 1.1.2 LetX+ be an ordered set. Then the following are equivalent. 1 There exists an injective order-preserving map f : X+ —> N. 2 There exists an injective order-preserving map g : X+ —> N such that the image g(X+) is a poset ideal of N. 2' Either #X+ g:X+->N.
< oo or there exists some order-preserving bijective map
3 X+ is countable, and #(—oo, A] < oo for any A £ X+. Proof. (By Y. Yoshinobu) 1=>2 Define g{x) := # ( / ( X + ) f~l [l,f(x)]). 2 => 2' is obvious, as any infinite poset ideal of N is N itself. 2' => 3 We may assume that X+ is infinite. In this case, X+ is obviously countable. As we have (-oo,A] C g^iihgW]), it follows that # ( - o o , A] < oo. We show 3=>1. We set X+ := {x € X+ | h t z = n} (n e No), and y(n) := (J"=o ^t^- As each X* is countable, we can label Xn
=
\ 3 ; n l ) 3-n2i 3-n3> • • • } •
As we have (Jn>o ^ ( n ) = X+ by assumption, it suffices to construct an injective order-preserving map gn : Y(n) —> N such that the restriction of gn to y(z) (i < n) is ft. We define such a gn inductively on n, so that the image of gn does not contain a multiple of 2" +1 . So assume that g{ (i < n) are already defined so that the image of g{ does not contain any multiple of 2 i+1 . We define (Sn)|y(n-i) = gn-i- We define gn(xnj) (j = 1,2,..., #X+) inductively on j . Assuming that gn{xni) is defined for / < j , we define Sn(Znj) to be the minimum natural number which is larger than gn(xni) (I < j) and gn-i(y) {y < xnj) and is divisible by 2" but not divisible by 2n+1 Now gn is defined, and it is easy to see that gn is an extension of & (i < n), injective and order-preserving, and the image of gn does not contain • any multiple of 2 n + 1 , as desired. (1.1.3)
Let A; be a field.
is a split Definition 1.1.4 We say that (C,X+, A, V) = (C,X£,AC,VC) highest weight coalgebra (resp. weak split highest weight coalgebra) over k if 0 C is a fc-coalgebra.
1. Highest weight theory over a field
159
1 X+ is an ordered set which satisfies the conditions in Lemma 1.1.2. 2 A = (AcW)\ex+ byX+.
is a family of finite dimensional C-comodules indexed
2* V = (Vc(A)) Aex+ is a family of finite dimensional C-comodules indexed byX+. 3 For any finite dimensional C-comodule V and any A € X+, if
for any /x > A, then Ext j,(A c (A), 10 = 0 for i = l,2 (resp. i = 1). 3* For any finite dimensional C-comodule V and any A £ X+, if
for any /J, > A, then Extj,(V,Vc(A)) = O for i = 1,2 (resp. i = 1). 4 If V is a finite dimensional C-comodule and Hom c (Ac(A), V) = 0 for any A e X+, then V = 0. 4* If V is a finite dimensional C-comodule and Hom c (V, Vc(A)) = 0 for any A G X+, then V = 0. 5 For A,/x 6 X+, Homc(Ac(A), V C ( M ) ) equals 0 if A ^ /x, and is onedimensional as a fc-space if A = /x. An element of X+ is called a dominant weight. The C-comodule A C(A) is called the Weyl module of highest weight A, and Vc(A) is called the induced module of highest weight A. (1.1.5) Let k be a field, and (C, X + , A , V ) a weak split highest weight coalgebra over A;. L e m m a 1.1.6 We have
for A, /i £ X+. If, moreover, (C, X + , A, V) is a split highest weight coalgebra, then we have Extc(A c (A), V C ( M ) ) = 0.
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III. Highest Weight Theory
Proof, We have either /x •£ A or /x ^ A. Assume that /x ^ A. Then as we have v ^ A for any v > /x, we have the desired formula by 5 and 3 * in Definition 1.1.4. If tx ^ A, then by 5 and 3, we have the result. • For a poset ideal IT of X+, we denote by M C (TT) the full subcategory of M consisting of C-comodules V such that Hom c (Ac0x), V) = 0 for any \i £ X+ with /x ^ 7T. c
Lemma 1.1.7 Let TT and TT' be poset ideals of X+. Then the following hold: 1 M c (0) = 0. 2 If IT C TT', then MC(TT) C
MC(TT').
3 M C (TT) is very thick (i.e., closed under subquotients and extensions) and closed under inductive limits in Mc. Proof. 1 and 2 are obvious. We prove 3. It is clear that M C (TT) is closed under extensions and subcomodules. Let Vi C V, with V" € M C (TT). Then Vi 6 M C (TT). AS for any A ^ TT, fi > A implies /i ^ TT, we have that Extc(A c (A),Vi) = 0 by Definition 1.1.4, 3. By the obvious long exact sequence argument on Ext, we have Homc(Ac(A), V/Vx) = 0, and hence V/Vi € M c (7r). This shows that M C (TT) is closed under quotients. Next, we show that M C (TT) is closed under inductive limits. Let (Vi) be an inductive system in MC(TT). Then for A ^ n, we have
Homc(Ac(A), 0 Vi) ^ 0 Hom c(Ac (A), Vi) = 0 iei
iei
by Lemma 1.1.9.3. Hence, ©j Vi 6 MC(TT). c
AS
lim V{ is a quotient of © { V;,
C
we have limVi € M (7r), and M (TT) is closed under inductive limits.
D
Lemma 1.1.8 Let A £ X+. Then top(A c (A)) = soc(V c (A)). Let us denote these isomorphic modules by Lc(X). Then Lc(X) is simple, and End c (£c(A)) = k. The set {LC(X) \ A £ X+} is a complete set of representatives of the isomorphism classes of simple objects ofM.c. Proof. As Hom c (A c (A), VC (A)) S k ^ 0, we have AC (A) ^ 0 and VC (A) ^ 0. Hence, top(Ac(A)) ^ 0 and soc(Vc(A)) ^ 0, because both of them are finite dimensional. Let L be a simple object which is a direct summand of the semisimple object top(Ac(A)). Then we have
Homc(L, Vc(n)) C Homc(Ac(A), V C (M)) = 0
1. Highest weight theory over a
field
161
for /x ^ A. As L is simple (in particular, non-zero), we have Hom c (L,soc(V c (A))) S Hom c (L, VC (A)) ^ 0 by Definition 1.1.4, 4*. Hence, when we set top(A c (A)) = Lx®L2@- • -®Lr, with each L< simple, then we have r < dim fc Hom c (top(A c (A)),soc(V c (A))) < dim* Hom c (A c (A), VC (A)) = 1. As r > 0, this shows r = 1, and hence top(Ac(A)) is simple. Similarly, utilizing condition 4 in Definition 1.1.4, we have that soc(Vc(A)) is also simple. As Homc(top(A(j(A)),soc(Vc(A))) = k as above, we must have Lc{\)
:= top(A c (A)) <* soc(V c (A))
and End c LC(A) ^ k. Assume A, (i € X + and A ^ /x. Then we have H o m c ( L c ( A ) , M / i ) ) C Hom c (A c (A), V c (/i)) = 0. This shows that a different A yields a non-isomorphic Lc(A). Lastly, assume that L is a simple object of M c . As L ^ 0, there exists some A € X + such that Home (Ac (A), L) ^ 0. Then clearly L is a homo• morphic image of Lc(A) = top(A c (A)), and L = Lc(X)L e m m a 1.1.9 Let it be a poset ideal of X+, and V a finite dimensional C-comodule. Then V € Mc(7r) if and only if Homc(^, Vc(/x)) = 0 for any fj, € X+ with fj. ^ 7r. Proof. As in (1.1.9.2), we denote the full subcategory of M C (TT) consisting of finite dimensional C-comodules by M C (TT)/. For \i € X+, we denote the full subcategory of M c consisting of finite dimensional C-comodules V such that Homc(K, VC(M)) = 0 for n g ir by C. By Lemma 1.1.7, we have that c M C (TT)/ is closed under subquotients and extensions in M . On the other hand, C is also closed under subquotients and extensions in M c . Indeed, C is obviously closed under extensions and quotients. Let Vi C V, with V,V/Vi 6 C. Then by 3 * of Definition 1.1.4, we have Ext c (V7Vi, V C ( M ) ) = 0 for /X ^ TT, and hence Hom c (Vi, V c (/i)) = 0. Thus, in order to prove the lemma, we may and shall assume that V = Lc(A) is a simple C-comodule. If A ^ TT, then we have HomcCMA), Vc(A)) D End c LC(X) ± 0, c
and hence Lc(A) belongs to neither M (n)f On the other hand, if A £ TT, then
Hom c (A c (A), LC (A)) ^ 0, nor C.
Hom c (L c (A), V c (/i)) C Hom c (A c (A), V C ( M )) = 0 D Hom c (A c (^),Lc(A)) for fi £
TT,
and hence Lc{\) belongs to both
M C (TT)/
and C.
•
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Highest Weight Theory
L e m m a 1.1.10 Let A,/x <E X+. If
or E x t c ( A c ( M ) , LC{X))
^ 0, then we have f i < X. For X e X+, r a d A C ( A )
and V c (A)/soc(V c (A)) belong to M c ( ( - o o , A)). VC(A) belong to M c ( ( - o o , A]).
Moreover, AC (A) and
Proof. Assume fi $ (-oo, A), or equivalently, V <jt X. Then for any v > fi, we have v ^ X and hence Extc(Lc{X), V C ( M ) ) = 0. As the sequence 0 = Hom c (A c (A), V c (/*)) -> Hom(rad(A c (A)),
is exact, rad(A c (A)) 6 M c ( ( - o o , A)) by Lemma 1.1.9. Similarly,
and Vc(A)/soc(Vc(A)) € M c ((—oo, A)) are proved using the dual argument. Next, VC (A) € M c ( ( - o o , A]) is obvious. As we have LC{X) C V C (A), it follows that LC{X) e M c ( ( - o o , A]). As M c ( ( - o o , A]) is closed under D extensions, we have A C (A) € M c ( ( - o o , A]). Lemma 1.1.11 Let n be a poset ideal of X+, and V e Mc. Then V € Mc(ir) if and only if any simple subquotient of V is isomorphic to Lc{X) vtith X € IT. Proof.
1.2 (1.2.1)
Easy.
Weak highest weight theory Let A: be a field.
Proposition 1.2.2 Let C be a k-coalgebra, and assume (C, X+, A, V,L) satisfies the following. a X+ satisfies the equivalent conditions in Lemma 1.1.2. b A = (A C (A)), V = (VC (A)) and L = (LC (A)) are families of finite dimensional C-comodules indexed by X+. c For each A € X+, Lc{X) is simple, and any simple C-comodule is isomorphic to some Lc(X). d top(A c (A)) £ Lc(X) £ soc(Vc(A)).
I. Highest weight theory over a field
163
e Any simple subquotient o/radAc(A) is isomorphic to Lc(fj) for some H < A. e* Any simple subquotient of Vc(A)/soc(Vc(A)) is isomorphic to LC(M) for some n < X. f A,yu € X+ and A ^ n imply Hom c (A c (A), V C ( M ) ) = 0. g A, fj, € X+ and i = 1 (resp. i = 1,2) imply Extj 7 (A c (A) ] V c (/x)) = 0. h / / A 6 X+, then End c LC{X) & k. Then we have that (C,X+,A, V) is a weak split highest weight coalgebra (resp. split highest weight coalgebra), and Lc(X) = top(A c (A)) for any X G X+. Conversely, if (C, X+,A, V) is a weak split highest weight coalgebra (resp. split highest weight coalgebra), then defining Lc(X) := top(Ac(A)), conditions a - h are satisfied by (C,X + , A, V, L). The converse was proved in the last subsection. We prove the first assertion. In the rest of this subsection, C is a fc-coalgebra, and (C, X+, A, V, L) is assumed to satisfy conditions a - f and condition g for i = 1 in the proposition. The axioms 1,2,2* in Definition 1.1.4 are obvious. The proof is divided into several steps. L e m m a 1.2.3 If X, fie X+ and A ^ /i, then LC(X) ^
Proof.
Lc(n).
Hom c (L c (A), Lc(n)) C Hom c (A c (A), V(/x)) = 0.
D
Lemma 1.2.4 // A,/x € X+ and \i ^ A, then Ext^(Ac(A),L c (/x)) = 0 for i = 1. //, moreover, we assume condition g in the proposition for i = 1,2, then we have Ext^(A c (A), Lc(n)) = 0 for i = 1,2. Proo/.
If ;/ < /i, then v ^ X. Hence, we have H o m c ( A c ( A ) , L c H ) C Hom c (A c (A), V c («/)) = 0.
This shows Hom c (A c (A), V c (//)/soc(V c ( M ))) = 0. As Extc(A c (A),V c (M)) = 0, it follows that Ext^(A c (A),L c (/x)) = 0 by an easy long exact sequence argument. If v < fi, then v ^ A, so we have Ext^(A c (A),L c (i/)) = 0. Hence, ExtJ 7 (A c (A), V C ( M ) / S O C ( V C ( M ) ) ) = 0.
Hence, if we also assume Ext^(Ac(A), V C ( M ) ) = 0>
tnen
Ex4(A c (A),L c (/i)) = 0 by an easy long exact sequence argument.
D
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III. Highest Weight Theory
Lemma 1.2.5 If \,n£ X+ and n ^ X, then ExVc(Lc{n), V C (A)) = 0 for i = l. If we also assume that condition g for i = 1,2 in the proposition is satisfied, then we have Ext' c (L c (/x), Vc(A)) = 0 for i = 1,2. Proof. Similar to Lemma 1.2.4.
•
Lemma 1.2.6 Let V be a finite dimensional C-comodule, and IT a poset ideal of X+. We have Homc(Ac(A), V) = 0 for any A ^ ir if and only if any composition factor ofV is isomorphic to Lc(n) for some n £ IT. Proof. The 'if part is obvious, as A $ ?r and /x € 7r imply Hom c (A c (A),
LC(M))
C Hom c (A c (A), V C ( M ) ) = 0.
We prove the 'only if part. We use induction on dim V. As the case V = 0 is obvious, we may assume that V ^ 0. If M is a simple subcomodule of V, then we have M = LC(M)I with M € v- As we have \i ^ A for any A ^ 7T, we have Extp(Ac(A),M) = 0 by Lemma 1.2.4. Hence, we have Homc(Ac(A), V/M) = 0. By induction assumption, we are done. • Lemma 1.2.7 The equivalent conditions in Lemma 1.2.6 are also equivalent to the condition: Homc(Vr, Vc(A)) = 0 for A ^ TT. Proof. Proved similarly to Lemma 1.2.6, using Lemma 1.2.5.
D
Now we continue the proof of the proposition. Axiom 3 in Definition 1.1.4 is obvious by Lemma 1.2.4 and Lemma 1.2.6 (applied to it := X+ \ (A,oo)). 3 * is obvious by Lemma 1.2.5 and Lemma 1.2.7. Axiom 4 (resp. axiom 4*) is nothing but the case TT = 0 in Lemma 1.2.6 (resp. Lemma 1.2.7). In axiom 5, f is assumed in the proposition. On the other hand, we have Homc(rad Ac(A), £c(A)) = 0 by e and Lemma 1.2.3. As we have Hom c (A c (A),Vc(A)/soc(V c (A))) = 0 applying Lemma 1.2.6 to it = X+ \ [A, oo), we have the two canonical maps End c L c (A) -* Hom c (A c (A),L c (A)) -> Hom c (A c (A), V C (A)) are isomorphisms. After we invoke assumption h, which has not been used so far, axiom 5 is also proved. Hence, (C, X+, A, V) is a weak split highest weight coalgebra (resp. split highest weight coalgebra). The fact Lc(A) = top(Ac(A)) is assumed in e. This completes the proof of the proposition. D
J. Highest weight theory over a Held
165
Definition 1.2.8 When conditions a-f and g for i = 1 (resp. i = 1,2) in the proposition are satisfied, we say that (C, X+, A, V, L) is a weak highest weight coalgebra (resp. highest weight coalgebra). If, moreover, h in the proposition is also satisfied, then we say that (C, X+,A, V,L) is split (by the proposition, this definition is justified). Definition 1.2.9 We say that (A,X+, A, V,L) is a weak quasi-hereditary algebra (resp. quasi-hereditary algebra) over k if A is a finite dimensional fc-algebra, and (A*,X+, A, V,L) is a weak highest weight coalgebra (resp. highest weight coalgebra) over k. (1.2.10) Let A; be a field, and (C,X+, A, V,L) a weak highest weight coalgebra over k. For a poset ideal -n of X+, we denote by Mc(ir) the full subcategory of c M consisting of C-comodules V such that Homc(Ac(A), V) = 0 if A ^ n. R e m a r k 1.2.11 Lemmas 1.1.7, 1.1.9, 1.1.10, and 1.1.11 in the last subsection are still valid in our situation. This is obvious from the argument so far. L e m m a 1.2.12 Let V G M , and n be a finite poset ideal of X+, n > 1. Assume the following:
and
a For any /z £ K, we have Ext' c (Ac(A0, V) = 0 (1 < i < n). b For fj, (=. 7T, we have Homc(Ac(//), V) = 0 unless \i is a maximal element of Tr. Then we have for any A € TT: 1 Extj;(L c (A), V) = 0 for 1 < i < n. 2 The canonical maps Hom c (V c (A), V) -* Hom c (L c (A), V) -> Hom c (A c (A), V) are all isomorphisms. 3 Homc(£
(0 < i < n).
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III. Highest Weight Theory
Combining this with Ext^(Ac(A), V) = 0 (1 < i < n), assertion 1 follows. As n > 1, we have Extc(W,V) = 0 and Homc(W,V) = 0 for W e (—oo,A)) with dimVF < oo. Considering the cases W = rad(Ac(A)) and W = Vc(A)/soc(Vc(A)), assertion 2 follows easily. 3 is obvious from 2 and assumption b. D (1.2.13) Let Jfc and (C,X + ,A,V,L) be as in (1.2.10). Let IT be a poset ideal of X+, and V € M c . We denote the sum of all C-subcomodules of V which belong to MC(TT) by V(n). Note that V(TT) is the largest Csubcomodule of V which belongs to M C (TT). Note also that V £ M°(n) if and only if V = V(n). Note also that ?(n) is a functor from M c to c c M C (TT), which is right adjoint to the canonical embedding M (7r) «-> M , see (1.1.10). Lemma 1.2.14 If (Vi) is a filtered inductive system of C-comodules, then the canonical map is an isomorphism. Proof. Follows immediately from Lemma 1.1.10.3.
•
For a finite dimensional C-comodule V, V* = Uom{V, k) is a left Ccomodule (hence is a right Cop-comodule), see (1.3.5). The duality Hom(?, A;) gives a contravariant equivalence between the category of finite dimensional C-comodules M^ and the category of finite dimensional left C-comodules C M/. Lemma 1.2.15 The quintuple
is a weak highest weight coalgebra over k. If (C, X+, A, V, L) is split, then so is (Cop,X+, V*,A',L*) ; where A':=(AC(\)')X€X+,
V:=(Vc(ArW,
and IS :=
(Lc(X)')xex*.
Proof. If V and W are finite dimensional C-comodules, then we have £(V, W) S ExtJc(V, W) S Ext^c"P(W', V)
s
for n = 0,1 by Lemma 1.1.5.7. The lemma follows from this.
•
Lemma 1.2.16 Let k! be an extension field of k, C a k-coalgebra, X+ an ordered set, and A and V families of C-comodules indexed by X+. Then (C, X+, A,V) is a weak split highest weight coalgebra over k if and only if (C',X+,A',V) is a weak split highest weight coalgebra over k', where C := C®k k', A' := (AC(A) ®* k')XeX+, and V := (VC(A) ®fc k')X€X+.
1. Highest weight theory over a field
167
Proof. For a finite dimensional C-comodule V and a C-comodule W, we have (n > 0) ExtJ(V, WO ®fc Jf = Ext£,(V ®* **, W ®* A*) by Proposition 1.3.6.20, 3. Hence, the 'if part is obvious. We prove the 'only if part. We check the conditions in Definition 1.1.4. The conditions 0,1,2 and 2* are obvious. Condition 5 is also obvious from the isomorphism above. If V is a finite dimensional C-comodule and W is a C-comodule, then we have Ext£(V, W) a Ex$.9kkf(V
®fc k', W)
by Proposition 1.3.6.20, 4. Hence, conditions 3 (for i = 1) and 4 are obvious. By Lemma 1.2.15, (C,X+, V*, A*) is a weak split highest weight coalgebra over k. Combining this with 3,4, we have 3*,4*. • Note that a base change of a non-split highest weight coalgebra is not necessarily a highest weight coalgebra again, as Lc(X) ®/t k' may not be simple. (1.2.17) Let jfc and (C, X+, A, V,L) be as in (1.2.10). As Mc is locally finite, any object admits an injective hull by Theorem 1.1.7.6. We denote the injective hull of Lc{X) by Qc(X). As LC(X) <-¥ Vc(A) is an essential mono, we have that Qc{X) is also an injective hull of Vc(A). The following two lemmas show that A and V are determined only by C, X+ and L. Hence, we may say that (X+,L), or an ordering X+ of the set L of simples of C, is a (weak) highest weight theory on C, and A is its set of Weyl modules, and V is its set of induced modules. Lemma 1.2.18 Let M be a finite dimensional C-comodule, and X 6 X+. Then the following are equivalent. 1 MSAc(A). 2 topM = Lc(X), any simple subquotient o / r a d M is of the form Lc(fi) with fi < A, and ExtJ;(M, Lc{v)) = 0 for any v e X+ with v ^ A. 3 topM = Lc(X), any simple subquotient of M is of the form Lc(fi) with H ? X, and Ext^(M, Lc{v)) = 0forueX+ with v < X. Proof. l=>2^-3 is obvious. We show 3 ^ 1 . By assumption, there exists an exact sequence 0 -» M' -> M -> LC(X) -> 0. As any simple subquotient of M' is of the form Lc(fi) with (j, ^ A, we have Extc(Ac(A), M') = 0 by Remark 1.2.11. Hence, the canonical map AC(A) —> LC(X) lifts to a map, say, ip : AC(A) —¥ M. As dimM < oo, we
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III. Highest Weight Theory
have top(Coker
Then the following
1 N £ VC(A). 2 dimAf < oo, socN = Lc(A), any composition factor of N/ socN is of the form Lc{fj) with n < X, and Ext c (Lc("), N) = 0 if v G X+ with v ^ A. 3 soc iV = Lc(\), any simple subquotient of N is of the form Lc(n) with H ^ A; and Extlc(Lc(v), N) = 0ifueX+ with v < A. • Lemma 1.2.20 For any A € X + , we /taw that End c (A c (A)),
and
End c (V c (A))
are division rings, and are isomorphic to one another as k-algebras. In particular, Ac(A) and Vc(A) are indecomposable. Proof. As Lc(A) is an simple object, Endc(-£/c(A)) is a division ring, by Schur's lemma. Next, as soc and top are fc-endofunctors of M^, we have fc-algebra maps soc : End c (V c (A)) -> Endc(Lc(A)) and top : End c (A c (A)) -> End c (L c (A)). Note that soc is given by a restriction, and top is induced by an induced map. Let
1. Highest weight theory over a Geld
1.3
169
Highest weight coalgebras and good comodules
(1.3.1)
Let (C, X+, A, V, L) be a highest weight coalgebra.
Proposition 1.3.2 For V € M c , the following are equivalent. 1 For any A e X+ and i > 0, we have Ext^(A c (A), V) = 0. 2 For any A € X+, we have Ext^(A c (A), V) = 0. 3 For any finite poset ideal •K of X+ and any maximal element A of IT, the canonical maps Hom c (A c (A), V) -> Hom c (A c (A), <- Eomc(Lc(X),V/V(n'))
<- Hom c (V c (A),
are all isomorphisms, and the canonical pairing Hom c (V c (A),
V/V(TT'))
® B VC(A)
->
1//K(TT')
is injective, and its image agrees with V(TT)/V(TT'), where ir' := IT \ {A}, and E := End c (V c (A)). Moreover, Hom c (V c (A) ) y/V(7r')) »a fi-/rec, and hence V(7r)/V(7r') is isomorphic to a direct sum of copies of Vc(A), as a C-comodule. 4 For any injective order-preserving map f : X+ —> N wit/i f(X+) a poset ideal of N,
i
b Fori > 1, Vi/Vi-i is isomorphic to a direct sum of copies of Vc(/~ 1 (i)). 5 The set of finite dimensional C-subcomodules of V and the set of Csubcomodules belonging to T(V) (see (I.I.11)) are cofinal. 6 V is a filtered inductive limit of objects of Proof. 1=>2 is obvious. We prove 2=*-3 by induction on #TT. AS V(0) = 0, it is easy to see that V(TT') £ ^ ( V ) , by induction assumption. Hence, by Lemma 1.1.6, Extc(A c (/i)i^( 7 r ')) = 0 for i = 1,2 and fj. G X+. As we are assuming 2, we have Ext^(A c (^), V/V(-K')) = 0 for any p e X+. On the other hand,
is an isomorphism for any \i £ IT', by the definition of ?(7r') and the fact Ac(^) € M C (TT'). Combining this observation with Extc(A c (^), V(n')) = 0, we have Hom c (A c (/j). V/V(TT')) = 0 for /x e IT1.
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III. Highest Weight Theory
Hence, by Lemma 1.2.12, Hom c (A c (A), V/Vtf))
«- Hom c (L c (A), V/V(ir')) <-Hom c (V c (A),K/\/(7r'))
axe both isomorphisms. By definition, we have V(ir') G MC(TT') and hence Hom c (A c (A),V(7r'))=0. As Ext^(A c(A),y(7r')) = 0, it follows that Hom c (A c (A), V) -> Hom c (A c (A), V/V(ir')) is also an isomorphism. Hence, the first part of 3 is proved. Next, we remark that by Lemma 1.2.20 and its proof, Lc(A) is an Emodule in a natural way. Also Hornc(Lc(A), ^/^ r ( 7r ')) i s E-iiee, and the pairing
p : Homc(Lc(A),
V/V(TT'))
®E LC{\)
-> V/Vtf)
(f®v^
fv)
is injective. In fact, when we set Imp = 0 i 6 / i / i with L{ = Lc(A), then we have Romc(Lc(\),V/V{ir'))
S* Hom c (L c (A),Imp) S 0 End c L c (A) ^ ® £ , i€l
i€l
and we have the desired assertion. In particular, Homc(Vc(A), V/V(TT')) is also E-iree. Next, we show that (/ : Hom c (V c (A), V/V(n')) ®E VC(A) is injective. Assume the contrary. Then Ker p' ^ 0, and this implies soc(Kerp') = soc(Homc(V c (A), V/V(it>)) ®E V C ( A ) ) n Kerp' ^ 0 by Lemma 1.1.10.7. On the other hand, soc(Homc(Vc(A), V/V{nf)) ®E VC(A)) = Hom c (V c (A), V/V(7r'))<8>Esoc(Vc(A)) S
Homc{Lc(\),V/V{n'))®ELc{\)
by Lemma 1.1.10.7 again, and hence p is not injective, which is a contradiction. Hence, p' is also injective. We set / := Imp', and we define J to be the pull-back of / by the projection V —• V/V(ir'). As / is a direct sum of copies of Vc(A), we
1. Highest weight theory over a
field
171
have / € Mc(n). As V(ir') G MC(TT'), it follows that J G Mc(ir). Hence, J C V(TT). As / is a direct sum of copies of Vc(A), Extlc(Ac(n),I) = 0 for i = 1,2 and /x G X+. As I,V(n') G .F(V), we have J G .F(V), and hence Ext'c (A c (M),i) = 0 for i = 1,2 and /x G X+. It follows that ^ A c ^ ) . V/J) = 0 for any /x € X+. Now assume that V(n)/J =fc 0. As V(w)/J € MC(TT), there exists some € 7T such that Since V(n)/J C V/J, we have Homc(Ac(M), V/J) ^ 0, too. Let /x e 7T be a minimal element among such, then Homc(Vc(A*), V/J) / 0 by Lemma 1.2.12. On the other hand, as 0 -> / -> V/V(TT') -> K/J -> 0
is exact and / is a direct sum of copies of Vc(A) and hence Ext^( Vc(^), /) = 0 (by Lemma 1.2.12), we conclude that Hom c (V c (/i),/) -)• Homc(V c(/i),\//K(7r')) is not surjective. By definition of /, we have /x ^ A, and /x € TT'. We have Homc(Ac(i/), /) = 0 for v < /x. On the other hand, by minimality of fi, we have Homc(Ac(v),V/J) = 0. These facts show Hom c (A c H,V/V(n')) = 0 for v < /x. Hence, again by Lemma 1.2.12, we. have Homc(Vc(M), V/V(n')) S Homc(Ac(/x), V / V ^ ) ) ^ 0. On the other hand, as n £ TT', we have that Homc(Ac(/i),T/(7r')) is an isomorphism. Combining this with Extp(Ac(/x), V(TT')) = 0, we have Homc(Ac(M),^/Vr(7r')) = 0, which is a contradiction. Hence, V(ir)/J = 0 is proved, and we have V(n) = J. It follows that V(TX)/V(TT') = I = Imp', and the desired assertion follows. We prove 3=>4. First, we show the existence of a required filtration. To prove this, we set V* := V(f~1([l, i}). By assumption, condition b is obvious. Hence, we have Vt e .F(V) for each i > 0. This shows Extc(Ac(A), V$) = 0 for any A € X+. And hence we have ExtJ?(Ac(A),UmV5) = 0. On the other hand, for any A G X+, any C-comodule map Ac(A) -> V factors through V/(A), hence it also factors through limVi. This shows Homc(Ac(A),limVi) -
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Highest Weight Theory
is surjective. Hence, Homc(Ac(A), V/limV;) = 0 for any A 6 X+, which shows V = lim Vi. The proof of the existence is complete. Next, we show the uniqueness, namely, Vt = V(f~l([l,i])) for such a filtration (Vj). As Vj/Vj-i is a direct sum of copies of Vc(/~ 1 (j)), we have Vt e M c (/- 1 ([l,*])) ) and hence VJ C Vtf-^i]). As H o m c ( A c ( / ( 0 ) , ^ 7 ^ - i ) = 0 for j ? I, we have Hamc(A c (/(0), W ) = 0
(j > i > I).
Hence, C Homc(A c (/(0), V/Vt) = 0 fort > I. On the other hand, as ^ / " ' ( [ M ] ) ) / ^ belongs to M c (/- 1 ([l,i])), we have Hom c (A c (/(0), ^ ( / - ^ [ l , t]))/V4) = 0 also for I > i. Thus, we have
for any A G X+, and this shows V{ = 4=>5 is obvious, because each V* belongs to .F(V), and we are assuming that there is an injective order-preserving map / : X+ —> N with f{X+) a poset ideal of N, see Lemma 1.1.2. 5=>6 This is clear, as V is the filtered inductive limit of its finite dimensional C-subcomodules. We prove 6=>1 using induction on i. As the assertion is obvious for i = 1,2, we may assume that i > 3. Let / be the injective hull of V. Then for any A e X+ and j > 1, we have Ext J c (A c(A), I/V) S Ext Jc+1(Ac(A), V). Hence, in particular, we have Ext^(Ac(A), I/V) = 0 for any A G X + . As we have already proved 2=>6, we know that I/V is a filtered inductive limit of objects of .F(V). By induction assumption, we have ExtJJx(Ac(A), I/V) = 0 for any A € X+, which shows Ext l c (A c(A), V) = 0. • We say that a C-comodule V is good if the equivalent conditions in Proposition 1.3.2 are satisfied. By the proposition, it is not so difficult to show that M c is a highest weight category [37] with X+ its set of weights, V its set of induced modules, and L = (Lc(ty)\ex+ its set of simple modules. Corollary 1.3.3 Let \,neX+
and i > 0. Then we have
unless A = \i and i = 0. Proof. The case i = 0 is contained in the definition of a highest weight coalgebra. If i > 0, then as we have VC(M) £ •?r(^7)) t n e required vanishing holds by the proposition. •
1. Highest weight theory over a Geld
173
Lemma 1.3.4 The full subcategory of good C-comodules is closed under extensions, monocokernels, direct summands and filtered inductive limits in Mc. Proof. Obvious from condition 1 in Proposition 1.3.2. Lemma 1.3.5 Let V be a good C-comodule. any poset ideal •K of X+.
D
Then V(TT) is also good for
Proof. The case #TT < oo is proved easily by induction on #7r. The general case follows from this, as we have V(n) = \\mV(p), where p runs through all finite poset ideals of TT. D Lemma 1.3.6 There exists a unique family of k-subcoalgebras (C^)n of C indexed by the set of finite poset ideals of X+, satisfying the following conditions: a C 0 = 0. b If-KCn',
thenCn C C^.
c If n is a finite poset ideal of X+ and A is its maximal element, then we have an isomorphism of (C,C)-bicomodules CJCn,
£ A C (A)' ® £ V C (A),
where ir' := w \ {A} and E := Endc(Vc(A)). Here we regard Ac(A) as an E-module through the canonical isomorphisms
E^
Endc(Lc(A)) ^ 4 End c (A c (A)).
In fact, CV := C(TT) satisfies the conditions, and we must define CV thus. We also have d limC7r = C. Proof. We prove conditions a,b,c,d after setting Cn := C(TT). Note that C(TT) is a A;-subcoalgebra of C by Lemma 1.3.6.10 (applied to B = M°(n) and j =?(TT)). Conditions a,b,d are obvious. We prove that condition c is satisfied by C(TT). AS condition 1 in Proposition 1.3.2 is satisfied by the injective C-comodule C, we have that C is good. We may utilize condition 3 in Proposition 1.3.2, and hence the composite AC(A)* ®E VC (A) £ Hom c (A c (A), C)
C/C(TT')) ® E V C ( A )
S Hom c (V c (A), C/C(TT'))
®E
VC(A)
<-+ C/C(TT')
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Highest Weight Theory
is an injective (C, C)-bicomodule map. As the image of this map agrees with C(7r)/C(7r'), c is also satisfied. We prove the uniqueness, i.e., Cn = C(n) for all TT, assuming a,b,c, using induction on #7r. If •n = 0, then the assertion is obvious. If n ^ 0, then we take a maximal element A of TT, and set TT' := TT \ {A}. Then we have CV' = C(TT') by induction assumption, and CV C C(TT), as C^/C^ is a direct sum of copies of Vc(A) as a C-comodule. By dimension counting, we have Cn = C(TT), as desired.
• +
Assume that IT is a poset ideal of X , not necessarily finite. Then as we have C(n) = limC(p), where p runs through all finite poset ideals of it, we have that C(ir) is a good subcoalgebra of C. We call C(ir) the Donkin subcoalgebra of C with respect to vr. If -K is finite, then C(TT) is finite dimensional. In this case, we denote the dual algebra C(TT)* of C(TT) by 5c(vr), and call it the Schur algebra of C with respect to •n. Lemma 1.3.7 Let V be a C-comodule. V € Proof.
Then V € Mc(7r) if and only if
Follows immediately from Lemma 1.3.6.10 (applied to B =
and j =?(TT)).
M C (TT) •
Lemma 1.3.8 Let A,/z £ X+, and assume /x ^ A. 77ien we /uw/e L c fo), VC (A)) = 0, Ext' c (V c (/i). Vc(A)) = 0,
Ext' c (A c (A), L C ( M ) ) = 0, anrf
Exf c (A c (A), A c (/i)) = 0
for i > 0. Proo/. We set TT = X+ \ (A,oo) and n = i, and apply Lemma 1.2.12. Then we have Ext* c (L c (/i), V C (A)) = 0 by Corollary 1.3.3. The vanishing Ext^(Ac(A), Lc{n)) = 0 is the dual assertion, and is proved similarly. The vanishings Exf c (V c (/x), V C (A)) = 0 and Ext' c (A c (A), A c (/x)) = 0 for i > 0 are obvious, as we have A C ( M ) , V C ( M ) & MC(X+ \ (A, oo)). D
1.4
Weak highest weight coalgebras and good filtrations
(1.4.1) Let k be a field, and (C, X+, A, V, L) a weak highest weight coalgebra over k. For a countable set Y, the symbol [Y] stands for the interval [1, # y ] if # y < oo, and [Y] = N if Y is countably infinite. Lemma 1.4.2 Let V € Mc.
Then the following are equivalent.
1. Highest weight theory over a Geld
175
1 V e ^"(V). 2 For any order-preserving bijective map f : X+ ->• [X+], there exist some n € [X+] and some filtration 0 = Vo C Vi C V2 C • • • C Vn = V such that for anyi, Vi/Vi-i is a finite direct sum of copies o/Vc(/ - 1 (i))3 V is finite dimensional, and for any finite poset ideal ir of X+ and any maximal element A of i\, we have that V(TT)/V(TT \ {A}) is a finite direct sum of copies of Vc(A). Proof. 1=^2 Using induction on i, we construct a filtration 0 = VoCV1C---CVicV, such t h a t for any j with 0 < j < z , V j / V j _ i i s a finite direct sum of copies
of Vcir 1 ^)), and V/Vi E ^({VcU^tf)) I* < J < #*})• This is enough to prove 1=>2. In fact, as V is finite dimensional, there are only finitely many (i £ X+ such that Lc(fi) appears in the composition factor of V. If we take such a \i with f{p) maximal, and we set n := /(/x), then we have V = Vn, as we have V/Vn € ^({Vcif^U)) IJ > fit1)})If i = 0, the required conditions are satisfied by letting Vo = 0. Assuming i > 0, and the construction is already done up to Vi_l5 we construct Vt. By induction assumption, there exists some filtration 0 = Wo C Wi C • • • C WT = V/Vi-i such that for each / = 1,2,... ,r, there exists some ji such that i < ji < # X and Wi/Wi-i = Vcif^Ui))Assume that the number of / such that ji = i is s. Then as we have Extc(Vdf^i), VcC/" 1^)) = 0 for j > i by Lemma 1.2.19, there is an exact sequence 0 - • ® Vet/" 1 *) -> V/VJ_i -)• W -4 0 such that W <E ^({VcC/'Hj)) M < 3 < # ^ » - Defining V{ to be the kernel of the composite V —> V/V^i —¥ W, the required conditions are satisfied. 2=>3 Obviously, V is finite dimensional. If n is a finite poset ideal of X+ with #TT = m and A is its maximal element, then applying Lemma 1.1.2 to the ordered sets ir \ {A} and X+ \ •n, we have that there exists an orderpreserving bijective map / : X+ -> [1,#X + ] such that f(n) = [l,m] and /(A) = m. By the assumption of 2, there exists some n > 0 and a filtration 0 = Vo C Vx C V2 C • • • C Vn = V
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III. Highest Weight Theory
such that Vi/Vi-i is a finite direct sum of copies of Vc(/ - 1 (i))- Letting Vi := V for i > n if necessary, we may assume that m < n. Then we have Vi = y(/- 1 [l,i]) for i = 0 , 1 , . . . ,n. In fact, each V ^ / " 1 ^ ) ) U < i) belongs to Mc(f~1[l,i]) and as M c (/~ 1 [l,i]) is closed under extensions, we have Vi e M c (/" 1 [l,i]), and hence V{ C V(/- 1 [l,i]). On the other hand, we have
Homc(Lc(r10')). V c (r 1 (0)) = Hom c (L c (r 1 0))>ic(r 1 (0)) = 0 for j < i < I. Hence, we have Hom c (Lc(/~ 1 0'))i^/^i) = 0 f° r J ^ *• This shows soc((V/Vri)(/"1[1>i])) = °> and h e n c e (V/Vi)(/-X[l.*l) = 0. As we have V{f-l[\,i])IVi c (V r /F i )(/- I [l,i]), it follows that V{f~1[\,i]) = Vt. Hence, V(n)/V(Tr\{\}) = Vm/Vm_i is a finite direct sum of copies of Vc(A). As 3=>2=>1 is obvious, the proof of the lemma is complete. D Corollary 1.4.3 Let W € M c . Then the following are equivalent. 1 For any finite poset ideal TT of X+, V = W(n) satisfies the equivalent conditions of the lemma. 2 For any finite poset ideal n ofX+ and its maximal element X, {A}) is a finite direct sum of copies of Vc(A).
W(TT)/W(TT\
3 For any order-preserving bijective map f : X+ —> [X+], there exists some filtration 0 cW0 CWi CW2 C • • • CW such that lim W{ = W and each Wi/Wi-i is a finite direct sum of copies o / V c l / " 1 ^ ) ) . 4 There exists a filtration (1.4.4)
0 = WQcWiCW2C
•••€]¥
ofW such that lim Wi = W, for i > 1, Wi/Wi-i is isomorphic to Vc(A) for some A G X+, and for any A € X+, there are only finitely many i such that W i / W ^ i ^ VC(A). We say that W 6 M c has a good filtration, if W satisfies the equivalent conditions in the corollary. We call a filtration (1.4.4) of W as in 4 of the corollary a good filtration of W. Assume that W £ M c has a good filtration. Then for any good filtration (1.4.4) of W and any A 6 X+, the number of i such that Wi/Wi-i = VC(A) agrees with dimHom c(Ac(A),VK)/dim(Endc(L c (A))), which is independent of the choice of good filtration. We denote this number by [W : VC(A)]. By definition, if V has a good filtration, then V(TT) and V/V(K) also have good nitrations for any finite poset ideal TT of X+. If (C, X+, A, V, L) is a highest weight coalgebra and W has a good filtration, then W is good.
1. Highest weight theory over a Beld
177
Lemma 1.4.5 The class of comodules with good filtrations is closed under finite direct sums and direct summands. Proof. As the functor ?(7r)/?(7r') is additive, it suffices to show that any direct summand of a finite direct sum of copies of Vc(A) is a finite direct sum of copies of Vc(A) for any A € X+. This is immediate from the indecomposability of Vc(A) (Lemma 1.2.20) and the Krull-Schmidt theorem D (Lemma 1.1.12.6). For a finite dimensional C-comodule V, we denote the number of appearances of Lc(A) in a composition series of V by (V : L C (A)). Obviously, we have (V : LC{X)) = dimHom c (V,Qc(A))/dimEnd c (MA)), where Qc(ty denotes the injective hull of Lc(X). Lemma 1.4.6 Assume that the C-comodule C has a good filtration. Then for any A € X+, QcW has a good filtration, and [QcW : VcM] = (Ac(/i) :
LcW)
for any /i e X+. In particular, we have [QcW '• Vc(A)] = 1, and [QcW '• = 0 for ix ? A. Proof. As UJ : Lc(A) -> Lc(A)®C is injective and Lc(A)®C is C-injective, we have that QcW is a direct summand of a finite direct sum of copies of C. By Lemma 1.4.5, QcW has a good filtration. The equality [QcW • V C ( M ) ] = dimHom c (A c (M),Qc(A))/dimEnd c (L c (A))
= is obvious. The last assertion follows from this.
(A C (M)
= LC(\)) D
T h e o r e m 1.4.7 Let (C,X+,A,V,L) be a weak highest weight coalgebra over a field k. Then the following are equivalent.
2 (C, X+, A, V,L) is a highest weight coalgebra. 3 The C-comodule C has a good filtration. 4 (Cop,X+, V,A*,L*) is a highest weight coalgebra.
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III. Highest Weight Theory
Proof. 1=^2=^3 is obvious. We prove 3=>1, using induction on i > 1. If i = l, then the assertion is obvious by the definition of weak highest weight coalgebra, so we may assume i > 2. By induction assumption, if V has a good filtration and 1 < j < i, then Ext^(A,V) = 0. Let A £ X+. When we set n := ( - c o , A], then we have a short exact sequence of C-comodules with good filtrations 0 ->
QC(A)(TT)
-» QC(X) -> Qc(A)/g c (A)(7r) -> 0.
By Lemma 1.4.6, we have Qc(A)(7r) =" V C (A). We have 1
)
=0
by induction assumption, and obviously we have Extp(A c (M))Qc(A)) = 0 for any /x G X + . This shows E x t j ^ A c M , VC(A)) = 0. Next, we show 2=>4. We already know that ( C o p , X + , V , A*,L*) is a weak highest weight coalgebra by Lemma 1.2.15. By Lemma 1.3.6, the C°pcomodule C has a good filtration. By 3=^-2, which is already proved, we have that (C op , X+, V*, A*, L*) is a highest weight coalgebra over k. We prove 4=>2. By 2=>-4, which is just proved, we have (C o p o p , X+, A " , V " , L " ) = (C, X + , A, V, L) is a highest weight coalgebra.
•
Corollary 1.4.8 Let k' be an extension field of k, C a k-coalgebra, X+ an ordered set, and A and V families of C-comodules indexed by X+. Then (C, X+, A,V) is a split highest weight coalgebra over k if and only if (C',X+, A', V ) is a split highest weight coalgebra over k', where C := C®k k', A' := (AC(A)')A€X+» and V := (VC(A)')A 6 X+- Moreover, we have ^c(A) (Si* A;' = Lc(A) /or any A £ X + in this case. Proof. Obvious by Lemma 1.2.16 and the theorem.
2
O
Donkin systems
2.1 U-acyclicity of flat complexes (2.1.1)
Let R be a commutative ring.
Lemma 2.1.2 (Universal coefficient theorem) LetW be anR-flat complex, and M an R-module. Assume that F is bounded above or flat.dim^ M < oo. Then there is a spectral sequence E™ = Tor* p (#«(F), M) =» Hp+q{¥ ® M).
2. Donkin systems
179
In particular, if flat.dim/j M < 1, then we have E™ = E™, and there is a short exact sequence
0 -> Hn{¥)
—r r
—> r
—r • • •
is said to be u-acyclic (universally acyclic) if for any /{-module M, we have Hl(¥ ® M) = 0 (i > 0) and the canonical map pM • H°{¥) ® M -> H°(W ® M) is an isomorphism. We list some consequences of the definition. Lemma 2.1.4 Let ¥ be a u-acyclic R-complex. Then R1 <S)¥ is a u-acyclic R'-complex for any base change R -> R!. Moreover, for any R-module M, ¥ ®M is u-acyclic. Proof. Easy.
D
Lemma 2.1.5 Let O->F4G-4H-»O be an exact sequence of R-complexes. Assume that ¥ is u-acyclic and / ' : F* -> G' is R-pure for i > 0. Then G is u-acyclic if and only if M is u-acyclic. Proof. Easy.
D
Lemma 2.1.6 Let (F*)A€A be a filtered inductive system of u-acyclic Rcomplexes. Then lim ¥\ is u-acyclic. Proof. Obvious.
180 (2.1.7)
III
Highest Weight Theory
Let R be a noetherian commutative ring.
Lemma 2.1.8 Let (R, m) be a local ring, and F a flat R-module. Then there is an exact sequence (2.1.9)
0 ^ P->F->G->-0
of R-modules such that P is R-free, G is R-flat, and G/mG = 0. Proof. Let B be an R/m-basis of F/mF, and consider the i?-free module P freely generated by B over R. Then the canonical map P —> P/mP = F/mF is lifted to some il-linear map
Proof. We take an exact sequence (2.1.9) as in the lemma. Then as dimK(m) P(m) = c and P is i?-free, we have P = Rc. For any p G Spec R, 0 -> P(p) -> F(p) -> G(p) -> 0 is exact by flatness of G, and we have G(p) = 0 for any p e Spec R by dimension counting. By Corollary 1.2.1.6, we have G = 0. Hence, we have • F = P = i?c. Lemma 2.1.11 Let jr : F° £+ f1 A F2 6e an 7l-/?at complex. Then we have the following: (F® fl/p) ts R-finite {resp. 0) /or any p G Spec/?, tfien / ^ ( F ® Af) is R-finite (resp. 0) /or any R-finite module M (resp. any R-module M). 2 // Hl{¥ ® fi/p) = 0 /or any p £ Spec fl, F° -+ F1 -> F 2 -»• Cokerd1 -> 0 is a u-acyclic R-flat complex, and Kerd° is an R-pure submodule of F°, hence it is R-flat.
2. Donkin systems
181
Proof. 1 If M is infinite, then M admits a filtration whose successive subquotients are of the form R/p with p € SpecR. As /^(F®?) is halfexact, HX(F ® M) is also infinite (resp. 0), as RMj and {0} are very thick in RM. As /^(F®?) preserves filtered inductive limits and {0} is also closed under filtered inductive limits, H1(¥ ® M) = 0 also for general M. 2 We set C := Cokerd1. As we have //X(F) = 0 by 1, we can take an i?-flat resolution of C of the form • F~ 2 -> F " 1 -> F° -4 F 1 -> F 2 -> C -• 0. By 1, we have Torf (C, M) = 0 for any fi-module M, and hence C is fl-flat. As 0 -+ Kerd0 -> F° -> F 1 -* F2 -> C -> 0 is exact and i?-flat modules are closed under epikemels, Ker d° is also i?-flat. As this sequence is a flat resolution of 0, the rest of the assertion follows. D (2.1.12) Let R be a noetherian commutative ring, and M an ii-module. We say that M is R-metafinite if there exist some noetherian commutative .R-algebra A and an .A-finite module structure of M which induces the original i?-module structure of M via restriction. Lemma 2.1.13 Let M be a metafinite R-module. Then for any R-finite module N, the tensor product N
an R-flat complex. Consider the following conditions: 1 For any p € Spec R, H{(F®R/p) = 0 (i > 0) and H°(F®R/p) is R-finite. 2 H°(TF) is R-finite protective, and F is u-acyclic. 3 H°(F) is R-finite, and /T(F) = 0 (i > 0).
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Highest Weight Theory
4 For any p G Spec R, we have Hi(¥(p)) = 0 for i > 0, and /ig.(p) := p) H°(¥(p)) is finite. The function hf is locally constant on Spec R. 5 Hi(¥(m)) = 0 (i > 0) for any m G Maxi?. Then we have the following. (2.1.15) 5<=4«=1<»2=>3. (2.1.16) If R is a regular ring, or¥ is bounded, then 3=»1. (2.1.17) IfF° is R-projective, then 4=>1. (2.1.18) Let n be a non-negative integer. Assume that Hl(¥(p)) = 0 (i > n), and Hn(¥ <8> R/p) is R-metafinite for any p G Spec R. Then Hi(¥ <8> M) = 0 (i > n) for any R-module M. In particular, if Hi(¥(p)) = 0 (i > 1) and Hl(¥ igi R/p) is R-metafinite for any p G Spec/?, then F is u-acyclic. (2.1.19) Let n be a non-negative integer. Assume that Hl(¥(m)) = 0 (i > n), and Hn(W®R/p) is R-finite for any p e SpecR. Then H^W^M) = 0 (i > n) for any R-module M. In particular, if H1(¥ ® R/p) is R-finite for any p G Spec R and H°(¥) is R-finite, then we have 5=>1. Proof. The implication 1=»2 is obvious by Lemma 2.1.11. It is trivial that 2 implies 1,3 and 4. The implication 4=>5 is also trivial. Hence, (2.1.15) follows. The assertion (2.1.16) follows easily from Lemma 2.1.2. We prove (2.1.17). For p G Spec R, as H°{¥®R/p) is an fl/p-submodule of the projective module F° ® R/p, we have H°(¥ (g) R/p) is i?/p-finite, or equivalently, .R-finite by Corollary 1.3.11.8. So it suffices to show that H{(¥
0 -> Hi{¥) ® R/Rx -> Hi(¥ ® R/Rx) -> Torf (Hi+l{¥), R/Rx) -> 0.
2. Donkin systems
183
Hence, for i > 2, we have soc#'(F) = As supp//'(F) C {m}, and the category of i?-modules whose support is contained in {m} is locally finite, we have Hx(¥) = 0 for i > 2. Hence, it suffices to show that Hl(F) = 0. By Lemma 2.1.11, we have that Kerd1 is an .R-pure submodule of F 1 , and is i?-flat. Hence, replacing F by the /2-flat complex
we may assume that F* = 0 (i > 2) without loss of generality. As F° is /?-projective and R is local, F° is R-hee by Kaplansky's theorem [92]. We take a basis B of F°. As dimK(m) i/°(F(m)) < oo, there exists some finite subset Bo of B such that i/°(F(m)) is contained in the /c(m)-span of Bo in F°(m) = «(m) • B. Now we set G° := R • Bo and Q := R • (B \ BQ). When we denote the composite map
by ip, we have that
gives an /?-flat complex G of length one, and we have a short exact sequence of i?-flat complexes 0 -> (id0 : ( ? ^ g ) - > F - ^ G ^ 0 . As n and 7r(p) (p 6 Spec R) are quasi-isomorphisms we may assume, replacing F by G, that F° is ii-finite free without loss of generality. As the sequence 0 -> #°(F(p)) -> F°(p) -> F\p) -»• 0 = H1(F(p)) is exact, dimK(P) F 1 (p) is finite and constant on Spec R by assumption. By Corollary 2.1.10, we have that F 1 is /l-finite, and hence so is i/^F). As f/^F)
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III. Highest Weight Theory
assume that R is a local domain and d > 0. By induction assumption, it suffices to show H* (F) = 0 for i > n. Note that supp£P(F) C {m}. By the same argument as in the proof of (2.1.17), H'(¥) = 0 for i > n is proved easily, using Lemma 2.1.2. We prove # n (F) = 0. As H := H"(F) is fl-metafinite by assumption, there is a noetherian commutative /?-algebra A such that H is /4-finite. We have H/xH C Hn{¥ ® R/Rx) = 0 for any x <E R \ {0}. This shows that x : H —> H is a surjective >l-linear map, which must be bijective. This shows that H is a K(0)-module. As we have H = H(0) 2* # n (F(0)) = 0, we are done. Now we prove the second assertion of (2.1.18). As we have H'(F®R/p) = 0 for i > 0, F is u-acyclic by Lemma 2.1.11. The assertion (2.1.19) is easier, and we omit the proof. D
2.2
The definition and the existence of a Donkin system
(2.2.1) Let R be a commutative ring, and C an .R-fiat coalgebra. For p € Spec .ft and an .R-module M, M(p) stands for M <8> /c(p). Definition 2.2.2 Let V and W be C-comodules. We say that W is uacyclic with respect to V if for any it-module M, the conditions 1 Extj,(V, W ® M) = 0 (i > 0), and 2 The canonical map p : Homc(V,W) ® M -> Homc(V,W ® M) is an isomorphism are satisfied. Lemma 2.2.3 Lei V be an R-finite R-projective C-comodule, and W a Ccomodule. Then W is u-acyclic with respect to V if and only if the complex of R-modules Cobarc(iy, V*) is u-acyclic. Proof. By Lemma 1.3.6.16, we have Ext'c(V, W ® M) S H\Coba.TC{W ® M, V ) ) & ifi(Cobarc(VK, V*) ® M) for any i?-module M and any i > 0.
D
Lemma 2.2.4 T/ie follovnng hold: 1 Let V and W be C-comodules. If W is u-acyclic with respect to V, then W<8>M is u-acyclic with respect to V for any R-module M. IfV is R-finite protective, then for any commutative R-algebra R —> R', W := W <8> R! is u-acyclic with respect toV = V ® R!, as a C" = C ® R'-comodule.
2. Donkin systems
185
2 The class of C-comodules which are u-acyclic with respect to V is closed under direct sums and direct summands. 3 Let 0 -t W\ ->• W2 -> W3 -> 0 be an exact sequence of C-comodules. If W\ is u-acyclic with respect to V and W\ •—> W2 is R-pure, then W3 is u-acyclic with respect to V if and only if W2 is u-acyclic with respect to V. 4 Assume that V is R-finite and R is noetherian. Then any filtered inductive limit of C-comodules which are u-acyclic with respect to V is again u-acyclic with respect to V. 5 For a fixed C-comodule W, the class of C-comodules V with respect to which W is u-acyclic is closed under extensions, direct summands, and epikernels. Proof. 1 Let AT be an fl-module. Then Extk(V, (W ® M) <8> N) = 0 (i > 0) is obvious. The canonical map p : Homc(Vr, W ® M)
which is an isomorphism. We prove the second statement of 1. By Lemma 2.2.3, Cobarc(Vy, V ) is u-acyclic. By Lemma 2.1.4, Cobarc(VF, V") ® R' is a u-acyclic complex over R'. By Lemma 1.3.6.15 and Lemma 2.2.3, we are done. Statement 2 is obvious. We prove 3. Let M be an arbitrary i?-module. As W\ •—> W2 is ii-pure,
is exact. Then by assumption, we have Ext' c (V, W2 ® M) S* Extj,(Vi W3 ® M) for i > 0. Now consider the commutative diagram: Homc(V, Wi) ® M —> Homc(V, W2) ® M -¥ Homc(V, W3) <8> M —> 0
4, a-
4-
4
0 —>• Homc(V, W\ ® M) —> Homc(y, W2 ® M) —> Homc(V, W3 igi M) —> 0 The rows are exact and the first vertical arrow is an isomorphism, since Wx is u-acyclic. By the five lemma, the second vertical arrow is an isomorphism
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if and only if the third is. Hence, we have W3 is u-acyclic with respect to V if and only if W2 is. Statement 4 follows immediately from Proposition 1.3.7.4. 5 It is clearly closed under direct summands. Now let 0 -+ Vx -» V2 -> V3 -> 0 be an exact sequence of C-comodules such that W is u-acyclic with respect to V3. Then for any i?-module M, we have Exf c (V 2 , W <8> M) a Extj^Vi, W <8> M)
(i > 0),
as W is u-acyclic with respect to V3. Moreover, we have a commutative diagram: Homc{V3, W) ® M -> Homc(V2, W) ® M -» Homc(Vi, W) ® M -> 0 0-> Homc(V3, W ® M) -4 Homc(V2, W® M) -> Homc(Vi, VK® M) -»• 0 The rows are u-acyclic with u-acyclic with is, the class in
exact and the first vertical map is an isomorphism, as W is respect to V3. By the five lemma again, we have that W is respect to V2 if and only if it is so with respect to Vj. That question is closed under extensions and epikernels. 0
Lemma 2.2.5 Let R be a noetherian ring, V an R-finite R-projective Ccomodule, and W an R-projective C-comodule. Assume that for any p G Spec R, we have Ext*, (p) (V(p),W(p)) = 0
(x>0)
and dimK(p) Homc( P )(V(p),iy(p)) is a locally constant function on SpecR with finite values. Then W is u-acyclic with respect to V, and Homc(W, V) is R-finite projective. Proof. By assumption, F := Cobar(W, V*) is an /?-flat complex, and F° = W ® V* is .R-projective. For p € Spec R, we have = #'(Cobar c ( p ) (W(p), ( V
by Lemma 1.3.6.15 and Lemma 1.3.6.16. By assumption and (2.1.17), we have that Cobarc(W, V*) is a u-acyclic complex. By Lemma 2.2.3, W is u-acyclic with respect to V. • Definition 2.2.6 We say that a triple {X+, A,V) is a semisplit highest weight theory over C, if the following are satisfied:
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187
D l X+ is an ordered set which satisfies the conditions in Lemma 1.1.2, and A = (A c (A)) A6 x+ and V = (Vc(A)) Ae x+ are families of .R-nnite /l-projective C-comodules, indexed by X+. D2 For any p € SpecR, (C{p),X+, A(p), V(p)) is a split highest weight coalgebra over /t(p). We also say that C, or better (C, X+, A,V), is a semisplit highest weight coalgebra over /?. We call X+ = X£ the set of dominant weights, Ac(A) the Weyl module of highest weight A, and Vc(A) the induced module of highest weight A. We say that (X+, A,V) is split if A C (A), VC(A) and R(X) := Hom c (A c (A),Vc(A)) are i?-free modules for all A G X+. Let (X+, A,V) be a semisplit highest weight theory over C. We say that C = (CV) is a Donkin system of C associated with ( X + , A , V ) , if (CV) is a family of i?-subcoalgebras of C indexed by all finite poset ideals n of X+, and the conditions a C 0 = 0, b ?r' C vr implies CV' C CV, c If 7r is a finite poset ideal of X+ and A is its maximal element, then C J C v s fl(A)* ® A C (A)'
D
Lemma 2.2.8 / / (X+, A, V) is a semisplit highest weight theory over C and R -> R! is a homomorphism of noetherian commutative rings, then (X+, A ' , V ) is a semisplit highest weight theory overC. If, moreover, (Cw) is a Donkin system over C, then (C'n) is a Donkin system of C, where (?)' denotes the functor ? ® R!.
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Highest Weight Theory
Proof. Condition D l is obvious. Condition D2 follows from Corollary 1.4.8. Note that the construction of R(X)* is compatible with the base change because of Lemma 2.2.7. Hence, the last assertion is obvious. D Lemma 2.2.9 Let {X+, A, V) be a semisplit highest weight theory over C. Then (X+, V*, A*) is a semisplit highest weight theory over Cop. If, moreover, (CT,) is a Donkin system of C, then (C°p) is a Donkin system ofCop. Proof. Condition D l is obvious. Condition D2 follows from Theorem 1.4.7. The last assertion is checked easily, as we have i?.(A)Cop = Homc<,P(Vc(A)*, A C (A)') a Hom c (A c (A), V C (A)) = R(X)C.
a (2.2.10)
The rest of this subsection is devoted to proving the following.
Theorem 2.2.11 Let R be a noetherian commutative ring, and C an Rprojective R-coalgebra. If (X+, A, V) is a semisplit highest weight theory overC, then there is a Donkin system (Cw) ofC associated with {X+, A, V). In the rest of this subsection, let R be a noetherian commutative ring, C an i?-projective .R-coalgebra, and (X+, A, V) a semisplit highest weight theory over C. We construct (Cn), in order to prove the (existence) theorem. The construction is inductive. For clarity, we state what we are actually going to prove. Proposition 2.2.12 Let n > 0. There exists a family (CV)#,r
2. Donkin systems
189
Note that a family (C^)#^
for i > 0 and fi ft A. It follows that Ext*, (p) (V c (A)(p), ( C / a - ) ( p ) ) = 0
(i > 0).
On the other hand, by Proposition 1.3.2, the dimensions of Hom c ( p ) (Vc(A)(p),(C/a-)(p)) * Hom c(p) (A c (A)(p),C(p)) are finite and locally constant on Spec R. By Lemma 2.2.5, we have that C/Cni is u-acyclic with respect to Vc(A). • The last assertion is obvious by Lemma 2.2.5 and Proposition 1.3.2. Lemma 2.2.14 The canonical maps
® R(X) are all isomorphisms of left C-comodules, where the <— is the map obtained by the composition AC(A) -> Vc(A) -
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Proof. It is obvious that these maps are left C-comodule maps. By Lemma 2.2.13, the modules in question are /^-finite projective modules with the same rank functions, and so these maps are isomorphisms, if they are surjective. This is checked after tensoring with n(p). The assertion is now obvious by Lemma 2.2.7, Lemma 2.2.13 and Proposition 1.3.2. D Lemma 2.2.15 The canonical pairing
p' : Homc(Vc(A), C/G') ® VC(A) -+ C/Gis an R-pure (in particular, injective) (C,C)-bicomodule map. Proof. As the modules in question are .R-projective, it suffices to check that the map in question is injective after tensoring with n(p) for all p € Spec R, by Lemma 1.2.1.4. As we have (C/G>){p) ^ C(p)/C(p)(ir'), this is clear by D Lemma 2.2.13 and Proposition 1.3.2. Now we define Cn to be the pull-back of Im p' by the projection C —> C/Cn>. By Lemma 2.2.15, we have C/G = Cokerp' is fl-flat. Hence, G is an .R-pure (C, C)-subbicomodule of C, namely, an /?-subcoalgebra. By Lemma 2.2.14 and Lemma 2.2.15, the isomorphism G/C^ = -R(A)* ® AC(A)* <8> VC (A) is obvious. Now we claim that the definition of G is independent of the choice of a maximal element A of TT. Then conditions a-d are obvious by construction, and the proof of Proposition 2.2.12 is complete. So let n ^ A be another maximal element of ir. We keep on using A to define G, to avoid ambiguity. Set TTQ := TT \ {A, /x}, TT0 := TT \ {/x}, and w' := 7T \ {A} (as above). These are finite poset ideals of n. By Lemma 2.2.5, it is easy to see that Extc(Vc(A), VC(AO) = 0 f° r i > 0. As we have Cn'/G'o S add VC(M) by induction assumption, we have
for i = 0,1. Consider the exact sequence
o -> G'lG; -* C/G', *•» C/G' ->• 0. By an easy long exact sequence argument, the canonical map
Homc(Vc(A),C/a<) -> Hom(V(A)C/a0 induced by p is an isomorphism. By the definition of Cn and the induction assumption, we have that the image p(CVo/C^ of GalG^ by P agrees with G/C^. This is equivalent to saying that Go + G' — Cn. Hence, even if we replace A by fj, and TT' by 7r0, the same construction implies Cv + Go — G- Hence, the definition of G is independent of choice of A, and the proof of Proposition 2.2.12 is complete. The proof of Theorem 2.2.11 is also complete. D
2. Donkin systems
191
2.3 Basic properties of the Donkin system (2.3.1) Let R be a noetherian commutative ring, C an /t-flat coalgebra, (X+, A,V) a semisplit highest weight theory over C, and (C*) a Donkin system of C associated with (X+, A, V). For a (possibly infinite) poset ideal TT of X, we define Cn := limC p , where p runs through all finite poset ideals of TT. It is easy to see that Cn is an 7?-flat i?-subcoalgebra of C. Lemma 2.3.2 For any poset ideal w of X+ and any p € Spec R, we have C,(p)
= C(P)(TT).
Proof. By Lemma 2.2.8, (CV(p)) is a Donkin system of C(p). By the uniqueness Lemma 1.3.6, the assertion is obvious if n is finite. As the base change preserves inductive limits, the lemma is also true for general TT. D
Lemma 2.3.3 Cx+ = C. Proof. As the canonical injection Cx+ '-* C is .R-pure and Cx+ is i?-flat, we have that C/Cx+ is i?-flat. By Lemma 2.3.2 and Lemma 1.3.6, d, we have (C/Cx+)(p) = 0 for any p € Speci?. Hence, by Corollary 1.2.1.6, we have C/Cx+ = 0 . D L e m m a 2.3.4 For any poset ideal TT of X+, we have that Cn is R-countable and IFP, and hence it is R-projective. In particular, C is R-countable, IFP, and R-projective. Proof. It is clear that Cn = limC p is IFP, and it is /^-countable, since IT is countable. It is .R-projective by Lemma 1.3.11.3. The last assertion is D obvious by Lemma 2.3.3. An object of the category add ^"(A) (resp. add^"(V)) is called a A-good (resp. V-good) C-comodule, see (I.I.11). Note that A-good comodules and V-good comodules are J?-finite .R-projective. If n is a finite poset ideal of X+, then CT, is V-good, and hence is .R-finite .R-projective. The dual algebra C* of CV is denoted by S*. Note that Sn is A-good. By Lemma 2.2.4, if V is A-good and W is a filtered inductive limit of V-good comodules, then W is u-acyclic with respect to V. Lemma 2.3.5 Let TT be a poset ideal of X+. are equivalent.
For V € M c , the following
1 V is a Cr-comodule. 2 For any X € X+ \ TT, we have Hom c (Ac(A), V) = 0.
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III. Highest Weight Theory
Proof. 1=^2 As V «-> V
D
Corollary 2.3.6 We have V C (A), AC(A) e Mc<-°°.*1. Proof. The assertion for Vc(A) is obvious by the lemma. Hence, by Lemma 2.2.9, we have AC(A)* e M 0?--.*!. This shows AC(A) = A C (A)" e Mc(-~.*l. D Corollary 2.3.7 If IT is a poset ideal of X+, then Mc* is closed under extensions in M.c. Proof. Follows immediately from the lemma.
D
Proposition 2.3.8 Let V be a C-comodule. Then the following are equivalent. 1 For any A G X+ and i > 0, we have Extj 7 (A c (A), V) = 0. 2 For any poset ideal TT of X+ and i > 0, we have Rl ind£* V = 0. 3 For any A G X+, we have ExtJ^ActA), V) = 0. 4 For any finite poset ideal TT of X+, we have R1 ind£* V = 0. 5 For any bijective order-preserving map f : X+ —>• [X+], there exists a filtration 0 = Vo C V\ C • • • C V such that the following conditions are satisfied:
2. Donkin systems
193
b Fori > 1, there exists an isomorphism of C-comodules Vt/Vi-i ^ R(\{i)T
® Hom c (A c (A(i)), V) ® VC(A(*)),
where \(i) := / ^ ( z ) and R(\(i))
:= Hom c (A c (A(i)), V c (A(i))).
7/
It is obvious that 0 = Vo C Vx C V2 C • • • C V. When we set \{i) := for i £ [-X^+], then we have an exact sequence 0 -4 R(\) ® V c (A(i))* ® A c (A(i)) -4 5 , w -> 5 ^ ! , -> 0. Set iVj := Homc (fi(A) ® V c (A(t))' ® A c (A(t)), K). Then 0 -> V4_i -> Vi -* N{ -)• i? 1 indc*"" 1 ^^) = 0 is exact. As 8i Hom c (A c (A(i)), V) (g) V c (A(i)) as C-comodules, condition b is satisfied. We prove condition a. Let W be an i?-finite C-subcomodule of V. Then the image of u) : W —> W <8> C is contained in W ® Cff for some finite poset ideal n of A"+, by Lemma 2.3.3. As n is contained in ir(i) for some i, we have W = indc"(>) W C VJ for some z. Hence, lim V^ contains any i?-finite
194
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Highest Weight Theory
C-subcomodule of V, and it agrees with V. Now the conditions a and b are verified for V* := res c * (t) indc " <>)(V), and the existence has been proved. 5=>1 As VC(M) is u-acyclic with respect to Ac(A) for any A , / i £ X+, we have Exf c (A c (A),V-/V$_ 1 ) = 0 (t,j>0). Hence, by Proposition 1.3.7.4, Ext' c (A c (A), V) s l i m E x t ^ A ^ A ) , V,) = 0
(* > 0).
Now we have that 1—5 are equivalent. We prove the uniqueness of the filtration in 5. To verify this, assuming that (V<) is a filtration which satisfies conditions a, b' and Vo = 0, it suffices to prove V* = ind c * w V by induction on i. If i = 0, then the assertion is obvious. Let i > 0, and assume that the uniqueness holds up to i - 1. By induction assumption, V^i is a C^y comodule, and by assumption b' and Corollary 2.3.6, V5/VJ_i is also a C^comodule. By Corollary 2.3.7, Vt is a CV^-comodule. Hence, V{ C ind c " <0 V. If j < i, then Hom c (A c (A(j)), V/Vi) = 0. This is because Hom c (A c (A(j)) ) y 1 /Vi)=0 for / > i (proved easily by induction on /) and limV; = V. On the other hand, if j > i, then Hom c (A c (A(j)),ind^ ( 0 V/VJ = 0. This is because ExtJ.(A c (A(j)), V<) = 0 by u-acyclicity of V c (/) for 1 < / < i with respect to A c (A(j)), and Homc(A c (A(j)),ind c T(l) V) = 0 because A(i) i 7T. By Lemma 2.3.5, indc' (i) V/Vi is a C(0)-comodule, and V{ = indc*(i) V. This completes the proof of the uniqueness. • Remark 2.3.9 When we set A^(A) := R(\)
® A C (A), V C (A)) S R(X)* <8) R(X) & R,
for any A £ X+. As we have ® AC(A)*
2. Donkin systems
195
Corollary 2.3.10 A Donkin system of C associated with (X+,A,V) is unique. Namely, if (C'n) is a Donkin system associated with (X+, A, V), then for any finite poset ideal TT of X+, we have 0^ = 0^. Proof. We have an order-preserving bijective map / : X+ —> [X+] such that /(TT) = [1, #TT] by Lemma 1.1.2 (applied to TT and X+ \ TT). As C[ := C'r{i) -1 (TT(I) := / [ l , i ] ) satisfies the conditions a and b' in Proposition 2.3.8 by the definition of Donkin system and Lemma 2.3.3, we have C[ = ind c " (t) C = C , w . Hence, C'T = C'#1T = C»(#») = Cn. D (2.3.11) Now we have the following by Theorem 2.2.11, Lemma 2.3.4, and Corollary 2.3.10. Theorem 2.3.12 Let R be a noetherian commutative ring, C an R-flat coalgebra, and (X+, A, V) a semisplit highest weight theory over C. Then the following are equivalent. 1 There is a Donkin system of C associated with (X+, A, V). 2 There is a unique Donkin system of C associated with (X+, A, V). 3 C is an R-projective module.
a (2.3.13) Let R be a noetherian commutative ring, C an .R-projective Rcoalgebra, and (X+, A,V) a semisplit highest weight theory over C. Let (CV) be the Donkin system of C associated with (X+, A, V). For a (possibly infinite) poset ideal of n of X+, we denote limC,, by Cn, where p runs through all finite poset ideals of n. We call Cn the Donkin subcoalgebra of C with respect to TT. For a finite poset ideal n of X+, we call the dual algebra Sn the Schur algebra of C with respect to ?r. These are determined only by (C,X+,A,V). Lemma 2.3.14 Let V be a C-comodule. Then the following are equivalent. 1 For any A € X+, V is u-acyclic with respect to Ac(A). 2 V is good, and V* := ind£" V •-> V is R-pure for any finite poset ideal TT ofX+. 3 There exists a filtration 0 = Vo C Vx C V2 C • • • C V
ofV such that each V{ is an R-pure submodule ofVi+i, limV^ = V, and for each i > I, there exist some A 6 X+ and an R-module M such that
Vi-i s VC(A) ® M.
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III. Highest Weight Theory
We say that V is u-good if the conditions above are satisfied. Proof. 1=>2 It is obvious that V is good. To prove the it-purity of Vn in V, we may assume that TT is finite. By Lemma 2.2.4, V is u-acyclic with respect to Sn. Hence, for any it-module M, V, ® M = H o r n e d , V) ® M S H o m c ( ^ , V ® M) = (V ® M ) , is a C-subcomodule of V ® M in a natural way. Hence, Vn '—¥ V is it-pure. 2=>3 We take and fix an order-preserving bijective map / : X+ -¥ [X+]. Then Vt = mdc{~1{>''] V is an it-pure submodule of V. Hence, V{ •—> Vi+i is it-pure. By Proposition 2.3.8, the filtration 0 = Vo C Vi C • • • is the desired one. 3=>1 For any A £ X+, Vi/VJ_i is u-acyclic with respect to Ac(A). As Vi-\ is an it-pure submodule of Vi, we have that Vi is u-acyclic with respect to Ac(A) by Lemma 2.2.4 and induction on i. Hence, V = limV^ is also u-acyclic with respect to Ac(A), again by Lemma 2.2.4.
•
The next is obvious. Lemma 2.3.15 The set of good C-comodules is closed under extensions, direct summands, monocokernels, andfilteredinductive limits. C and Vc(A) (A € X+) are u-good. The set of u-good C-comodules is closed under direct summands and filtered inductive limits. If
is an exact sequence of C-comodules, W\ is u-good and i is R-pure, then W2 is u-good if and only if W$ is u-good. If V is a good (resp. u-good) Ccomodule and TT is a (possibly infinite) poset ideal of X+, then ind^" V and V/ind£* V are good (resp. u-good). IfV is u-good, then V®M is u-good for any R-module M. IfVis u-good and R—tR'isa noetherian commutative R-algebra, then V ® R' is a u-good C ® R'-comodule. Lemma 2.3.16 For any poset ideal TT of X+ and V € M c ", we have & indg" V = 0 ( t > 0 ) . Proof. By Corollary 1.3.6.19, Lemma 1.3.6.17 and Lemma 1.3.6.16, we have isomorphisms g' V R* indg' V SS Cotor^(V Cotor^(V, C) Cn) ££ limCotor'(V(p) limCotor' c (V(p), C) Cp) 8* 8* lim/T lim/T indg indg" V(p), where p runs through finite poset ideals of TT. SO replacing TT by each p, we may assume that n is finite.
2. Donkin systems
197
As Ci, is u-good, it is u-acyclic with respect to Sn. Hence, the cobar resolution Cobarc, V of V as a CV-comodule is an indg* -acyclic resolution of V as a C-comodule. On the other hand, we have indg*(CobarC)r V) = Cobar c , V, and hence we have R' indg* V = 0 (i > 0). Theorem 2.3.17 Let n be a poset ideal of X+, comodules. Then the canonical map
D and V and W be C w-
is an isomorphism for i > 0. Proof. Consider the isomorphism Homc(V, ?) = Homc,(Vr, ?) o indg* of functors on M c . As indg* has an exact left adjoint resg", it preserves injectives. Hence, we have an isomorphism of derived functors R+ Hom c(V, ?) £ R+ Home, (V, ?) o R+ indg'. As we have R+ indg" W = W by Lemma 2.3.16, we have the desired isomorphism, taking the cohomology. • Corollary 2.3.18 If n is a poset ideal of X+, then (n, A(TT), V(TT)) is a semisplit highest weight theory over Cn, and its associated Donkin system is {Cp)p, where A(TT) = ( A C ( A ) ) A € ^ and V(TT) = (V c (A)) Aew , and p runs through all finite poset ideals of -n. Corollary 2.3.19 IfV and W are R-finite C-comodules, then Ext^(V, W) is R-finite for i > 0. Proof. As we have C = Cx+, there exists a finite poset ideal ir of X+ such that both V and W are C^-comodules. By the theorem and the i?-linear equivalence s,M = M c ", we have /^-isomorphisms
^ F , W) S Ex4(K, W) S Ext^(K, W). As S* is an fl-finite algebra, Ext^(V, W) is .R-finite, and so is Ext^K, W).
• Lemma 2.3.20 If V and W are R-finite C-comodules, then the canonical map ip{ : Ext^ c (V, W) -> Extj^V, W) is an isomorphism for i > 0.
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III. Highest Weight Theory
Proof. We may assume that i > 1. There exists some finite poset ideal IT of X+ such that both V and W are Cr-comodules. For a finite poset ideal p of X+ containing IT, we have M,(V,
W) S Extfc,(V, W) = Exfc(V, W)
since M ^ = s p M; contains enough 5p-projectives and by Theorem 2.3.17. When we denote the canonical map limExt' cp(V, W) -» ExtJVIc(y,iy) by ip*, then the last paragraph shows that y>1 o ij}1 is an isomorphism. So it suffices to show that ip% is surjective. Let a £ ExtlMc(V, W), and 0 - • W -> Mj ->
•Mi->V-»0
be a representative of a. Then as there is some p D IT such that ©^ M, is a Cp-comodule, a is already realized as an exact sequence of Cp-comodules. Hence, ip1 is surjective. D
3
Ringel's theory over a field
3.1 Ringel's approximation over a field (3.1.1) over k.
Let k be a field, and (C, X+, A, V, L) a highest weight coalgebra
Lemma 3.1.2 If X € X+, then we have inj.dimc Vc(A) < cohtA. Proof. We may assume that coht A < oo, and we prove the lemma by induction on n := coht A. Consider the exact sequence 0 -> VC(A) -> QC(A) -> QcW/VcW
-» 0.
Then by Lemma 1.4.6, Qc(A)/Vc(A) has a good filtration, and we have [Qc(A)/V c(A) : VC(M)] = 0 if p. ^ A. In particular, we have Vc(A) = QcW if n = 0, and the assertion holds. Assume n > 0. If p, > A, then coht/z < n. Hence, for any finite dimensional C-comodule V, we have Ext£(V, Qc(A)/Vc(A)) = 0 by induction assumption. As QC(A) is C-injective, Ext£+1(V, VC(A)) = 0. As M c is locally noetherian, inj.dimc Vc(A) < n by Lemma 1.1.9.4. D Lemma 3.1.3 If X & X+, then inj.dimcLc(A) < rank^V"1" 4- ht A. In particular, gl.dim C < 2 rank X+.
3. Ringel's theory over a held
199
Proof. We may assume that r a n k X + < oo. We prove the first assertion by induction on n = ht A. The sequence 0 -> LC(A) -> V C (A) -> V C (A)/L C (A) -> 0 is exact, and any composition factor of Vc(A)/Lc(A) is of the form Lc(fi), with fi < X. Hence, Lc(A) = Vc(A) if n = 0, and the assertion holds by Lemma 3.1.2. If n > 0, then inj.dimcLc(fj) £ r a n k X + + n — 1 for /t < A by induction assumption. Combining this with inj.dim c Vc(A) < rankX"1", we are done. We prove the last assertion. By the first part and Proposition 1.3.7.4, Ext2c™nkX++1(V,W)
=0
for any finite dimensional C-comodule V and any C-comodule W. By Lemma 1.1.9.4, inj.dim c W < 2 r a n k X + . O We set Ac •= M?, yc := ^"(V), Xc := ^ ( A ) , and w c := <*c n 3^cLemma 3.1.4 A C-comodule V belongs to yc = .F(V) i/ and only if V is good and finite dimensional. Proof. Obvious by Proposition 1.3.2.
D
Lemma 3.1.5 3fc is closed under extensions, monocokernels, and direct summands. Xc is closed under extensions, epikernels, and direct summands. Proof. Thefirstassertion is obvious by Lemma 3.1.4. We prove the second assertion. By Theorem 1.4.7, (Cop,X+, V*, A",L*) is a highest weight coalgebra, and for V G Ac, we have V € Xc if and only if V* e Jfc°p by definition. By the first part, we are done. D Lemma 3.1.6 If V £ Ac, then Ac-resol.dimV < oo. Hence, we have Xc — •AcProof. There exists a finite poset ideal 7r of X+ such that V 6 M C (TT). By Lemma 3.1.3 applied to C(TT), there is an Sc(7r)-finite 5c(7r)-projective resolution P —> V of V of length at most 2 rank TT. AS we have Sc(ft) € Xc, each term of P also belongs to Xc by Lemma 3.1.5, and we are done. D Lemma 3.1.7 Let V 6 Ac and A € X+. Set X:={W€Ac\
ExtJ,(Ac(A), W) = 0}.
Then there exists an exact sequence (3.1.8)
a:0-»V-Ut/->D->0
in Ac such that U £ X, D G add{Ac(A)}, and i is left minimal. If a is such an exact sequence, then i is the left minimal X-approximation of V, and in particular, such an a is unique up to isomorphisms of complexes.
200
III. Highest Weight Theory
For the left minimality and left minimal -^-approximation, see (1.1.12). We call the exact sequence a in the lemma the minimal X-extension of V. Proof. We denote the finite dimensional divisionfc-algebraEndc(Ac(A)) by E, see Lemma 1.2.20. By Corollary 2.3.19, Ext1c(Ac(X), V) is finite dimensional over k, and is alsofinitedimensional as a right E- vector space. We take a basis a\,...,ar of it as a right £-space. We set D = 0 j = 1 Ac(A), and we define the exact sequence (3.1.8) to be the element of Extc(£>, V) which corresponds to a := {a\,...,aT) by the canonical isomorphism Extlc(D, V) S ® ExtJ,(Ac(A), V). i
As the induced map a': Homc(Ac(A), D) -> Ext^(Ac(A), V) is right Elinear, a' is surjective by the choice of a (hence it is also an isomorphism by dimension counting). As Ext^(Ac(A),.D) = 0, we have Extc(Ac(A),L/) = 0, in other words, U G X. By the definition of X, we have Extlc(D, X) = 0, and hence i is a left -f-approximation by the dual assertion of Lemma 1.1.12.1. We show that i is left minimal. Let ip G Endc U with tpi = i, and ip : D —> D be the map induced by
3. Ringel's theory over a held
201
Proof. Left to the reader as an exercise.
D
Lemma 3.1.11 IfV€ Ac, then there is an exact sequence Q-+V UY -+X ->0 in Ac such that Y € 3fc, X e Xc, and Us a left minimal yc-approximation, uniquely. More precisely, such an exact sequence is obtained as follows. 0 There exists a finite poset ideal n of X+ such that Extc(Ac{p), V) = 0 for fi $L IT. Fix such a finite poset ideal TT, and set n := #7r. 1 Fix an order-preserving bijective map f : n -> [l,n]. Set X(i) := / - 1 ( i ) . 2 Define exact sequences a(i) :0->V^%
Y(i) -> X{i) -> 0
fori G [l,n+l] by descending induction oni. Seta(n + 1) to be the exact sequence 0-+V^V->0^>0. Assume that i S [1, n], and a(i + 1) is already defined. Let 0(i) : 0 - + y ( i + l ) ^ Y(i) -> D(i) -> 0 be the minimal X(i)-extension of Y(i + 1). Set p(i) := n(i) o p(i + 1), and X(i) := Cokerp(i). Now a(i) is defined. 3 Forie [1, n + 1], we have X(i) 6 ^"({Ac(A(j)) | i < j < n}) and p(i) is left minimal. 4 7/t€[l,n + l] andXtX+Xf-HllJ-l}),
then Ext^(Ac(A),y(i)) = 0.
5 a(l) is the left minimal yc-approximation, and X(l) G Proof. 0 Take n so that V e MCM. Then the desired property is satisfied. 1 is always possible. 2 makes sense, by the uniqueness of the minimal A(i)extension. We prove 3 by descending induction on i. If i = n+1, then the assertion is trivial, so assume that i € [1, n]. Note that D(i) 6 add Ac(A(i)). As there is an exact sequence 0 -> X(i + 1) -> X(i) -> jD(t) -> 0, the first assertion follows. We are to prove that p(i) is minimal. So let (p : y(i) -+ y(i) be a C-comodule map such that tpp(i) = p(i). As we have A(j) ^ (—oo, A(i)]
202
III. Highest Weight Theory
for j > i, we have Uomc{X(i + l),D(i)) = 0 by the first assertion. The composite
Y(i +1) - ^ Y(i) £> Y(i) -> D(i) is zero on V = Imp(i + 1). The induced map X(i + 1) = Coker/9(i + 1) -> D(i) is also zero, so the composite above is zero. It follows that ip maps Y(i + l) to Y(i + 1). As p(i + l) is left minimal, the restriction rjj :=
Theorem 3.1.12 (Ringel) (Xc,yc,^c) text in Ac-
is on Auslander-Buchweitz con-
Proof. As we know the condition Ac = <*c by Lemma 3.1.6, it suffices to check that {Xc,yc,vc) is a weak Auslander-Buchweitz context. Namely, it suffices to check the conditions AB1-AB3 in Theorem 1.1.12.10. A B l is contained in Lemma 3.1.5. AB2 is also trivial by Lemma 3.1.5 and Ac = Xc. As w c = Xc H ^ c is the definition, it remains to prove that we is an injective cogenerator of Xc. Let V € Xc- Then Yv G yc- On the other hand, as Xv £ Xc and Xc is closed under extensions, Yv e XcHence, Yv £ we with Xv £ Xc, and we is a cogenerator of Xc- As Extlc(Xc, yc) = 0 (i > 1) and CJC C yc, ^c is -*c-injective. d Note that as End c V is a finite dimensional fc-algebra (hence is semiperfect) for any V £ Ac, we have that any V £ Ac admits a unique minimal Ac-approximation and a minimal ^c-hull.
3.2
Tilting modules over a field
(3.2.1) Let ik be a field, and (C, X+, A, V, L) a highest weight coalgebra over k. Set Xc := ^(A), yc := ^ ( V ) , and wc := Xc fl ^ c
3. Ringel's theory over a field
203
L e m m a 3.2.2 Let X £ X+, 0 -> AC(A) -» TC(A) -> XC (A) -> 0 6e £/ie minimal yc-hull of Ac(X), and 0 -> Yc{\) ->• T^(A) -> V C (A) -> 0 6e
of Vc(A). 77ien we ftaue:
(3.2.3) Tc(A) is an indecomposable object of we(3.2.4) (TC(A) : LC(A)) = 1, and (TC(A) : L C ( M ) ) ^ 0 implies n < A. (3.2.5) i4ny indecomposable object of uc is isomorphic to 7c(A) for some A. More precisely, ifT is an indecomposable object ofwc, then the set {n € X+ | (T : Lc{fj)) ^ 0} Aos a roarimum element, say A, and we have (3.2.6) 7£(A) S TC(X). (3.2.7) Assume that C is finite dimensional, and let A denote the quasihereditary algebra C*. When we set T := © Ae x+ ^c(A), then T is a basic tilting-cotilting module of A, and we have a d d T = u>A. Proof. Set n := (-oo,A). Then Ext£(Ac(M)» A C (A)) = 0 for \i $ n. By Lemma 3.1.11 XC (A) e ^ ( { A C ( M ) IM 6 w}). Hence, (XC (A) : L c (/i)) = 0 for n <£. A. This proves (3.2.4). Next, we decompose TC(A) = Ti©- • -©T,., with T{ indecomposable. This is possible, see Lemma 1.1.12.6. As (Tc(A) : Lc(X)) = 1, we may assume
(Ti(X) : LC(X))
= 1 a n d (T>(X) : LC(X))
= 0 ( i > 2). T h e n a s
Hom c (A c (A),T i ) = 0
(* > 2),
we must have i = 1, by the left minimality of Ac(A) -> Tc(X). (3.2.3) holds. We prove (3.2.5). Set
Hence,
J := add{T c (A) | A G X+}. Then by Corollary 1.1.12.13 applied to Xo := A, u/ is an injective cogenerator of Xc. By Theorem 1.1.12.10, 2, w' = addw' = w. Now the first assertion follows from the Krull-Schmidt theorem, Lemma 1.1.12.6. The last assertion is an immediate consequence of the first assertion and (3.2.4). We prove (3.2.6). The exact sequence
o -> v c (A)' -* rc(xy -> Yc(xy -> o
204
III. Highest Weight Theory
is a minimal ^ - h u l l of the Weyl module Vc(A)* for the highest weight coalgebra
(C op ,X + ,V,A*,Z/). Hence, TQ(X) G uc is indecomposable, and the set {n G X+ \ {TC(X) : Lc(fj)) 7^ 0} has A as its maximum element. Hence, we have TQ(X) = Tc(X) by (3.2.5). We prove (3.2.7). It is clear by Theorem 1.4.10.22 that T is the (uniquely determined) basic cotilting module of A which satisfies addT = 01,4. On the other hand, we have addT* = <jAoP, and T is also a tilting module. D We call Tc(A) the indecomposable tilting module of C of highest weight A. Proposition 3.2.8 Let k be afield, C a k-coalgebra, and (X+, A, V) a split highest weight theory over C. Let IT be a poset ideal of X+, and F:0-> V ->X -*Y -40 a sequence in Mc(n)f. Let k1 be an extension field of k, and ?' denote the functor ? ® k!. Then the following are equivalent. \ ¥ is a minimal yc(-n)-hull ofV in
Mc(n)f.
2 F is a minimal yc-hull ofV in Mj. 3 F* is a minimal Xc»p-approximation ofV* in M^°P. 4 W is a minimal yc>-hull ofV in Mf. Proof. Note that F is exact if and only if F* is exact if and only if F' is exact. For X € Mc(ir)f, X G Xc if and only if X € XC(V) if and only if X* E yc«t>. As we have
pf', Vc(A)) 3 Ext^(X, VC(A))' for any A e X+, and (?)' is faithful exact, X G Xc if and only if X' G Xc>. As we have a similar equivalence for Y, we have the desired equivalence, except for the assertions on the minimality. As for minimality, 1*»2^3'*=4 is obvious. We prove 2^»4. By Lemma 3.1.11, it suffices to prove that, for V € Ac, if G : 0 -4 V -» W -> D -4 0 is a minimal A-extension of V, then G' is a minimal A-extension of V. By assumption, we have Ext^(A c (A),lV) = 0 and D £ A c (A) e r , where
4. Ringel's theory over a commutative ring
205
r = dimfcExtJ7(Ac(A),V) (note that k = End c A c (A), as we consider a split highest weight coalgebra). Then we have E x t ^ A ^ A ) , W) s ExtJ7(Ac(A))iy)' = 0, and D' 3 Ac<(A)®r. As we have dim*. Ext^(Ac .(A), V") = dim*- Ext^(Ac(A), V)' = r, G' is a minimal A-extension of V by Lemma 3.1.9.
•
Corollary 3.2.9 Under the assumptions of the proposition, we have the following for A G X+. 1 For any poset ideal n of X+ such that A G ir, we have Tc(n){X) = Tc{\). 2 Tc(\y = TCoP(\). 3 If k' is an extensionfieldof k, then we have 7c®fc'(A) = 7c(A) ® A;'. Remark 3.2.10 All the results in (IIM) and (III.3) are well-known as the theory of highest weight category and the theory of quasi-hereditary algebras, see [37], [45], [44], and [52]. See the appendix of [52], which contains substantial historical remarks. We have used thorough coalgebra-comodule approach, which is useful in generalizing the base ring to an arbitrary commutative noetherian ring.
4
Ringel's theory over a commutative ring
4.1 Tilting modules over a commutative ring (4.1.1) Let R be a noetherian commutative ring, and C an i?-projective .R-coalgebra. Let (X+, A, V) be a semisplit highest weight theory over C, and (Cr)* i t s associated Donkin system. We set Ac •= Lemma 4.1.2 Let A be a left noetherian R-algebra, R' an R-flat commutative R-algebra, M e AMf, and N G ,iM. Then the canonical R!-linear map ExVA(M,N)' -> Ext?A,{M',N') is an isomorphism for i>0, where (?)' denotes the exact functor ? ® R'. Proof. The case i = 0 and M = A is trivial. The case i = 0 follows from the five lemma. Let F be a projective resolution of M with each term i4-finite. Then we have #i(Hom/i(F,./V))' S //i(Hom/i(F,yV)') a
Hi(HamA.(F,N1)),
and as F' is an /l'-projective resolution of M' by flatness, we are done.
•
206
III. Highest Weight Theory
Lemma 4.1.3 ([38, Lemma 3.3.2]) Let A be an R-finite projective Ralgebra, and J an R-finite projective A-module. If J(m) is A(m)-projective for any m 6 Max R, then J is A-projective. Proof. By Lemma 4.1.2, we may assume that (R, m) is a complete local ring. Then as A is semiperfect, there is an A-projective cover
• Lemma 4.1.6 Let IT be a finite poset ideal of X+, and V an R-finite Sc(n)module, and r > 0. / / proj.dim fl V < r, then we have proj.dim Sc( w) V < 2rank7r + r. In particular, we have proj.dim Sc( w ) V < oo. Proof. Let F :
>Fn^Fn-!^
• Fy -*• Fo -> V -*• 0
be an Sc(7r)-projective resolution of V with each term 5c(7r)-finite. We set fij := Imdj. Then as fir is il-finite projective, replacing V by Qr (and r by 0), we may assume that r = 0, namely, V is ii-projective. Then for each m € Max R, 0 -> ft2rank*(m) -> F 2rankff _i(m) ->
• Fi(m) -> F 0 (m) -»• V(m) -> 0
is exact. By Lemma 3.1.3, gl.dimSc(7r)(m) < 2 rank 7r, and hence fi2rankir(nt) is Sc(ft)(m)-projective. By Lemma 4.1.3, fl2rankir is 5c(7r)-projective, and hence proj.dim5c( B\ V < 2 rank n. •
4. Ringel's theory over a commutative ring
207
Theorem 4.1.7 Let V and W be R-finite R-projective C-comodules. Then we have the following. 1 Let n be a non-negative integer. //Ext^ m )(V(m),iy(m)) = 0 (i > n) for any m e Max/?, then we have ExtQ9Rf(V <8> R', W
b W is u-acyclic with respect to V, and Homc(V, W) is R-finite protective. c For any m e MaxR, we have Extj 7(m) (V(m), W(m)) = 0 (i > 0). Proof. Let n be a finite poset ideal of X+ such that both V and W are CV-comodules, or 5c(7r)-modules. By Lemma 4.1.6, there is an Sc(?r)-nnite projective resolution F of V. Set G := Homs c(ir )(F, W). Then G is a perfect .R-complex (i.e., a bounded complex of i?-modules with each term R-finite projective). As can be seen easily, utilizing Theorem 2.3.17, we have
for any commutative .R-algebra R', any /?'-module M, and any i. All the assertions follow immediately from Proposition 2.1.14, applied to G. D
Corollary 4.1.8 Let V be an R-finite R-projective C-comodule. Then the following are equivalent. 1 V i s good. 2 V is u-good. 2' K 2'* V*€ 3 V is V-good, i.e., V 6 add JF(V). 3* V is A-sood, i.e., V* G add^(A C o P ). 4 For any m € Maxi?, V^m) is good. 4* For any m e Maxfl, V ( m ) e J"(ACop(m)).
208
III. Highest Weight Theory
Proof. The equivalence 1<=>2«=>4 is an immediate consequence of the theorem. 2=>2' Assume that V is u-good. Then there is a finite filtration 0 = Wo C W1 C • • • C Wn = V of V such that each Wj is an /Z-pure C-subcomodule of V, and Wi/Wi_i = Ac (/*(*)). v) ® %*(*))* f o r e a c h »• S e t
Then Mj is fl-finite. On the other hand, Wi/Wi-i is it-finite projective, because of the finite-projectivity of V and the purity of Wi and Wj_i. As Vc(M(*))m is a non-zero Rm-iree module for any m 6 Max/?, Mt is also infinite projective. Hence, 2' is proved. We prove 2' =>3. As V C .F(V), it follows that addV C add^(V). As addJ r (V) is closed under extensions (1.1.11), we have .F(addV) C add^(V). 3=>1 This is clear, since objects of V are good, and good comodules are closed under extensions and direct summands. 3«-3*, 2' <&2'* and 4«-4* are obvious. a Lemma 4.1.9 LetV e Ac and X € X+. We set X := {W e Ac | Ext^(Ac(A), W) = 0}. Then 1 There exists an exact sequence (4.1.10)
a : 0->V UU ^ D^yO
in Ac such that U € X and D is a finite direct sum of copies of Ac(A). 2 //, moreover, (R, m) is local, then we can take i to be left minimal in 1. Then i is a left minimal X-approximation of V, and in particular, such an a is unique up to isomorphisms of complexes. We call an exact sequence a which satisfies condition 1 a X-extension of V. If, moreover, i in the exact sequence is left minimal, then it is called the minimal X-extension of V. Proof. Set E := End c (Ac (A)). By Theorem 4.1.7, AC(A) is u-acyclic with respect to itself, and the canonical map R-¥ E = Endc(Ac(A)) is an isomorphism.
4. Ringel's theory over a commutative ring
209
Note that Ext^(A c (A), V) is R-hnite by Corollary 2.3.19. Let au..., ar be a generating set of Extlc(Ac(X),V) as an i?-module. If {R,m) is local, we take this as a minimal set of generators. We set D = ©[ = 1 Ac(A), and we define the extension (4.1.10) to be the element corresponding to a = (a\,... ,ar) by the canonical isomorphism
As the Yoneda product from the left a' : Hom c (A c (A),D) -> ExtJ,(A c (A), V) is ^-linear, it is surjective by the choice of a. As we have Ext^(Ac(A), D) = 0, we have ExtJ7 (Ac (A),f7) = 0, in other words, U G X. Now 1 has been proved. To prove 2, we keep the assumption that (R, m) is local. As we have Extc(D, X) = 0, it follows that i is a left ^-approximation by the dual of Lemma 1.1.12.1. It remains to prove i is left minimal. Let ip € Endc U and pi = i, and let i\> : D —>• D be the map induced by
^
ExtJ,(Ac(A),V) 4- id
Hom c (A c (A),D)
^
E x t ^ A ^ A ) , V).
By the definition of minimal basis and by the fact R = E, a':RT*
Hom c (A c (A),Z?) -> ExtJ ? (A c (A) > V)
is a projective cover (as an .ft-module, or an E-module). Hence, rp' is an • isomorphism. This proves 2. Utilizing Lemma 4.1.9 and Corollary 4.1.5, we can verify the following, as in the proof of Lemma 3.1.11. Proposition 4.1.11 Let it be a finite poset ideal of X+, and V 6 AcAssume that ExtJ;(A c (/i), V) = 0 for /j, £ •n. Then there exists an exact sequence (4.1.12) a:0^vUY^X->0 in Ac such that Y is good and X G T({Ac(n) \n € 7r}). //, moreover, (R, m) is local, then we can take i to be a left minimal good approximation. Such an exact sequence is unique, up to isomorphisms of exact sequences. D
210
III. Highest Weight Theory
When (R, m) is local, we denote the left minimal good approximation (4.1.12) of V, which uniquely exists by the proposition, by
for V £ Ac- Moreover, for A € X+, we denote YAci-x] by TC(A).
Lemma 4.1.13 Let V G Ac, and -K be a finite poset ideal of X+. Assume Extc(Ac(At), V) = 0 for fj. E X+\n. Then we have Ext^(A c (^), V) = 0 (i > 2) for v G X+, provided v ^ 7r or v is a maximal element of n. Proof. By Proposition 4.1.11, there exists an exact sequence such that Y is good, and X e J"({AC(A) | A e n}). As X G M c ", it follows that Ext' c (A c (i>),X) = 0 (i > 1) by Corollary 4.1.5. On the other hand, Exfc(Ac(i'),Y) = 0 (i > 1), as Y is good. Hence, we have the required property by an obvious long exact sequence argument. Q Proposition 4.1.14 Assume that (R,m) is local. Let V be an R-finite Rprojective C-comodule, and ir a finite poset ideal of X+ such that ExtJ7(Ac(At),K)=0
(^€X + \7r).
If u ^ -n or u is a maximal element of it, then for the minimal v-extension
ofV, we have a(m) is a minimal v-extension ofV(m). Proof. By Lemma 3.1.9, it suffices to show that the number of minimal generators of Extc(A c (^), V) as an /^-module agrees with dimK(m) Let p be a finite poset ideal of X+ such that both i / G p and V is an Sc(p)module. Let F be an Sc(p)-finite projective resolution of V, which exists by Lemma 4.1.6. As Hom Sc(^)(F, V) is an .R-perfect complex, there is a spectral sequence Ep2'« = Tor« p (Ext 9 c (A c (f), V), K (
^
9
By Lemma 4.1.13, E%' = 0 (q ^ 0,1). Moreover, obviously E^q = 0 for p > 0. Hence, E0/ = E^1. As p + q = 1 and Ep2'q ^ 0 imply (p, q) = (0,1), we have
E x t ^ A c H O n ) , V(m)) ^ E°J S E0/ « E^c(Ac(u), This proves the lemma.
V) ® «(m). •
4. Ringel's theory over a commutative ring
211
Corollary 4.1.15 Let (R, m) be a local ring, and V an R-finite projective C-comodule. Then we have Yv(m) S Yv^ and Xv{m) S Xv<-ml Corollary 4.1.16 Let (R, m) be a local ring, and A € X+. Then we have Tc(A)(m) = Tc(m)(A). In particular, 7c(A) is indecomposable. Proof. The first assertion is an immediate consequence of Corollary 4.1.15. As Tc(m)(A) is indecomposable by (3.2.3), 7c(A) must be also indecomposable by the first assertion and Nakayama's lemma. • (4.1.17)
Now we set ;yPr0 -={V eAc\ proj.dimfl V < oo and V is good},
:= add^(A), and d£° := Xg° n ) T - By Corollary 4.1.8, we have A>Pro = jT(addA).
Lemma 4.1.18 Let f : R-> R! be a homomorphism of commutative noetherian rings. Then we have the following. 1 Xg° ® R' C S 2 / / / is flat, then j£ r o ® R'C 3 If f is faithfully flat, then for V € Ac, we have V e X%° (resp. V € ycvo, V e wpcm) if and only if V ® R' E X^R, (resp. V ® 7?' €
Proo/.
Follows easily from Corollary 4.1.8 and Corollary 1.4.8.
D
Lemma 4.1.19 We have X^™ = {V 6 Ac \ proj.dim^V < oo}. Proof. As any object of XQTO is /?-projective, any object of XQTO is of finite
.R-projective dimension. Conversely, if V 6 Ac and proj.dimHV < oo, then there exists a finite poset ideal n of X+ such that V G M c*. By Lemma 4.1.6, proj.dimSc(^ V < oo. As add SC(TT) C XQ°, it follows that ^ r o -resol.diml/
r o
and T\ good for each X £ X+, then we have UQ° =
212
III.
Highest Weight Theory
Proof. It is obvious that X^0 is closed under extensions and direct summands. We prove that X^"0 is closed under epikernels. Let
be an exact sequence in Ac such that X, Xi £ XQK'. As X\ is fi-projective, we have that 0 -> X[ -> X* -> V* -> 0 is exact, and A\* and X* are u-good Cop-comodules by Corollary 4.1.8. As V* is infinite projective, we have that V* is a u-good C op-comodule by Lemma 2.3.15. By Corollary 4.1.8 again, V £ Xg0. Good comodules are closed under extensions, direct summands, and monocokernels. On the other hand, comodules with finite .R-projective dimension are also closed under extensions, direct summands, and monocokernels. Hence, ycT0 is closed under extensions, direct summands, and monocokernels. By Lemma 4.1.19, we have yc™ C Xg°. Hence, conditions AB1 and AB2 in Theorem 1.1.12.10 hold. We verify condition AB3. Let A £ X+. Then by Proposition 4.1.11, there is an exact sequence 0 -> AC(A) -> Tx -> Xx -> 0 with T\ good and Xx £ XQTO. In the sequel, we fix such exact sequences for all A £ X+. We set T = {TA | A £ X+}. Then we have addT C u%°. In fact, as Xx, AC(A) £ X%ro, we have Tx £ Xg°. As Tx is good and i?-finite projective, TA £ ycro. Next, we show that addT is an injective cogenerator of Xg*°. In fact, as addT C yg0, it follows that addT is A^ro-injective. As we have X%ro = add^ 7 (A), we conclude that addT is an injective cogenerator of XQT0, by Corollary 1.1.12.13. By Lemma 1.1.12.14, ( A g ^ ^ g ^ a d d T ) is a weak Auslander-Buchweitz • context, and we have addT = uicro. Corollary 4.1.21 Let (X+, A, V) be a semisplit highest weight theory over an R-projective R-coalgebra C. If R is a regular ring, then we have that the triple {Xc*0^^0,^0) is an Auslander-Buchweitz context in AcProof. Follows immediately from Lemma 4.1.19, Theorem 4.1.20 and Lemma 1.4.10.17. D Corollary 4.1.22 If (R, m) is a local ring, then we have wp7ro = a d d { T c ( A ) | A £ X + } . Proof. Trivial.
•
4. Ringel's theory over a commutative ring
4.2
213
Minimal Ringel's approximations over local rings
(4.2.1) Let (i?,m) be a noetherian local commutative ring. Let C be an .R-projective .R-coalgebra, (X+, A, V) a semisplit highest weight theory over C, and (Cv) its associated Donkin system. Lemma 4.2.2 Let R be Henselian local. Then any object o/w£.ro is uniquely a finite direct sum of copies of TC{X) for X G X+. In particular, any indecomposable object T of U(fo is isomorphic to 7c(A) for a unique A. An object T ofuJg0 is isomorphic to TC(X) if and only ifT(m) = Tc{m)(X). Proof. Any object of Ac has a semiperfect endomorphism ring by Corollary 2.3.19. The first assertion follows immediately from Corollary 4.1.22, Corollary 4.1.16, and Lemma 1.1.12.6. The rest of the assertions are trivial.
• Proposition 4.2.3 Any object of <JQ° is uniquely a finite direct sum of TC(X) (A € X+). In particular, any indecomposable object of w£ro is isomorphic to Tc(A) for a unique A. An object T o/w^T0 is isomorphic to Tc(X) if and only ifT(m) £ T c(m) (A). Proof. It suffices to prove the first assertion. Since Tc(A)(m) = TC(m)(A) is indecomposable for A G X+ by Corollary 4.1.16 and (3.2.3), the uniqueness assertion is obvious by the KrullSchmidt theorem (Lemma 1.1.12.6). Next, we prove that any object V in w£ro = add{Tc(A) | A E X+} is a direct sum of ?c(A). We prove the existence of a decomposition. It suffices to prove that if V 6 u>c with V j= 0, then there is some A 6 X+ and a split mono Tc(X) -> V. If so, then there is a decomposition V = Vi 0 TC(AO). And if VI 7^ 0, then there is a decomposition Vi = T c (Ai) © V2, and if V2 ^ 0, then © V3. Continuing thus, we have a sequence of C-subcomodules 0 C TC(AO) C TC(AO) © Tc(Xi) C TC(AO) © Tc(Xi) © TC(A2) C • • • of V. This is an increasing sequence of C-subcomodules of V, and eventually we have V^+i = 0 for some r. This shows that we have a decomposition V = Tc(X0) © • • • © Tc{XT) for some r, and completes the proof. Let R be the m-adic completion of R, and set ? = ? ® R. By Lemma 4.2.2, fcjx) = Td{X), since fcjx) G u£ r0 by Lemma 4.1.18 and Tc(X)(mR) = TC{X) ® R ® {R/mR) s T c (A)(m).
214
III.
Highest Weight Theory
As V ^ 0 and V G w?ro, there is a split mono (p : TC(A) *-4 V for some A G X+ by Lemma 4.2.2. Set T := TC(A), and f := 7b(A). Let ^ : V -> T be a C-comodule map such that •ipip = id. As V is u-acyclic with respect to T and vice versa, we can write <£ = £ i ¥>t ® Q, with with Vj e Eomc(V,T) V?i <E Hom c (T,y) and a € fl, and ^ = Ejipj®^ and cj- € /?. As End<j T is local, there are some i and j such that ^ j ^ is an isomorphism. As ? is faithful exact, %pj(pi is also an isomorphism, and hence
R'/n
by Corollary 4.1.16 and Corollary 3.2.9. By the proposition, we have the isomorphism. • (4.2.5) Let (i?,m) and C be as in (4.2.1). Let (X+, A, V) be a semisplit highest weight theory over C. Theorem 4.2.6 Let
F : 0 - > V UX
^V
-^0
be a sequence in Ac- Let R be the m-adic completion of R, and ? denote the functor ? <S> R. Then the following are equivalent. 1 F is an XQ°-approximation
ofV, andp is right minimal.
2 F is an XQTO-approximation of V, and there is no non-zero direct summand of X contained in Imi. 3 F is an X^° -approximation ofV, andp is right minimal. c pro 4 F is an X^°-approximation of V, and there is no non-zero direct summand of X contained in Imi. In particular, ifV £ Xc, then there is a unique minimal XQ™-approximation. Similarly, there is a unique minimal y^°-hull of V, and it agrees with the left minimal good approximation.
4. Ringel's theory over a commutative ring
215
Proof. First note that F is exact if and only if F is. By Lemma 4.1.18, F is an ;t pro -approximation of V if and only if F is an ^"^-approximation of V. Hence, the only problem is the minimality. The implications 1=>2 and 4<=>3 follow from Proposition 1.1.12.8. We prove 3=>1. Assume that
J = E J"i ® c«
(Ji G Homc(Tc(A), K), cs G A)
9 = £ 0j ® 4
(^ e Homc(X, TC(A)), c't 6 A).
and t
Then we have Es,t9t^s® c s c t = ^ - As End^(T c (A)®^) is local, there exist some s, £ such that q'tij's is an isomorphism. Then lmij's is a non-zero direct summand of X contained in Imi, which is a contradiction. The last assertion is proved easily, and the proof is left to the reader. •
4.3 (4.3.1)
Cohen—Macaulay analogue of u-good module Let R be a commutative noetherian ring.
Definition 4.3.2 Let M and N be .R-modules. We say that an i?-linear m a p ip : M —> N is semipure \i
/^-module L which is locally of finite flat dimension, see (1.4.10.18). Note that an i?-semipure map is injective, and any R-pure map is Rsemipure. If R is a regular ring, then an i?-semipure map is i?-pure. Lemma 4.3.3 Let R be a Cohen-Macaulay ring, N a maximal CohenMacaulay R-module, M an R-finite module, and
216
III. Highest Weight Theory
2 For any m G Max 7?, there exists some parameter ideal (i.e., an ideal generated by a system of parameters) J of Rm such that > N ® Rm/J is injective. 3 (p is injective, and Coker ip is maximal Cohen-Macaulay. Proof. 1=>2 is obvious. 2=3-3 We may assume that (R, m) is local. Set d := dim .ft, C := Coker (p, I := Imtp, and K := Ker<^. As ip
=0
by Corollary 1.2.9.2. As 7/J7 -» Af/JiV is injective, Torf (C, 7?/J) = 0. By Lemma 1.2.9.6, depth C = d, and C is maximal Cohen-Macaulay. Hence, 7 is also maximal Cohen-Macaulay. Hence, Torf(7, R/J) = 0, and 0 -> K/JK -> M / J M -> 7/J7 -> 0 is exact. As M/JM —> I/JI is an isomorphism, K/JK = 0. By Nakayama's lemma, K = 0. 3=>1 If L is an 7i-module locally of finite flat dimension, then Torf (C, L) = 0 by Theorem 1.4.10.19. The assertion follows immediately. D Definition 4.3.4 We say that an 7?-complex
(4.3.5)
F : 0 -> F° A F 1 A • • •
is su-acyclic if for any .R-module L locally of finite flat dimension, H'(W (g> L) = 0 (i > 0) and the canonical map 77°(F) ® L -> 7f°(F ® L) is an isomorphism. Lemma 4.3.6 Le< R be a Cohen-Macaulay ring, and F (4.3.5) be an Rcomplex. Assume that Fl is maximal Cohen-Macaulay for i > 0. Then the following are equivalent.
4. Ringel's theory over a commutative ring 1 Hi(¥)=0
217
(i>0).
2 F is su-acyclic, and H°(¥) is a maximal Cohen-Macaulay R-module. 3 For any m € Max R, there exists some parameter ideal J of Rm such that H{(F®Rm/J) = 0 (i>0). Proof. 1=>2 We may assume that (R, m) is local. Let L be an .R-module with flat.dimRL < oo. We take a bounded flat resolution G of L. By Theorem 1.4.10.19, we have T o r f ^ . L ) = 0 (i > 0, j > 0), so F
is an isomorphism, and Torf(/f°(F),L) = 0 for i > 0. Considering the special case where L = R/J with J a parameter ideal of R, we conclude that H°(F) is maximal Cohen-Macaulay. 2=>3 is obvious. 3=>1 We may assume (R, m) is local. We use induction onrf = dimi?. If d = 0, then we have J = 0, and the assertion is obvious. Let d > 0, and J = (x\,...,Xd), and set x := x\. Then by induction assumption (applied to R/xR and F/xF), we have H^F/xF) = 0 (i > 0). Hence, for i > 0, the multiplication by x a; : tf*(F) -+ fT(F) is surjective. By Nakayama's lemma, we have Hl(F) = 0 (i > 0).
D
(4.3.7) Let i? be a noetherian commutative ring, C an i?-projective coalgebra, (X+, A, V) a semisplit highest weight theory over C, and (CV) its associated Donkin system. Definition 4.3.8 Let V and W be C-comodules. We say that W is suacyclic with respect to V if
2 The canonical map Hom c (V, W) <S> M —> Homc(Vr, W
218
III.
Highest Weight Theory
1 Let V and W be C-comodules. If W is su-acyclic with respect to V, then W ® M is su-acyclic with respect to V for any R-flat module M. If, moreover, V is R-finite projective, then W = W
•
Lemma 4.3.10 Let V be a C-comodule. Then the following are equivalent. 1 For any X E X+, V is su-acyclic with respect to
Ac{\).
2 V is good, and for any finite poset ideal n of X+, we have Vn := ind£" V <—> V is R-semipure. 3 There exists a filtration 0 = Vo C V1 C V2 C • • • C V of V such that lirnVj = V, and for any i > 1, V< is an R-semipure submodule of V^+i, and there exists X £ X+ and an R-module M such that Vi/V^ S VC (A)
4. Ringel's theory over a commutative ring
219
Proof. Let n be a finite poset ideal of X+ such that both V and W are C*comodules. Then Cobarc,(H^, V*) is an .R-complex with each term maximal Cohen-Macaulay. By Lemma 4.3.6, we have the desired assertions. • Corollary 4.3.12 Assume R is Cohen-Macaulay. Let V be a C-comodule which is maximal Cohen-Macaulay as an R-module. Then the following are equivalent. 1 V is good. 2 V is su-good. 3 V S .F(V® XR), where XR denotes the full subcategory of RM consisting of maximal Cohen-Macaulay R-modules.
4.4
Cohen—Macaulay Ringel's approximation
(4.4.1) Let R be a commutative noetherian ring with a pointwise dualizing module KR. In particular, R is Cohen-Macaulay by Bass's conjecture (1.2.5.1). We set AR := RM/, and define XR to be the full subcategory of AR consisting of maximal Cohen-Macaulay .R-modules, and yR to be the full subcategory of AR consisting of V € AR such that inj.dimRp Vp < oo for any p € Spec/?. We set u>R := addK R . By Proposition 1.4.10.11, {XR,yR,LjR) is an Auslander-Buchweitz context in AR. We denote the full subcategory of AR consisting of finite projective .R-modules by XR°. We set VR := X^0, i.e., VR denotes the full subcategory of AR consisting of /^-finite modules of finite projective dimension. Let C be an .R-projective .R-coalgebra, and (X+, A, V) a semisplit highest weight theory over C, and (Cn) its associated Donkin system. We define Xc to be the full subcategory of Ac consisting of V 6 Ac such that V € XR (as an .R-module) and Hom(V, KR) is a good C op-comodule. We also define yc to be the full subcategory of Ac consisting of V E Ac such that V e yR (as an .R-module) and 1lom(KR, V) is good. We set uic '•= Xc D ^ c - We define Vc to be the full subcategory of Ac consisting of V e Ac such that V e VR (as an .R-module) and V is good. The following is trivial by Proposition 1.4.10.11. Lemma 4.4.2 The functor Hom(?, KR) is an exact equivalence from the exact category Xc to the exact category of maximal Cohen-Macaulay good Cop-comodules, and Hom(?, KR) itself is its quasi-inverse. The functor r\om(KR, ?) is an exact equivalence from yc to Vc, andl®KR is its quasiinverse. Hence by Corollary 4.3.12, we have:
220
III
Highest Weight Theory
Corollary 4.4.3 Xc = T{& <S> XR), and Xc is closed under extensions, direct summands, and epikernels in Ac- The full subcategory yc is closed under extensions, direct summands, and monocokernels in AcL e m m a 4.4.4 Ac = XcProof. Let V € Ac- Then there exists a finite poset ideal n of X+ such that V e Ac,- We set 5 := SC{K). AS V € XR, we have h < oo, where h :— w/j-proj.dim V. Take an exact sequence of S-modules 0 -> V -> FA_i ->
> Fj -» F o -> V -> 0
such that each Ff 5-finite projective. Then each F, is also i?-finite Rprojective, and hence V is maximal Cohen-Macaulay. As each F{ belongs to Xc, it suffices to show V € XcSet r := rankTr, and take an exact sequence 0 -> V" -> F 2r _i ->
• Fj -> F o -> V" -» 0
of S-modules with each Ft 5-finite projective. By Lemma 4.1.6, we have proj.dimSoP Vc(A)* < 2r for A 6 7r. Hence Ext^p(Vc(A)*, Hom(V, KR)) = 0 for A € n and i > 2 r + l . As 0 -> Hom(V", A"fi) -> Hom(F 0 , /C fi ) - ^ H o m ( F i , >• H o m ( F 2 r _ 1 , / f a ) -)• Hom(V", /f f l ) -»• 0
is exact and each Hom(F,, KR) is a good C°p-comodule, we have ExtU(V c (A)%Ham(V" l /ffl)) = 0 for A G 7T and i > 1. Hence, Hom(V", KR) is a good C° p-comodule. Hence, V" € Ac, and the lemma is proved. • Theorem 4.4.5 Let R be a Cohen-Macaulay ring with a pointwise dualizing module KR, C an R-projective R-coalgebra, and (X+, A, V) a semisplit highest weight theory over C. Then (Xc, yc^c) *s an Auslander-Buchweitz context of Ac, and u>c = KR Proof. The conditions A B 1 , A B 2 in Theorem 1.1.12.10 are obvious by Corollary 4.4.3 and Lemma 4.4.4. Next, we prove uic = KR <8> w c ro. First, we show KR
4. Ringel's theory over a commutative ring
221
have KR®V € Xc- On the other hand, as V is infinite i?-projective, we have
Uom{KR, KR®V) = V, and hence KR®V
e yc. Hence KR ® ucro C wc.
ro
We show u)C C KR ® w£ . If T e w c , then as T e 3>C) we have T^KR® Eom(KR, T). So it suffices to show that Hom(A'fi) T) € u£ r °. In fact, Hom(/(' fl) T) is .R-projective and good, and hence Hom(K R , T) € ycm. On the other hand, as Uom(KR,T)*
a Hom(Hom(i<'fl, T), ^ Hom(KR (8) H o r n e t , T), /f H ) S* Hom(T,
it follows that Hom(A'/{, T)* is an /?-finite il-projective good Cop -comodule. By Corollary 4.1.8, Uom(KR,T) € Xg°. Now uc = KR®wg.ro is proved. It remains to prove the condition AB3. As uc = Xc n ^ c by definition, it suffices to prove that u>c is an injective cogenerator of XcIn fact, it \ £ X+, V e XR, and M € w£ro, then there is a spectral sequence E™ = Ext£(A c (A), Ext9R(K, *TH ® M)) => Ext^ +9 (A C (A) <8> V, # * ® M) by Proposition 1.3.5.13. As Ext R (V, KR ® M) = 0 (q > 0), we have isomorphisms Ext' c (A c (A)
of V. Let 0 -> AC(A) ^ TA -> XA -> 0 r
is be a 3£ °-hull of A C (A). Then the map j ® i : AC (A) ®V ^>TX®YV injective. When we set L := Coker j ® i, then there is an exact sequence 0 -4 AC(A) ® Xv -* L -> Xx ® Yv -> 0. Hence, L G
222
4.5
III. Highest Weight Theory
Applications to split reductive groups
(4.5.1) Let R be a noetherian commutative ring, and G = Spec// a split reductive group over R. We use the notation, the terminology and the conventions in (1.4.5) and (1.4.7). Lemma 4.5.2 If R is afield, then (H, XQ, AG, V G ) is a split highest weight coalgebra over R, where A G := (AG (A))AeX + and VG := (V G (A)) Ae;r t-. G
G
Proof. First, consider the case that R = k is an algebraically closed field. We set La := (L G (A)) AeX +. It suffices to show the conditions a - h (g is for i = 1,2) in Proposition 1.2.2. By Lemma 1.4.5.10, the condition a is obvious. The conditions b - g are obvious from (1.4.7). Hence, (XQ, A G , V G , £ G ) is a highest weight theory over H. By Lemma 1.2.20, End G .k G (A) is a finite dimensional division fc-algebra, and hence it agrees with fc. So h is also true. We prove the case that R is a general field. Let R' be the algebraic closure of R. By Proposition 1.3.6.20, 2, we have VG(A) <8>R R' = V G 8 f f (A), and AG(A) ®R R' = A G ® R ' ( A ) . By Corollary 1.4.8, we are done. D As is described in [90, p. 248], this fact, Serre's duality, and Borel-BottWeil formula together yield a simple proof of the following well-known fact. Corollary 4.5.3 If R is a field of characteristic zero, then any G-module is a direct sum of LQ(X). Ext G (V, W) = 0 for any G-modules V and W in this case. Proof. Recall (1.4.7.11) that we have AG(A) = VG(A) for any dominant weight A. The composite map LG(A) = soc(V G(A)) -> VG(A) S AG(A) -» top(A G (A)) S LG(A) = 0. This shows V G ( A ) = is non-zero, because (rad(A G(A)) : LQ(\)) AG(A) = LG(A). Hence, we have ExtG{LG{X), LG(fi)) = 0 for all A,/j£ X£. As any simple G-module is isomorphic to some LG(A), we have Ext G (V, W) = 0 for finite dimensional G-modules V and W. By Proposition 1.3.7.4, this is true even if W is not finite dimensional. By Lemma 1.1.9.4, any G-module is injective. Hence, the last assertion is now clear. As is proved by induction on length, any finite dimensional G-module is semisimple. By Lemma 1.1.10.5, any G-module is semisimple, which completes the proof of the corollary. • Lemma 4.5.4 For A 6 X%, we have Ri i n d ^ A ) = 0 (i > 0), and ind%(Rx) is R-finite R-free. For any commutative R-algebra R!, we have
4. Ringel's theory over a commutative ring
223
Proof. It suffices to prove that the complex F := CobarR[B](R\, H) is u-acyclic with H°(¥) R-finite free. As the construction of F is compatible with the base change, we may assume that R = Z. Note that F is an fi-flat complex. Note also that F° = R\ ® H = H is /?-projective by Theorem II.2.2.8. Hence, we can apply (2.1.17) to F. It suffices to prove that //'(F®/c(p)) = 0 (i > 0) for p G SpecR, and that dimK(p) H°(W® «(p)) is finite and constant. Let p G Spec/?, and it be the algebraic closure of /t(p). By Kempf's vanishing (1.4.7.6), we have ® B ( P ) fc £ 1T(F ®fc)S # indg®t ^ = 0 for i > 0. Hence, we have i/'(F <8> «(p)) = 0 (i > 0). By (1.4.7.9), dimK(p) H°(F ® K(p)) = dim* # ° ( F ® fc) = is independent of k and is
finite.
•
Hence, we have the following. Lemma 4.5.5 (H,XQ, A G , V C ) is a spZii highest weight coalgebra defined over Z. By Theorem II.?.2.8 and Theorem 2.2.11, we have: Lemma 4.5.6 There is a unique Donkin system {Hn) of H associated with Let 7r be a poset ideal of XQ. We set Ca{^) '•= H* = lim// p , where p p
runs through all finite poset ideals of TT. We call CG{^) the Donkin subcoalgebra of G with respect to TT. We say that a G-module V belongs to TT if V is a Co (Tr)-comodule. If 7T is finite, then we denote CG{K)* by SG(TT) and call it the Schur algebra of G with respect to TT. We denote the weak Auslander-Buchweitz context (<^ rc \ 3 ^ ° , w^ro) of cM/by / -L/pro ->;pro pro-, (•^G i J ' G ' W G )•
Definition 4.5.7 We call an object of uiGTO a tilting G-module. The name 'tilting' here is not the same as the tilting module explained in (1.4.9), but the origin of this word is in (3.2.7), and these are connected in some sense, see [50]. If R has a pointwise dualizing module KR, then we denote the AuslanderBuchweitz context (XH,yH,ujH) of G M/ by (XG,yG, UJC).
224 (4.5.8)
III.
Highest Weight Theory
The following theorem was proved by O. Mathieu [107].
Theorem 4.5.9 Assume that R is an algebraically closed field. If V and W are finite dimensional good G-modules, then V ®W is good. Corollary 4.5.10 Let R be an arbitrary noetherian commutative ring. Let V and W be G-modules. Then the following hold: llfVis
R-finite R-projective good and W is good, then V ® W is good.
2 If V, W e X%°, then V
4.6
Good modules of a general linear group
In this subsection, we give some remarks on good modules of G = GL(n, R).
4. Ringel's theory over a commutative ring
225
(4.6.1) We follow the notation, terminology and conventions of (1.4.6). We say that a sequence of non-negative integers A = (Aj, A2,...) is a partition if Ai > A2 > • • •, and An = 0 for n > 0 . For a partition A, the transpose A = (Ai, A2,...) of A is defined by
It is easy to see that A is a partition, and we have A = A. The sum £^ A* is called the degree of A. The minimum non-negative integer r such that A r+1 = 0 is called the length of A. By definition, the length of A agrees with Ai. The set of partitions of length at most n, viewed as a subset of Z" = X(T), is denoted by F(G). Note that F(G) is a poset ideal of X+. A G-module which belongs to F = F(G) is called a polynomial representation. We have C G ( F ) = R[End(Rn)], and GG (F) is a subbialgebra of H. Example 4.6.2 Set A := (r, 0 , 0 , . . . , 0) € Zn = X{T), and
Then 7r(r) agrees with the set of partitions of degree r with length at most n. Moreover, we have F = []r>o 7r(r). The Schur algebra SG(7r(r)) with respect to 7r(r) is nothing but the classical Schur algebra studied by I. Schur. See [62] for more. (4.6.3) For a partition A, Akin-Buchsbaum-Weyman [4] defined the Schur functor L\. It is a universally free functor of type (1,0) defined over Z. The module Lx(Rn) is a G-module by (1.4.8). If Ai > n, then Lx{Rn) = 0, while if Ai < n, then we have L\(Rn) = V G ( A ) . More generally, if X is a scheme and Q a locally free coherent C?x-module, then L\{Q) is a GL(Q)-module. Note also that L\(Q) is a locally free coherent C?x-module. Akin-Buchsbaum-Weyman [4] defined the Schur map d\ : A A ~* Lx, which is a universal map between universally free functors, where A A i s defined by Al An AA V := A V ® • • • ® A V. Note that d\ is surjective. Note also that the following holds. Proposition 4.6.4 Set A :=
{AA#"I^
e f}.
Then we have add A =
We show the outline of the proof of the proposition. First we must prove that AA Rn ls tilting. As A* V is .ft-finite free for any i, AA R™ is also i?-finite free. So we may assume that R is an algebraically closed field. By Corollary 4.5.10, it suffices to show that A* V is tilting for each i. In fact,
226
III.
Highest Weight Theory
we have A' V = VG(wi) = A G (wi), where a>j is the partition £i -\ |-e». As (jj is minimal in X£, the proof is reduced to proving A1 V = ^ G ^ t ) , which is proved easily using (1.4.7.10). As AA ^™ i s tilting and is a polynomial representation, we have add A C w c™(r) • Hence, it suffices to show that add A is an injective cogenerator of XQ°, to prove the proposition. By Corollary 1.1.12.13, it suffices to show that for any A € f, there is an exact sequence 0 -> AG(A) ->
with Dx e
AA
Rn -> Dx -> 0
Xg(r).
We claim that Hom G (A G (A), AA #") = R, and Hom G (A G (^), AA # n ) = 0 for fj, £ A. As we know that AA ^ " is u-good, the proof of the claim is reduced to the case where R = C, and this is well-known and checked at the character level, see [104]. If R is a field, then AA # " - TG(\) 8 T'x, where T'x is a finite direct sum of T G(/i) with // < A. Note that it follows, if R is a field, that any non-zero map from AG(A) to AA -ft" factors through TG(A), and is a ^crO(rN-hull (hence is injective). Now take a free generator d'x of Hom G (A G (A), /\\Rn) = R as an Rmodule. Then for any m 6 Max/Z, the base change d'x(m) is injective, and hence Dx •= Cokerd'A is i?-projective. As Dx(m) is A-good for any m, Dx is also A-good. As Dx is a polynomial representation, Dx G
Remark 4.6.5 In [4], an explicit construction of d'x : AG(A) —> AxRn is given. In fact, d'x is constructed as a universal map of universally free functors there (but the notation is different).
Example 4.6.6 As we have seen, an .R-finite .R-projective good G-module is u-good, see Corollary 4.1.8. However, this is not true without the Rprojectivity assumption. Set R := Z, G := GL2, and V := I?. Let ? denote the functor ® Z Z/2Z. There is a short exact sequence of G-modules 0 -» V^ i+ S2V Z+ A2 V -> 0, where p(viV2) = V\ A v2, and V^1^ is the first Frobenius twisting of V, see [90, (1.9.10)]. The image Imi is the Z/2Z-span {x\,x^), where X\,X2 is a basis of V. Taking the pull-back of this exact sequence by the canonical
4. Ringel's theory over a commutative ring
227
projection n : f\2 V -> f\2 V, we have a commutative diagram of G-modules 0 0 0
t - ). v-a) t
•i->
-s
->
t 0
0
0
t
t
52v tp M
^^
A 2 ^ --»•
0
t--^ A V^
t2
-> A V^ --> A
2
t
t
0
0
— -> 0
^ -•>
0
all of whose rows and columns are exact. As S2V = Vg(2), the exactness of the middle column shows that M is good. Note that Imi = soc Vg(2). It follows that HomG(A2 V, 5 2 V) S Hom<5(A2 V, S2V) = Homc(A2 V", soc(5 2 V)) = 0. Hence, HomG(A 2 V, M)- * HomG (A2 V, A2 K)" On the other hand, p is an isomorphism, and Hom G(A2 V, M) S Hom G(A2 V, S2V) = 0. Hence, M is good, but is not u-good.
Chapter IV Approximations of Equivariant Modules and their Applications 1
Approximations of (G, A)-modules
Throughout this section, R is a noetherian commutative ring, G = Spec H is an i?-flat /2-group scheme, and A is an i?-flat noetherian commutative G-algebra.
1.1
Graded G-algebras
(1.1.1) Let us assume that G has a fixed closed subgroup G m C Z(G), where Z(G) is the center of G. Then any G-module is a Gm -module by restriction, and hence it is a Z-graded /?-module by (1.4.3). In the sequel, the grading of a G-module is the one obtained in this way, unless otherwise specified. For a G-module V, V{ denotes the degree i component of V, so that V = 0 j € z Vi as an i?-module. Any G-linear map preserves the grading of G-modules. L e m m a 1.1.2 Let V € G M and i € Z. Then Vi is a G-submodule of V, and V = © i € Z Vi is a direct sum decomposition of V as a G-module. Proof. For any commutative .ft-algebra A, g € G(A), v £ Vt and t G Gm(A) = A*, we have t(g(v
In particular, when we set A = H
and define g e G(A) = Homfl-ai6(#, H ® R[Gm])
byh>->h®landte
Gm(A) = Hom fl - a i g (.R[G m ],// ® i?[Gm]) by s K> 1
230
IV. Approximations of Equivariant Modules
above shows wv{v) £ V^ ® H. Hence, Vt is an //-subcomodule of V, that is, a G-submodule of V. D (1.1.3) As A is a G-algebra, it is a Gm-algebra, and is a Z-graded Ralgebra. From now on, until the end of this subsection, A is assumed to be positively graded, i.e., Ai = 0 (i < 0) and R —> yl0 is an isomorphism. We set OT := © i > 0 A{. Note that OT is a G-ideal of A, and we have A/M = R. We also assume that G is R smooth with connected geometric fibers. We denote by Vo the full subcategory of G M consisting of it-finite Rprojective G-modules. By V, we denote the full subcategory of G,/iM consisting of v4-finite j4-projective (G, j4)-modules. L e m m a 1.1.4 We have V = !F(A ® Vo). Namely, for any A-finite Aprojective (G, A) -module P, there exists a filtration 0 = P (0 ) C P ( 1 ) C • • • C P{r) = P
(r > 0)
of P such that for each 1 < i < r, there exists Q^ £ Vo such that P(i)/P(i_i) = A
s:= £
rank* ,
If s = 0, then we have P/VJIP — 0. As 9Jt is contained in the G m -radical of A, we have P = 0 by the equivariant Nakayama, Lemma II.2.3.18. So this case is obvious. We assume s > 0, and P ^ 0. When we decompose P = 0 j e z Pj with respect to the grading of P , we have Pj = 0 for j
ityan: p j0 ^ (4 ® p io ) ®A A/m -»• P / O T P
is an isomorphism from Pj 0 to the degree j 0 component (P/971P)JO = P,o of P/DJIP, which obviously G-splits (in particular, i?-splits). Hence, when we set C := Cokerv?, then C/9DTG is it = .4/SH-finite i?-projective. Another consequence is that Torf (C, A/DJl) = 0. Hence, by an easy spectral sequence argument, we have Torf (C, V) = 0 for any j4/9Jt-module V.
1. Approximations of (G,A)-modules
231
Since A is positively graded, any graded maximal ideal m of A contains 9JI. It follows that Torf m(G m, «(m)) = 0, and hence Cm is .4m-finite free for any graded maximal ideal m of A. This shows that for any graded prime ideal ?P of A, Gip is ^ - f i n i t e projective. By Corollary II.2.4.3, C is .4-finite ^4-projective. Now it is easy to see that
232
IV. Approximations of Equivariant Modules
Lemma 1.1.8 If A is R-smooth, then the Yoneda product ExVs(A, A) ®A Exfcj(i4, ,4) -> Ext' s +i (i4, A) is (G,A)-linear, and the Yoneda algebra Exts(A,A) exterior algebra /\(I/I2)* as a (G, A)-algebra.
is isomorphic to the
Proof. By (1.5.2.11), the Yoneda product is (U, S)-linear, hence it is (G, A)linear. From the short exact sequence 0 -> / -> S -¥ S/I -> 0, we have a long exact sequence of (G, 5)-modules Hom s (S//, S/I) ^ Hom s (5, S/I) ->• Hom s (/, S/I) -> Ext^S/I,
S/I) -> 0,
and hence Ext^(5/7,5//) S (7/72)*. As Homs(S/I,S/I) s* 5/7 = .4, it suffices to show that the Yoneda algebra E = Exts(S/I,S/I) is a skew commutative S/7-algebra, and the induced map A(I/7 2 r = A(Ext^(5/7,5/7)) -> £ is an isomorphism. Now the problem is not related to the G-action any more, and the question is local. Hence, we may forget the G-action, and may assume that (5, n) is a noetherian local ring, and I is generated by a regular sequence 7 = ( a i , . . . , ah) by assumption and Theorem 1.2.7.10, 2. Although the assertion is now more or less well-known, we briefly illustrate how it is proved. Let F be the Koszul complex Kos(ai,..., a/,; 5) = f\F, where F := Rh. Then F is a finite-free S-resolution of A = S/I. For any r] £ /\l F*, -we have a map $(r/) : F -> ¥[i] given by
E ^ - l ) " det(gu(fav))%vfa{i+1) 0
A • • • A / ff(r)
(r > i) (otherwise),
where the sum is taken over all cr £ &r such that al < • • • < ai and <x(i+l) < •••< ar. An easy calculation shows that $(77 AT/) = $(77) O$(T7'). Note also that the boundary map F —>• F[l] is given by $(e*), where e* = Yli=i a^,*, where e j , . . . , ej is the standard basis of F*. This shows $(77) o d = $(7? A e*) = ( - l ) ^ ( e * A 77) = ( - l ) ' d o $(77), and hence $(77) : F -» W[i] is a chain map. It is trivial that $(1) = id, so $ gives a degree-preserving 5-algebra map
1. Approximations of (G', A)-modules
233
where Y := Homs(F,F), and Z'(Y) denotes the subalgebra of cocycles in the DG algebra Y (the multiplication of Y is given by the composition). See [21] for DG algebras. To prove that the Yoneda algebra is skew commutative and is generated by degree one, it suffices to prove the composite map
is surjective. Let fj £ Extjj(4, A) = // i (Hom s (F, S/I)) = A' F*®S/I. Then we have ! ) 6 A ' f which restricts to fj. Obviously, $(??) : F -* F[i] followed by the augmentation ¥[i] —» S/I[i] represents fj in
=
Z°(Eoms(¥,S/I\i])).
Hence the map in question is surjective.
•
Lemma 1.1.9 The following hold. 1 If J is a G-ideal of A and J is perfect of codimension h', then we have KA/j S ExtA'(A/J,KA) as (G, A/J)-modules. 2 If A is R-smooth, then we have KA =
LOA/R
® KR
O*
(G, A)-modules.
3 The (G,A)-module KA is determined only by A and KR up to isomorphisms of (G, A)-modules, and is independent of the choice of S and I. That is, if S' is an R-smooth positively graded G-algebra and I' is a Gideal of S' such that A = S'/I', then KA agrees with the (G,A)-module K'A constructed in the same way using S' and I'. Proof. As the question is componentwise, we may connected. We prove 1. As R is Cohen-Macaulay Cohen-Macaulay i?-module, S is Cohen-Macaulay Cohen-Macaulay 5-module. By Corollary 1.4.10.20,
assume that Spec R is and KR is a maximal and Ks is a maximal the spectral sequence
££•' = ExtpA(A/J, Ext>s(A, Ks)) =» Extps+q(A/J,
Ks)
in Proposition II. 1.1.14 collapses, and hence we have KA/J = Exths+h'(A/J,KS)
=
ExtA'(A/J,KA).
Next, we prove 2. As A is .R-smooth, / is a local complete intersection ideal of codimension h by Theorem 1.2.7.10. By Theorem 1.2.7.6, we have an exact sequence of (G, .<4)-modules o ->• i j t1 -> ns/R ®SA->
nA/R -> o.
234
IV. Approximations of Equivariant Modules
Taking the top exterior power and utilizing Lemma 1.1.8, we have an isomorphism Ks ®s A S {wAIR ® KR) ®A Exths(S/I, S/I)\ When we denote the canonical projection S —> S/I by p, then we have a canonical (G, yl)-linear map Exths(S/I,p) : Exths(S/I,S)
->
Exths(S/I,S/I).
As we have proj.dim s 5/7 = h, this is surjective. As both Ext^(S/I, S) and Ext^(S /1, S/1) are rank-one S//-projective, the map Extg(5//,p) is an isomorphism. Moreover, the Yoneda product Ks ®s Ext£(S/J, S) -> Exths{S/I, Ks) = KA is an isomorphism, as we have Ext' s (S//, 5) = 0 {i^h)
and
Torf (Ext^(5/7, S), Ks) = 0 (i # 0).
Hence, we have ^
S ExtS(5/7, S) ®s Ks ^ {Ks ®s A) ®A Exths{S/I, S/I) £ wA/R ® /ffl. 3 follows from 1,2.
•
We call KA the G-equivariant pointwise dualizing module of A Corollary 1.1.10 Assume that SpecR is connected, and A has relative dimension d over R. Then as a G-module, we have ExtdA(R,KA) = KR, where R = A/ffl. Lemma 1.1.11 If K is a (G,A)-module whose underlying A-module is pointwise dualizing, then there is a unique rank-one R-projective G-module A such that K = KA ® A. Conversely, for any rank-one R-projective Gmodule A, KA ® k is pointwise dualizing as an A-module. Proof. The converse is trivial. As HomA(KA, K) is a rank-one j4-projective (G, j4)-module, there is a rank-one .R-projective G-module A such that EomA(KA,K)^A®A. Hence, KA = ttomA(EomA(KA, K), K) S Hom(.4 ® A, K) S K ® A*, and we have K = KA®A. Such a A is unique, because A = E.omA(KA, A/Wl.
K)®A D
1. Approximations of (G, A)-modules
1.2
235
Reductive group actions on graded algebras
Throughout this subsection, let R be a noetherian commutative ring, G a split reductive i?-group scheme over R, and let us consider a closed subgroup G m of G contained in the center Z(G) of G. Let A be a noetherian positively graded i?-flat G-algebra. As in the last subsection, 9JlA = 9Jt stands for the G-ideal ® i > 0 A{ of A. (1.2.1) We fix T C B C G and A C £ as in (1.4.5.6), and follow the notation in (1.4.5). Then as we have G m C Z(G)° C T at each geometric fiber, we have Gm C T over R, where (?)° denotes the connected component containing the unit element. This shows that a canonical map between the character groups |?| : X(T) —> X(Gm) = Z is defined as a restriction. For any A G X(T), we call |A| the degree of A. The set l?!" 1 ^) of weights of T of degree r is denoted by XT. For IT C X(T), we denote n D XT by TTT. For a poset ideal n of X+, AG,A{K) denotes the category of A-finite AG,A(X+). (G, A)-modu\es belonging to n as G-modules. Let AG,A stand for Note that AG,A{^) is a very thick full subcategory of AG,A = G,A^/We say that a poset ideal T of X+ is t-closed if Cc(T) is an /?-subalgebra (hence it is a subbialgebra) of H = R[G]. If F is t-closed and V and W are CG(F)-comodules, then so is V ® W. We say that a poset ideal F of X+ is t-closed with respect to A if A
Let F be a poset ideal of X+ which is t-closed with respect to A.
L e m m a 1.2.3 For V € gM/, the following are equivalent. 1 V £ M^ G ( r ) . 2 A®V
Proof. to A
GAGA^)-
1=^2 This is obvious, as F is assumed to be t-closed with respect
2=>1 If A ® V E AG,A(T)>
then
VS£(A®V)®A
A/mA is a quotient of
A
For V e AG,A(T)
and s,teZ(s<
D
t), we set
vM ••= vs e vs+l © • • • e
vt.
For s,t € Z (s < t) and n C X+, we denote by (*,s,t,A)Mf(r) the full subcategory of ^ ^ ( F ) consisting of M G AG,A(T) such that M[Sitj belongs to 7T. Note that (7r]Sjt;/i)M/(r) is a very thick full subcategory of ^ ^ ( F ) .
236
IV. Approximations of Equivariant Modules
Let 7T be a finite poset ideal of X+ = XQ. The Schur algebra of G with respect to IT is denoted by SG(TT) = S(TT). By definition, (?)(s,t] is an exact functor from ^iStt.A)Mf(T) to S(TT)M/. Remark 1.2.5 The conjugate action of G on itself is trivial on G m C Z(G). Hence, it follows that any root of G is of degree 0. It follows that if /z < A, then \fi\ = |A|. Lemma 1.2.6 Let p be a finite poset ideal of F, and V an R-finite Gmodule which is a quotient of a finite direct sum of copies of A. Then there is a finite poset ideal TT of T such that V
ii 7T is a finite poset ideal of F iii If M € G M belongs to IT, then (^4 <8> M)[S]t] belongs to TT. Lemma 1.2.8 Let s,t € Z (s < t), and p be a finite subset of F suc/i i/iat p = ps\J ps+i U • • • U /9f T/ien t/iere is an (s, t; A, T)-closed set containing p. Proof. If A € p and £t < A, then we have |/i| = |A| by Remark 1.2.5. Hence, we have s < |/x| < t. Replacing p by UAep(~°°! ^]> w e m a y assume that p is a finite poset ideal of F. We prove the lemma by induction on w = t — s. liw — 0, then n := p is (s, t; A, F)-closed, and we are done. In fact, if M belongs to TT, then M = Ms. As A is positively graded, (A
1. Approximations of (G, A) -modules
237
As M[S|t_!] belongs to n', (A
Hom s w (K, M[.it]) ^ Horn,,,, M ) M / ( r)(^ ® V, M)
is an isomorphism.
Lemma 1.2.10 Let -K be an (s,t;A,r)-closed set. Then for V e g^j M £ (w,s,t;/))M/(r) and i > 0, we have (7r)(V,
MM)
= E x t ; w j i M ) M / ( r ) ( ^ ® V, M).
Proof. If P is a finite 5(7r)-projective module, then A (g> P is a projective object of ^iStt.A)Mf(T) by the isomorphism (1.2.9). Hence, if F is an 5(7r)projective resolution of V, then ^4®F is a (^iS]t;j4)M/(r)-projective resolution of A <8> V, since yl is .R-flat. Hence, we have
/ M M ) S Jtfi( ^
® F, M)) S Ext^ a t ; / t ) M / (r)(^ ® V, M).
• Lemma 1.2.11 Let TT be an (s,t;A,T)-closed set, N € ( B , s, t;j 4)M/(r), and V 6e an S{ir)-projective module. Then we have E x t ^ (r)(A (g) V, N) = 0 (i > 0). Proof. Let a be an element of ExtlAc A^(A exact sequence (1.2.12)
o-tN-tMi-t
<S> V,N) represented by an
>Mi-*A®V-*0
in AG,A(T). By Lemma 1.2.8, there is an (s,<;^4,r)-closed set TT' D 7T and [Mi © • • • © Mi\[3
TT;
such that
238
IV. Approximations of Equivariant Modules By Theorem III.2.3.17 and Lemma 1.2.10, we have
S E x t ^ V , iV(s,t]) * Exf s w (V, JV M ) = 0 for i > 0. Hence, a is already zero in Ext1 , is zero in
M,(r)(-^ ®V,N).
Hence, it
D
Lemma 1.2.13 For M,N <S
AG,A{F),
there exists anX £ ^ ™ ( r ) such that
i There is a surjective (G, A)-linear map A
^
® X, N) s Ext{,(X, N) ^ Ext^X, N[s,t]) S Exfs(7r)(X, NM) = 0
for i > 0.
D
Theorem 1.2.14 Let M,N € A?,/i(r). T/ien the canonical map
is an isomorphism for i > 0. Proof. We use induction on i. The case i — 0 is trivial. If i > 0, then we take A ® X —> M as in Lemma 1.2.13, and we have a short exact sequence
Then we have an associated map between the two long exact sequences of Ext, and by the choice of A®X —> M, we are done by induction assumption applied to 0. •
1. Approximations of (G, A)-modules
239
Lemma 1.2.15 Let M,N £ AG,A- Then Ext^A(M, N) is R-finite for i > 0. Proof. By Lemma 1.2.13, there exists an exact sequence
such that Xj £ X%° and Ext*G A{A
1.3
Relative Ringel's approximation
(1.3.1) In this subsection, we follow the notation in the last subsection. So let R be a noetherian commutative ring, G a split reductive /?-group scheme with a fixed G m C Z(G), and A a noetherian /J-flat positively graded Galgebra. We set VJlA '•= ©j>i Ai. Let F be a finite poset ideal of X+ which is t-closed with respect to A. Moreover, in this subsection, we assume that A is good as a G-module. Lemma 1.3.2 Let V be a good G-module, and M a good (G, A)-module. If V is R-finite R-projective or M is R-flat A-finite, then M
240
IV. Approximations of Equivariant Modules
Lemma 1.3.6 Let F € V{T). Then the following hold: 1 F is good <^=> F/mF is V-good *=> F e F(A® {y%° nV0(T))). 2 EomA(F,A) is good <=» F/mF is A-good <=> F € Proo/. By Lemma 1.1.4 and Lemma 1.2.3, we may assume that F = with V € V0(T). Then as we have V = F/VJIAF and HomA(A ®V,A)^ A ® V*, assertion 2 is reduced to 1 by Corollary III.4.1.8. We prove the first equivalence in 1. The 4= direction is obvious by Lemma 1.3.2. The => is also obvious, because V = Ao <8> V is a, direct summand of A
= iM e -AcA?) | M is >l-projective, and HomA(M,A) is good}, := {N e AG,AF) I ^ i s g° o d - and proj.dim^ iV < oo}, and u%°A(T) := ^ r ° (r) D ^ ( r ) . If T = X+, then we may drop (r), and may write # £ ° , y g ° , wg~ , V, Vo, and so on. Lemma 1.3.7 The full subcategory XQ^T) of AG,A(T) is closed under extensions, direct summands, and epikernels. The full subcategory yc°A(T) of Ac,A(T) ™ closed under extensions, direct summands, and monocokernels. Proof. This is trivial.
•
Lemma 1.3.8 If X £ X^rA and Y is a good (G,A)-module, then we have Proof. By Lemma 1.3.6, we may assume that X = A®V, with V e X%'° = add^"(Ac). By Proposition II.1.1.14, we have
JA ® V, Y) S Ext&V, Y) = 0 for i > 0.
•
Proposition 1.3.9 We have w ^ ( r ) = A ® wg^(r), ond wg^(r) is an injective cogenerator of ^J° Proof. By Lemma 1.3.6, the inclusion A ® w^™(r) C w ^ ( r ) follows. We prove the inclusion in the opposite direction. Let T e w ^ ( F ) . By Lemma 1.1.4, there is a filtration 0 = T(o) C • • • C T(r) = T
1. Approximations of (G,A)-modules
241
of T such that T^/T^-i) = A
for 1 < z, j < r. Hence, the filtration splits, and we have T = 0 { A ® Q^ = A®Q. This shows T £ A ® Wcc(r)Now we prove that w ^ ( F ) is an injective cogenerator of A^™(r). Let X = ^4 <S> V, with V G A^™/rj. As w£,™/n is an injective cogenerator of Theorem III.4.1.20, there is an exact sequence of G-modules
such that Q G wg£(r) and V e Xj£(r). Then
is exact, and we have A ® Q e wg^(r) and .4 ® V e Ag^(r). In view of Lemma 1.3.6, Lemma 1.3.7 and Lemma 1.3.8, we have that W(?oA(r) is an injective cogenerator of XQ^(T) by Corollary 1.1.12.13. D Lemma 1.3.10 We have Xg^T)
= {M £ Ac A(T) | proj.dim^ M < oo} D
S5 Proo/.
By Lemma 1.2.13, for any M £ AG,A(T), there is an A ® X £ and a surjective (G,A)-\ineax map / : A ® X ->• M. Then as Ker/ belongs to F, we may proceed by induction on proj.dim^ M, and it suffices to prove that if M £ P(F), then M € X^T). As we have XC,A(T)
we prove this assertion by induction on a := ;fP™r) -resol.dimM/m A M = ^ ° ( r ) -proj.dim
M/mAM.
If s = 0, then M/OT^M is A-good, and we have M £ XgA(T) by Lemma 1.3.6. So this case is obvious. Let us consider the case s > 0. Then there is an exact sequence
such that X £ Xg°{r). Then M' £ V(T), and 0 -> M'/VJtAM' -> X -> M/MAM
->• 0
242
IV. Approximations of Equivariant Modules
is exact. As y£°(r) -proj.dim M'/mAM'
= s-1, M' £ xf^T)
T
assumption. As A®X £ X^ A{T), we have M e XgA(r), is proved.
by induction
and the assertion •
Combining the results above, we conclude the following. Theorem 1.3.11 The triple
is a weak Auslander-Buchweitz context of AG,A(T).
Moreover, we have
AG,A(T) | proj.dim^ M < 00}.
In particular, if A is a regular ring, then (XQTA(r),y^°A(r),u^°A(T)) Auslander-Buchweitz context of AC,A(T).
is an
Corollary 1.3.12 Let h be a non-negative integer, M 6 AG,A and 0 ^ M. Then the following are equivalent. 1 A>(E^(F)-resol.dim(M) = h and grade A M > h. 2 M is a G-module belonging to T, M is perfect of codimension h, and ExtA(M,A) is good. Proof. We prove 1=>2. Let
be an A"^(r)-resolution of M of length h. As this is an >l-projective resolution of M, we have h > proj.dim^ M > grade^ M > h, and we have that M is perfect of codimension h. As FQ belongs to F, M belongs to F. Moreover, as H o m ^ F , ^ ) ^ ] is a resolution of ExtA(M, A) and HomA(Fi, A) is good for each i, ExtA(M, A) is good, as good modules are closed under monocokernels. We prove 2=>1. By the theorem, we have M S A"^(F). Hence, there is an exact sequence (1.3.13)
0 -)• N -» F h _! ->
)-Fi->F0->M^-0
such that Fi £ X%mA(F) for i = 0 , 1 , . . . , h - 1. It suffices to prove N € *G™(r)- A s w e n a v e proj.dim^ M = h, it is clear that N e V{F). Hence,
1. Approximations of (G, A)-modules
243
the sequence (1.3.13) is an i4-projective resolution of M. quence 0 -> Fo* -> F* ->
Hence, the se-
> Fh*_j -> iV* -> E x t ^ M , 4) -» 0
is exact, where (?)* = EomA(?,A). As each F* and ExtJ^(M, A) is good, N* is also good, since good G-modules are closed under extensions and monocokernels. Hence, we have N £ <*G™(r). D Corollary 1.3.14 Let h be a non-negative integer, M £ Then the following are equivalent.
AG,A>
an
d 0 ^ M.
1 Wc r ^(r)-resol.dim(M) = h and grade^M > h. 2 M is a good G-module belonging toT, M is perfect of codimension h, and ExthA{M,A) is good. Proof. 1=>2 M is perfect of codimension h and ExtA(M,A) is good by Corollary 1.3.12. On the other hand, as any object in w ^ ( r ) is good and good modules are closed under monocokernels, it follows that M is good. 2=>1 As we have M 6 y%°A(T),
wg£(r)-resol.dim(M) = Ag£(r)-resol.dim(M) = h by Corollary 1.3.12 and Theorem 1.1.12.10, 7.
D
(1.3.15) Let us denote the full subcategory of good G-modules in G M by Q. For a G-module V, it is easy to see that A G -inj.dim V = Q -cores.dim V. Hence, Q is closed under extensions, epikernels, monocokernels, and direct summands in Lemma 1.3.16 If M is an A-finite (G, A)-module with proj.dim^ M < oo, then M £ Q as a G-module. Proof. As Q is closed under monocokernels and M € XQ'AI w e m a v assume that M € V. As Q is also closed under extensions, we may assume that M = A
AG,A{^),
where R is the m-adic completion of R, and ? — ? <S> R.
244
JV. Approximations of Equivariant Modules
1 F is an XQA(T) -approximation ofV, andp is right minimal. 2 F is an XQ°A{Y)-approximation ofV, and there is no non-zero direct summand of X contained inlmi. 3 ¥ is an X?T0Ar)-approximation
ofV, and p is right minimal.
4 F is an X?TC^(T)-approximation ofV, and there is no non-zero direct summand of X contained
inlmi.
of V, if In particular, there is a unique minimal XQ°A(T)-approximation proj.dim^ V < oo. Similarly, there is a unique minimal yQ°A(F)-hull ofV, if pro).dimA V < oo. Proof. As Endg^ X is infinite by Lemma 1.2.15, it is semiperfect. Hence, the theorem is proved similarly to Theorem III.4.2.6. D Corollary 1.3.18 Assume that (R,m) is local. If M G ^ c ^ ( r ) , then there is an uiQ°A(r)-resolutionW of M such that for any (possibly infinite) ^°A(T)resolutionG of M, F is a direct summand ofG as a (G, A) -complex. Such an ¥ is unique up to isomorphisms of (G, A)-complexes, and the length of¥ is finite and is equal to Wg.r^(F)-resol.dim(M). Proof. As the uniqueness is obvious, we prove the existence. We use induction on s := w£;r<^(r)-resol.dim(M). If s = 0, then we set F = • • • -> 0 -> 0 -> M -> 0. If s > 0, then we take a minimal
o-^r -> x -> M -*o of M. We set FQ := X, and we denote the unique o)g.r^(F)-resolution with the desired property of Y, which exists by induction assumption, by F'. We define F to be the augmented complex F' -> F o -> 0, which is obviously an cjgr^(F)-resolution of M, and is of length s by induction assumption. Let be an arbitrary Wgr<^(r)-resolution of M. It suffices to prove that F is a direct summand of G as a (G, ,4)-complex. We set U := Im9{. As each homogeneous component of U is fl-finite, it belongs to some finite poset ideal TT of T. As we have gl.dim S(TT) < oo for any finite poset ideal TT of F and U admits a (possibly infinite) good resolution
• • • -> G i+ i -+ d -> h -> 0,
1. Approximations of (G, A)-modules
245
each homogeneous component of I{ is good, and hence /* is good. This shows that a:0-»/i->Go->M->0 is an XQ°A(r)-approximation of M. If s = 0, then a splits, and hence F is a direct summand of G. Now assume that s > 0. By the theorem, a is isomorphic to the direct sum (3 © idiv for some W 6 <^G°A(^)> where /?:0->Ji-»Fo->M->0
(Ji
:=lmdi=Y)
and id w : 0 -4 W -> W -> 0 -)• 0. As the kernel Gi of the composite map Gi -4 7i —> IV is an extension of h by Ji, it is good. This shows that G\ —> W splits, and we have that Gi = G\ © W, and hence Gi € wgr^(F). Now we know that G is isomorphic to the direct sum of idiv and G' :
• G2 -¥ G\ -+ Fo -> 0.
Hence, it suffices to show that F is a direct summand of G'. However, this is clear by induction assumption applied to J\ and our construction. •
1.4 Relative Cohen—Macaulay Ringel's approximation (1.4.1) Let R be a noetherian commutative ring, and G a split reductive /Z-group scheme with a fixed closed subgroup Gm C Z(G). Let S be a noetherian i?-flat positively graded G-algebra. We assume that 5 is a regular ring and good (as a G-module). As S is faithfully flat over R, R is also regular. We set 9Jts := ©j>i St. Let / be a G-ideal of 5. We assume that / is perfect of codimension h, and we set A := S/I. Moreover, we assume that A is faithfully flat over R and good as a G-module. We define WlA :— 0 ^ At. Note that KR := R is a pointwise dualizing module of R. Corresponding to this pointwise dualizing module, the equivariant pointwise dualizing modules Ks and KA of S and A, respectively, are determined uniquely, see (1.1.7). Moreover, we assume that KA = Extg(j4, Ks) is also good. The settings above will be effective until the end of this subsection. Note that A is positively graded by assumption. Lemma 1.4.2 Ks is good.
246
IV. Approximations of Equivariant Modules
Proof. As 5 is Gorenstein, K$ is rank-one projective as an S-module. Hence, by Lemma 1.3.3, K$ is good. • Hence, 5 = 5/0 also satisfies the general assumption for A = S/I. Let us denote by XA the full subcategory of C,A^ consisting of (G, A)modules which are maximal Cohen-Macaulay as ^-modules. Let 3^4 denote the full subcategory of G,/IM consisting of /1-finite (G, .4)-modules such that inj.dim^ Mp < oo for any p G Spec ,4, and set uA := XA n 3^j. Note that for W G AG,A, we have W G U>A if and only if M, as an ^4-module (forget the (G, A)-mod\i\e structure), lies in addKA (see (1.4.10)). Note that any object in XA is .R-flat. Now we define the full subcategories Xc,Ai 3fc,/i! a^d U>C,A by XG,A
and
LJG,A
•= {M £ AG,A I M e XA and MomA{M,KA) is good}, •= {N € AG,A I N e yA and Honu(/C4, TV) is good}, —XcA
L e m m a 1.4.3 ? ®A KA : V ^> "VA is an equivalence of exact categories, with Hom/i(/C/i, ?) its quasi-inverse. Proof. Obvious by (1.4.10.16).
D
L e m m a 1.4.4 Hom,i(?, KA) is a duality of XA- That is, Hom/if?, KA) is a contravariant equivalence of exact categories from XA to itself. The quasiinverse of Hom(?, KA) is Hom(?, KA) itself. Proof. Note that the standard isomorphism M -> H o m ^ H o m ^ M , KA), KA)
(m ^ (/ ^
fm))
is (G, y4)-linear. By this observation and (1.4.10.15), the lemma is obvious.
• Lemma 1.4.5 The following hold: 1 For M € XA, M is good if and only if Homj4 (M, KA) £ XC,A if a"d only if there exists some L G XQ,A such that M = Hom / i(L, KA)2 For N £V, N is good if and only if N <8>^ KA G yG,A if and only if there exists some L G yG,A such that N = H o m ^ A ^ , L). 3 XGA is closed under extensions, direct summands, and epikernels in
G,/IM.
4 yG,A is closed under extensions, direct summands, and monocokernels in
1. Approximations of (G, A)-modules
247
5 If V is a A-good G-module and X 6 XGA,
then X ® V € XG,A-
6 If W is a V-good G-module and Y G yG
then Y
7 If T is a tilting G-module and Z 6 wGt,t, i/ien Z
^
9 A Proo/. 1,2 follow from Lemma 1.4.3 and Lemma 1.4.4. 3,4 follow easily from 1,2, Lemma 1.4.3, and Lemma 1.4.4. 5 As X <8> V is maximal Cohen-Macaulay and H o m ^ X ® V, KA) ^ Hom(X, tf,i) ® V , we have X ® V £ Ac,/i by Lemma 1.3.2. 6 We set Z := H o m A ( ^ , y ) . Then we have Z € yGmA = uG'°A. Let F be an wG"^-resolution of Z of finite length. As F is an -4-finite projective resolution of Z, F ® W is a finite ,4-projective resolution of Z g) W. As each term of F <8> W is good by Lemma 1.3.2, it follows that Z ® W is also good of finite .4-projective dimension. As we have HomA(KA, Y
Then as M e is >l-finite AWe set A^ = a d d ^ as an
N = EomA(F ®A KA, KA) S Hom^(F, KomA{KA, KA)) * EomA(F, A). By Lemma 1.3.6, T = F/WIAF is tilting. By Lemma 1.3.9, F S A®T. follows that M ^ (i4
It D
248
IV. Approximations of Equivariant Modules
Corollary 1.4.7 / / R is local and X € X+, then KA <8> TG(X) is indecomposable. Any object in UG,A is uniquely a direct sum of (G, A)-modules of the form KA®TG{X). Proof. As TG(X) ^ EomA(KA, KA ® TG(X)) ® A/WlA is indecomposable, we have that KA ®TG(X) is also indecomposable by the G-equivariant Nakayama's lemma. The second assertion is obvious by the proposition. D
Lemma 1.4.8 If X £ XGA and T is a V-good G-module, then we have ExtlGA(X,KA
ErtG>A{X ® r , KA) « ExtG(T',EomA(X,
KA))
by Proposition II.1.1.14. As T* is A-good and Hom^pf, KA) is good, the last Ext-module vanishes for i > 0. •
Proposition 1.4.9 We have A?,/i = %G,AProof. Let M e A(iiA. By Lemma 1.2.13, for any s > 0, there exists an exact sequence (1.4.10)
0 - > X ->• A®VS-+
> A®V!-+
A®V0-+
M
^0
such that each Vj is A-good, and hence A
> KA ® Vs' ->• H o m / i(X, ATA) ^ 0
in XA. As each /fyi ® V^* is good and s > Q -cores.dim(A^) — 1, we have that D Hom / i(X, KA) is good, and hence X £ AfG A . Proposition 1.4.11
UJG:A
is an injective cogenerator of XGA.
J. Approximations of (G,A)-modules
249
Proof. There exists a rank-one i?-projective G-module A such that Ks = 5<8> A. Then Ext£(i4,S) = KA® A* is good. Applying Corollary 1.3.14 to A 6 G,S^/> we have that there exists an exact sequence 0 -> Th -> TA_! ->
>Ti^T0^
A-+0
such that T{ € wg'5 (0 < i < h). We set J := Imd. Then J is good and fl-flat. Hence, Extg S (5,J) = ^(G, J) = 0, and we have a commutative diagram of (G, 5)-modules 0 - > / - > S - > . 4 - » 0
ir
ii
4=
O ^ J ^ T o ^ A ^ Q . We set y := Coker j = Coker j ' . As j 0 S/9Jls is an il-split mono, we have Torf(S/9Rs,y) = 0 and y / 9 J l s y is fi-flat. Hence, j is injective and Y' is 5-finite 5-projective, by an argument similar to the proof of Lemma 1.1.4. As 5 and To are good, Y' is good. Note also that f is injective. For M £ XG,A> w e set AT = HomA(M, KA). As 5 is regular, the triple {^GS^yG"side's) i s an Auslander-Buchweitz context in Ac,s- Hence, the (G, 5)-module N has an
where X is an 5-finite 5-projective module with X/msX A-good, and Y good. As Y and N are good, X is also good. Hence, we have X € UJQ°S, and there is a tilting G-module V such that X = 5 ® V. As Af is an i4-module, we have a commutative diagram of (G, 5)-modules 0 ->• I®V
-> S®V
-> A®V
-> 0
By the snake lemma, we have that p is surjective, i is injective, and Z := KerpS* Coker i. We show that Z is good. Consider the push-out diagram
0 -> 0 -¥
0
0
4
4
I0V U if® i v J 0 V -+
4 Y' 0V
4 0
Y 4 Y"
4
^
y®v 4 0
-->
z
-> 0
-> z
- > 0.
4=
250
IV. Approximations of Equivariant Modules
Then as Y' and V are good and V is infinite projective, we have that Y'®V is good by Lemma 1.3.2. As Y is good, Y" is also good. As J ® V and Y" are good, Z is also good. Now consider the exact sequence of maximal Cohen-Macaulay (G, A)modules
0-> Z ->A®V ^ N ->0. Taking the canonical dual, 0 -> M -> /C4 ® V* -> Homyl(Z, / d ) -> 0 is exact. As we have KA ® V* G WG,A and Hom/i(Z, / d ) € <*G,/I, it follows that LJC,A is a cogenerator of XQ,A- AS we know that U>C,A is Ak^-injective by Lemma 1.4.8, we have that WC,A is an injective cogenerator of XG,A- a Combining the results obtained so far, we have: Theorem 1.4.12
2
(XQA^GA
A) *S
o,n Auslander-Buchweitz context of
An application to determinantal rings
Let R be a commutative noetherian ring.
2.1 Resolutions of determinantal rings — the problem and its background Let m, n and t be integers such that 1 < t < m < n, and V = Rm and W = Rn. The symmetric algebra 5 := Sym(l^
0 -> Fh % F h _! ->
• Fi ^> 5 -»• S/It -)• 0
S/It with "good properties." Even if we restrict ourselves to consider this problem only, there is a long history about it.
2. An application to determinantal rings
(2.1.3)
251
Projective dimension and Cohen-Macaulay property
Ignoring the chronological order, we review an important result due to M. Hochster and J. A. Eagon [84]. Theorem 2.1.4 (Hochster—Eagon) The S-module S/It is perfect of codimension (m — £ + l)(n — t+1). If R is a normal domain, then so is S/It. Moreover, S/It is a free R-module. In particular, we must have h > (m — t+l)(n —1 + 1) in (2.1.2), whatever the resolution is. As 5 is locally Cohen-Macaulay over R, it follows that if R is CohenMacaulay, then so is S/It, by the theorem. We will give a sketch of a proof of the main part of Theorem 2.1.4 in (2.2). As S/It is an R-iree module, if F is an 5-free resolution of S/It, then for any commutative .R-algebra R', the base change R' <E)R F is an S'-free resolution of S'/I't, where 5 ' = R'®RS ^ R'[xi:j} and I[ = ItS'. In particular, if we have a free resolution (2.1.2) over the ring of integers Z, then we have a free resolution over arbitrary R by base change. However, the construction problem over Z is still a difficult problem, as we will see later.
(2.1.5)
Graded minimal free resolution
Letting each variable i y be of degree one, S is a graded polynomial algebra over R, and It is a graded ideal of S generated by elements of degree t. We say that the resolution (2.1.2) is graded if it is not only a finite free Sresolution, but also a complex in the abelian category of graded S-modules (that is, each Fi is graded and each dt is degree-preserving). We denote the ideal of 5 generated by all variables a^- by Tls. In other words, 9Jls = h- We say that the resolution (2.1.2) is minimal if di
is a graded minimal free resolution of S/It, then we have
rank Ft = dimfl Torf {S/Iu S/Wls). In particular, we have sup{i | Fi ^ 0} = (m — t + l)(n - t + 1).
252
IV. Approximations of Equivariant Modules
Note that the category of graded 5-modules is equivalent to the category of (T, S)-modules, where T = G m acts on 5 via degx^ = 1. Hence, 1 and 2 are nothing but a special case of Corollary 1.3.18. 3 is a local version of Lemma 1.2.8.3. The number rankF* = dim fl Torf(S// t ,5/OT s ) is called the zth Betti number of S/It (in the graded sense). Returning to the case of a general noetherian base ring R, if F is a graded minimal free resolution of S/It over R, then for any commutative i?-algebra R', the base change R' ®fl F is a graded minimal free resolution of S'/I't. So the construction over Z is important. However, the following negative result is known. Theorem 2.1.7 ([72, 73, 132]) IfR = Z, then there exists a graded minimal free resolution of S/It if and only ift = \ or m — t < 2.
(2.1.8)
Equivariant resolution
Let G denote the split reductive .R-group scheme G = GL(V) x GL{W). As V <8> W is a G-module in a natural way, the symmetric algebra S = Sym(V <S> W) is a G-algebra. It is easy to see that It is a G-ideal. A finite free 5-resolution F of S/It is called G-equivariant if F is a (G, S)module resolution. Any (G,S)-module has the grading corresponding to the morphism G m <—> Z(G) (t t-> (£idy,idw)) of .R-group schemes, and this grading is the one given by degi^ = 1. In particular, any G-equivariant finite free 5-resolution is graded. As 5 is /J-flat and positively graded, we may utilize the general theory in (1.2). An explicit construction of a G-equivariant minimal free resolution over R = Z is known, in the following cases. Example 2.1.9 The case t = 1. The Koszul complex (1.2.9) Kos(xij; 5): 0 -> S ® A m n (y ® W) ->
> S ® A'(V ® w) -* s -»• S/h -> 0
is a G-equivariant minimal 5-free resolution of S/It. Example 2.1.10 The case t = m. J. A. Eagon and D. G. Northcott [53] constructed a graded minimal 5-free resolution of S/Im, explicitly. D. A. Buchsbaum [29] gave a description of the Eagon-Northcott resolution as a (G, 5)-resolution, utilizing divided power representations. Buchsbaum's description shows that the Eagon-Northcott resolution has the structure of an ^^(Fj-resolution, where F is the t-closed subset of XQ which is uniquely characterized by GG(F) = fl[End(K) x End(W)]. In the two extreme cases described above, the graded minimal 5-free resolution is linear. The construction of D. Eisenbud and S. Goto [54] shows
2. An applicationtodeterminantal rings
253
that if the graded minimal free resolution of S/It is linear over any field, then the graded minimal free resolution of S/It exists and is linear over Z, and has a G-equivariant structure. By Theorem 2.1.7, there is a graded minimal 5-free resolution of S/It also for t = m— 1, m—2. K. Akin, D. A. Buchsbaum and J. Weyman [3] gave an explicit description of the minimal free resolution for the case t = m — 1, but it is not known that the minimal free resolution has a G-equivariant structure. The case t = m - 2 is more difficult. The question whether the graded minimal 5-free resolution of S/It has a G-equivariant structure in the case t = m — 1, m — 2, is open. (2.1.11) Buchsbaum-Rim resolution Almost at the same time as the construction of the Eagon-Northcott resolution (Example 2.1.10), D. A. Buchsbaum [28] independently constructed a finite free G-equivariant 5-resolution 0 -> Fn_m+1
->
> Fi ->• S -> S/Im -> 0,
of S/Im, such that each F* is of the form 0
5 <8» As(1) V ® A5(2) V ® • • • ® A s(i ~ J) V ® A
for i > 1. Although this resolution is not minimal, it has a good property in the sense that each Fi is a direct sum of tensor products of exterior powers of V and W. Exterior powers have some good properties which divided powers do not have. For example, the universal quotient Q of the Grassmann variety enjoys the property that the higher cohomology of A* Q vanishes for i > 0 by Kempf's vanishing theorem. The resolution is known as a generalized Koszul complex ox the Buchsbaum-Rim resolution [28, 31, 32]. (2.1.12) Lascoux—Pragacz—Weyman resolution If R = Q, then G = GL(V) x GL(W) is linearly reductive. It follows that the minimal free 5-resolution (2.1.2) of S/It is uniquely G-equivariant. A. Lascoux [99] gave a complete description of the (G, 5)-module Fi for any i for arbitrary m, n and t. He also gave a candidate for the boundary map di- P. Pragacz and J. Weyman [125] gave a complete description of the G-equivariant resolution later. So the minimal G-equivariant resolution of S/It is called the Lascoux-Pragacz-Weyman resolution (sometimes called Lascoux's resolution). K. Akin and J. Weyman study a larger action than that of G, namely the Lie super algebra gl(m\n) action on the resolution, see [5]. Note that if R D Q, then the base change of the Lascoux-PragaczWeyman resolution from Q is a G-equivariant minimal free resolution of
s/it.
254
IV. Approximations of Equivariant Modules
2.2 Buchsbaum—Rim type resolutions Against the background reviewed in the last subsection, Buchsbaum and Weyman studied the following problem [30]. Problem 2.2.1 Construct a finite free S-resolution F : 0 -> Fh ->
> Fx -> S -> S/It -> 0
of S/It explicitly, for arbitrary t, which satisfies the following conditions. 1 F is G-equivariant. 2 h = proj.dims S/It = (m-t
+ l)(n -t + 1).
3 For any i > I, Fi = S ® T(i) as a (G, S)-module, where T(i) is a direct sum of tensor products of G-modules of the form f\% V
2. An application to determinants! rings
255
Theorem 2.2.4 S, It, and Extg(S/It, S) are good as G-modules, where h:=(m-t + l){n-t + l). The goodness of S and It is classical, and known as the straightening formula, whose complete proof is found in [4]. The problem is the goodness of Exths{S/It, S).
We prove that Theorem 2.2.4 really implies Theorem 2.2.3. First note that we may assume that R is noetherian, as a base change of a BuchsbaumRim type resolution is again of Buchsbaum-Rim type. Set F =: F m x F n , where Ft is the set of partitions whose lengths are at most i. Then F is a t-closed subset of XQ. In fact, = CGL{V)(rm) ® C GL(v) (F n ) S R[End(V)) ® R[End(W)} is a subbialgebra of R[G] (III.4.6). Moreover, 5 = Sym(V
On the other hand, we have where C\ := CGL{v)(^m) and C% :— ^Gh(w)^n)- Hence, the lemma follows from Proposition III.4.6.4. D Now we know that Theorem 2.2.4 implies Theorem 2.2.3. If R is a regular ring, then 5 and A := S/It satisfy the general assumptions of (1.4) by Theorem 2.2.4. Hence, we have the following, by Theorem 1.4.12. Corollary 2.2.6 Set A := S/It. If R is a regular ring, then the triple is an Auslander-Buchweitz context.
256 (2.2.7)
IV. Approximations of Equivariant Modules Now we prove Theorem 2.2.4.
Lemma 2.2.8 The construction ofExt^(S/It,S) is compatible with a base change. Namely, for any morphism of noetherian commutative rings R -> R!, there is a (G, S')-isomorphism R! ® Exths(S/It, 5) S Exths,(S'/rt, S'), where (?)' = R! ® ?, and I't —t ItS' is an isomorphism. Proof. Obviously, we may assume R = Z. The map I[ —> ItS' is an isomorphism, since S/It is fl-flat (Theorem 2.1.4). There is a G-equivariant finite free S-resolution F of S/It whose length is h by Lemma 1.2.13. Then Hom5(F, S)[h] is a (G, S)-resolution of Ext£(S// t) 5) by Theorem 2.1.4. As we have Homs(F, 5)[ft]' = Homs<(F', 5')[ft] and F' is an S'-projective (G, S')D resolution of S'/I't, we are done. Hence, to prove Theorem 2.2.4, it suffices to prove that Extg(S/It, S) is u-good, assuming that R = Z. By the lemma again, we may assume that R is an algebraically closedfield,since Extg(S/It, S) is a direct sum of Z-finite Z-free Gz-modules.
2.3 Kempf's construction (2.3.1) In this subsection, we assume that R = k is an algebraically closed field, and we prove Theorem 2.3.2 Exths{S/Iu S) is good. This completes the proof of Theorem 2.2.4 (for general noetherian commutative R). We also prove Theorem 2.1.4 for the case that R = k is an algebraically closed field. As S/It is an iZ-free module (see, e.g., [4]), the general case of Theorem 2.1.4 follows easily. The following construction was given by G. Kempf, and used by Lascoux [99] and Roberts-Weyman [132] to study resolutions of determinantal rings effectively. (2.3.3) Set X := Eom(V,W*) = SpecS, and let Y denote the closed subscheme Spec/1 = Spec 5/7 of X. We denote the Grassmann variety [96] of (t — l)-quotients of V by G. There is a maximal parabolic subgroup P of GL(V) such that G = GL{V)/P. Letting GL{W) act trivially, G is a fc-smooth projective G-variety. Let
2. An application to determinants rings
257
be the tautological exact sequence of G. Note that rank Q = t — 1, and this is an exact sequence of G-bundles. We set Z := U ® W*. This is also a G-bundle over G. From the tautological exact sequence, we have an exact sequence
of G-bundles over G. We denote the projection from X x G to X (resp. G) bypi (resp. p2). The following is a variation of Kempf's vanishing theorem (1.4.7.6). Proposition 2.3.4 Let M = M(Vi,V2,V3) be a universal module functor overk of type (3,0), and assume that M is a filtered inductive limit of objects of add T({Lx{Vi) ® L^(K2) ® LV(V3) | A, n and v are partitions}). Then we have #'(G, M(V, Q, W)) = 0 (* > 0), and H°(G, M{V, Q, W)) is a good G-module. Moreover, the G-linear map M{V, V, W) s H°(G, M(V, V, W)) -> H°(G, M(V, Q, W)) induced by the canonical map M(V, V, W) —> M(V, Q, W) is surjective. This proposition was proved by Roberts-Weyman [132], and used effectively. Since p2 is affine and {p2)*Oz — Sym(Q
258
IV. Approximations of Equivariant Modules
Proposition 1.2.11.13). We have already seen the vanishing RlntOz = 0 (i > 0). As pii is proper, IT is also proper. We show that TT is birational in order to prove that TT is a resolution of singularities. For anyfc-scheme/ : W -> Specfc, we have X(W) = Hom(f*V,f*W*). By the definition of Grassmann variety [96], G(W) = {a : 0 -> 11' U f'V A Q' -» 01 rank Q' = t - 1}/ ~ , where we say that a ~ fS for two exact sequences
and P : 0 -> Tlx i > /*V ^> Qx ->• 0 if there is an isomorphism p : TZ' = TZi such that ii o p = i'. In the sequel, we consider that a = j3 for simplicity, if a ~ /?. Hence, we have Z(W) = {(a, VO | a e G(W), ip : Q' ^
W*}.
The morphism i : Z 4 X x G is given by the natural transformation {a,ip) H-> (a,xpp') via Yoneda's lemma (Lemma 1.1.1.6). Hence, ?r is given by (a,^) ^ VP'Next, we set U := Y \ Spec S/h-i- Note that we have U(W) = {
2. An application to determinantal rings
259
Proposition 2.3.6 The canonical sheaf w-z •= A t o p ^z//t *s isornorphic to ( A ' ° P V)ts>it~l)
® (A t o p l y ) ® ^ " 1 '
as a G-bundle over Z. Proof. As is well-known [90, p. 229], we have Q.G/k = Q* ® H. Hence, we have = Atop(Q* ®H) S (Atop = (A top g)® ( ~ m) ® (A top
v)®^'1).
As we have Vtz/G ^ Z* ^ Q ® W, it follows that WZ/G = (Atop Q)®n ® (A top H^)®('-1). By Lemma 1.2.11.12, we are done.
•
Again by Proposition 2.3.4, we have RlTrtwz = 0 (i > 0), and 7r.wz is good. By Proposition 1.2.11.13, Y is Cohen-Macaulay, and Theorem 2.1.4 for the case where R is an algebraically closed field follows. Moreover, there is an ^-isomorphism (not necessarily a (G, .^-isomorphism) TT.UZ ss cjy = Exths(S/It, Ks) s* Exths(S/It,
S).
As TT.WZ is good, Ext5(5// t ,/('s) is also good, by Lemma 1.1.11. Hence, Theorem 2.3.2 is proved.
Glossary the cardinality of a set X, xv the set of positive integers, xv No
the set of non-negative integers, xv
G*
the group of invertible elements of a semigroup G, xv
A*
the unit group of a ring A, xv the category of Ox-modules, xv
Qco(A")
the category of quasi-coherent Ox-modules, Xv
Coh(X)
the category of coherent Ox-modules, xv the field Ap/pA9 for p € Speed, xvi the fiber of «(p) in X, xvi
Max .A
the set of maximal ideals of A, xvi
suppM
the support of M, xvi
MinM
the set of minimal primes of M, xvi
M
the quasi-coherent sheaf associated to M, xvi
K(X)
the residue field Ox,x/ntx, xvi
R[X]
the coordinate ring of an affine ii-scheme X, xvi
AM
the category of (left) /4-modules, xvi
Set
the category of (small) sets, 1
Grp
the category of groups, 1
Ab
the category of abelian groups, 1
ob(C)
the set of objects of the category C, 1
C(M, N)
the set of morphisms from M to N in the category C, 1
Homc(M,N)
the same as C(M, N), 1 the opposite category of C, 1
262
Glossary
Func(.4, B)
the category of functors from A to B, 2
Nat(F, G)
the set of natural transformations from F to G, 2
cokeri
the canonical map from the target of i to Cokeri, 6
kerp
the canonical map from Kerp to the source of p, 6
Sexrt(.Aop, RM) the category of contravariant left exact R-functors from A to fiM, 8 C(A) +
the category of chain complexes in A, 9
C (A)
the category of chain complexes bounded below in A, 9
C~ (A)
the category of chain complexes bounded above in A, 9
Cb(A)
the category of bounded chain complexes in A, 9
¥[n]
the complex F shifted by n, 9
Hom^(F, G)
the Hom-complex from F to G, 9
K(A)
the homotopy category of unbounded complexes in A, 10
1
the full subcategory of K1 (A) consisting of exact sequences,
E (A)
10 1
D {A)
the derived category of A, 10
ExtJ
the nth extension group in the exact category determined by F, 15
RpQ
the ith F-right derived functor of Q, 17
sh(T, C)
the set of sheaves on T with values in C, 21
a(.F)
the sheafification of T, 22
Vf
the full subcategory of noetherian objects of V, 24
soc A
the socle of A, 27
rad A
the radical of A, 27
top A
the top of A, 27
addA^
the smallest full subcategory containing X closed under finite direct sums and direct summands, 28 the smallest full subcategory containing X closed under
T{X)
extensions, 28 X
the full subcategory of objects with finite
X -resol.dim A
the A'-resolution dimension of A, 28
263
Glossary
the full subcategory of objects with finite A'-coresolutions, 28
X
X -cores.dim A the A"-coresolution dimension of A, 28 X-m)A\mA X
I
the A'-injective dimension of A, 29 the full subcategory of •t-injective objects, 29
X-proj.dim A
the A'-projective dimension of A, 29
1
the full subcategory of -V-projective objects, 29
X
rad A
the Jacobson radical of a ring A, 30
flat.dim/j M
the /?-flat dimension of M, 40
W{M)
the quasi-coherent faisceau induced by M, 44
Na
stands for W(N), 44
ann M
the annihilator of M, 46
depthR(I, M)
the /-depth of M, 46
codim/j M
the codimension (the height of the annihilator) of M, 47
ht/
the height of/, 47
grade M
the grade of M, 47
emb.dim /?
the embedding dimension of R, 49
Ty(X, T)
the group of sections of T over X with supports in F , 51
//y(X, ?)
the ith local cohomology functor of X with supports in Y, 51
dim v ,(i)
the relative dimension of at x, 52
/?/*(M)
the ith Betti number of M, 56
l
fi R(M)
the ith Bass number of M, 56
type M
the Cohen-Macaulay type of M, 57
IY
the dualizing complex of Y, 63
Gr/ M
the graded module of M associated to / , 67
Mc
the category of right C-comodules, 75
C
the category of left C-comodules, 75
Homc(M, M')
the set of C-comodule maps from M to M', 75
Cobarc(M)
the cobar resolution of a C-comodule M, 79
M
/V S
c
L
the cotensor product of TV" and L, 81
264 Cotor^(M, N)
Glossary the zth cotorsion module of M and N, 84
Cobarc(M, N) the double cobar complex of M and N, 84 A°
the dual coalgebra of A, 90
A#U H
the smash product of A and U, 99
BM
the category of ( # , B)-Hopf modules, 100
Sch/X
the category of X-schemes, 101
GM
the category of G-modules, 103
Gm,x
stands for O$ = GL(1,X), 105
Gu
the set of unipotent elements of G, 106
EG
the set of roots of G, 107
W(G)
the Weyl group of G, 109
Ac
the base of the root system EG of G, 109
E£
the set of positive roots of G, 109
E5
the set of negative roots of G, 109
l(w)
the length of w, 109
Wo
the longest element (of the current Weyl group), 109
XQ
the set of dominant weights of G, 110
A*
stands for -w0X, 110
R\
the rank-one iMree S-module whose restriction to T (resp. U) is A (resp. trivial), 111
VG(A)
the induced module of G of highest weight A, 111
AG(A)
the Weyl module of highest weight A, 111
LG(A)
the simple G-module of highest weight A, 112
UMF(r, s; X)
the category of universal module functors of type (r, s) over X, 114
G,AM
the category of (G, ^4)-modules, 127
(f*
the restriction with respect to tp, 131
the inflation with respect to tp, 131
Y*
the image closure of the action Y x G -> X, 139
1*
the generic point of {x} , 140
Hy G
the hyperalgebra of G, 144
Glossary
265
G-sch
the category of G-schemes, 146
G,xM
the category of aff-(G, 0x)-modules, 151
htp
the height of p, 158
cohtp
the coheight of p, 158
Ac(A)
the Weyl module of highest weight A, 159
Vc(A)
the induced module of highest weight A, 159
C
M (TT)
the category of C-comodules belonging to n, 160
Lc(A)
the simple comodule of highest weight A, 160
V(?r)
the sum of all G-subcomodules of V which belong to M c (7r), 166
QC(A)
the injective hull of LC (A), 167
[W : VC(A)]
the number of V C (A) in any good filtration of W, 176
(V : LC(A))
the Jordan-Holder multiplicity of LC(A) in V, 177
R(X)
stands for Homc (A c (A), VC(A)), 187
5,
the Schur algebra with respect to it, 195
Ac
stands for Mc, 199
yc
stands for ^(V), 199
Xc
stands for ^(A), 199
ijjc
stands for Xc n yc, 199
Tc(A)
the indecomposable tilting module of C of highest weight A, 204
GG(7T)
the Donkin subcoalgebra of G with respect to n, 223
5G(T)
the Schur algebra of G with respect to n, 223
A
the transpose of a partition A, 225
XG,A(T)
see the corresponding page, 240
Q
the category of good G-modules, 243
It
= It{xij), the determinantal ideal generated by all t-minors of (xij), 250
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pp. 119-138, in Japanese.
Index preadditive, 2 (AB5) condition, 18, 75, 100, 127 adjoint representation, 105 admissible morphism, 6 antipode map, 74 artinian object, 24 associated faisceau, 101, 102 Auslander-Buchweitz context, 34, 202, 212, 220, 250 weak, 34, 211, 242 B base, 109 basic, 120 Bass series, 56 Betti number, 56, 252 bialgebra, 72 bialgebra map, 73 bicomodule, 76 Borel subgroup, 106 negative, 110 bounded, 9 bounded above, 9 bounded below, 9 Buchsbaum, D. A., xiv, 50, 252254 C canonical module, 63 canonical sheaf, 63 category Ab-, 2 additive, 2 artinian, 24 derived, 10 filtered, 4 noetherian, 24
R-,2 U-,1 character group, 105 coalgebra, 72 coalgebra map, 73 coassociativity, 72 cocommutative, 72 codimension, 48 cofinal, 4 cofree, 16 cogenerator, 32 Cohen factorization, 53 Cohen-Macaulay, 48 approximation, 120 maximal, 48, 117 Cohen-Macaulay type, 57 comodule, 75 comodule algebra, 99, 104 comodule map, 75 complete intersection, 54 complete intersection ideal, 48 contravariantly finite, 29 coproduct, 72 cotensor product, 81 cotilting bimodule, 116 cotorsion module, 84 counit of adjunction, 3 counit law, 72 counit map, 72 covariantly finite, 29 covering, 21 D degree, 225, 235
278 A-good, 191 dense subspace, 88 depth, 46, 48 depth sensitivity, 58 descent datum, 45 determinantal ideal, 250 direct image, 23 division ring, 27 dominant order, 110 dominant weight, 159, 187 Donkin subcoalgebra, 174, 195, 223 Donkin system, 187 dual algebra, 87 dual coalgebra, 90 dual Hopf algebra, 98 dualizing bimodule, 116 dualizing complex, 59 fundamental, 59 normalized, 60 E embedding dimension, 49 equidimensional morphism, 52 essential, 19 etale, 52 etale topology, 23 exact category, 5 F F-admissible, 15 F-exact, 15 F-injective, 16 F-right derived functor, 17 faisceau, 23 faithfully flat, 37 filtered inductive limit, 4 filtered inductive system, 4 filtered projective system, 4 final subcategory, 4 finite free complex, 43 flat, 37 flat complex, 43 flat dimension, 40 flat morphism, 51 formal character, 112
Index fppf sheaf, 23 fppf topology, 23 free complex, 43 functor additive, 2 constant, 3 continuous, 23 exact, 7 .R-linear, 2 universal module, 113 universally free, 113 universally projective, 113 G (G, ,4)-algebra, 127 (G, ^j-module, 127 G-algebra, 104 G-algebra map, 127 G-equivariant, 252 G-faisceau, 146 G-generator, 19 small family of, 19, 20, 24 G-ideal, 127 G-invariance functor, 127 G-linear map, 103 G-maximal ideal, 151 G-module, 102 rational, 102 (G, £>x)-niodule coherent, 147 quasi-coherent, 147 G-radical, 151 G-ring, 54 Gabriel-Quillen embedding, 8 generalized hyperalgebra, 98 generalized Koszul complex, 253 geometric point, xv good, 172, 193 good filtration, 176 Gorenstein, 49 Gorenstein dimension, 65 Gorenstein ideal, 48 Gorenstein module, 57 Grassmann variety, 256
279
Index Grothendieck, 19, 20 Grothendieck topology, 21 group-like element, 73 H Henselian, 30 highest weight coalgebra, 165 semisplit, 187 weak, 165 highest weight theory, 167 semisplit, 186 Hochster, M., 251 homomorphism Cohen-Macaulay, 54 complete intersection, 54 Gorenstein, 54 local, 53 Hopf algebra, 74 Hopf module, 100 hyperalgebra, 144 I IFP, 93, 138 indecomposable, 31 induced module, 111, 159, 187 induction, 104 inductive limit, 4 infinitesimally flat, 144 injective cogenerator, 33 injective hull, 19 invariance, 97 inverse image, 23 J Jacobson radical, 30 K /C-injective, 12, 126 /C-injective resolution, 12 Karoubian, 7 Koszul complex, 58 L A-extension, 208 minimal, 200, 208 left minimal, 29 left
Levi subgroup, 110 Lie algebra, 105 local ring, 30 locally artinian, 24 locally finite, 24 locally noetherian, 24 longest element, 109 M M-sequence, 46 maximal, 47 maximal torus, 106, 107 split, 107 MCM, 48 metafinite, 181 minimal, 251 minimal free complex, 56 minimal prime, xvi Mittag-Leffler, 42 Mittag-Leffler condition, 41 module trivial, 96 module algebra, 99 morphism compactifiable, 61 N V-good, 191 New Intersection Theorem, 47 noetherian object, 24 normal, 50 normally flat, 67 O opposite coalgebra, 76 Ox-module, xv P parabolic subgroup, 106 maximal, 106, 256 partition, 225 perfect, 251 perfect complex, 43 perfect ideal, 48 perfect module, 48 Poincare series, 56 pointwise dualizing, 60, 117
280 G-equivariant, 234 poset ideal, 157 positively graded, 230 presheaf, 3 presheaf inverse image, 23 product map, 72 projective complex, 43 projective cover, 19, 30 projective limit, 4 pure, 38 pure submodule, 38 Q quasi-coherent, 44 quasi-finite, 52 quasi-hereditary algebra, 165 weak, 165 quasi-isomorphism, 10 R /?-coalgebra, 72 radical, 27, 106 rank, 105, 106 rational, 93 rational part, 91 rational resolution, 65 rational singularity, 65 reductive, 106, 107 split, 107 refinement, 21 regular, 49 regular morphism, 51 relative dimension, 52 relatively acyclic, 17 representable, 3, 101 representation polynomial, 225 resolution Buchsbaum-Rim, 253 Eagon-Northcott, 252 Lascoux-Pragacz-Weyman, 253 Lascoux's, 253 of Buchsbaum-Rim type, 254 restriction, 104 right minimal, 29
Index right ^-approximation, 29 root negative, 109 positive, 109 S saturated, 7 saturation, 9 Schur algebra, 174, 195, 223 Schur functor, 225 semigroup scheme, 73 semiperfect, 30 semipure, 215 semisaturated, 7 semisaturation, 9 semisimple, 26, 27 Serre's (Ri) condition, 50 Serre's (Si) condition, 50 Sharp's conjecture, 60 sheaf, 21 of Kahler differentials, 52 sheafification, 22 short exact sequence, 6 simple object, 26 site, 20 skeletally small, 2 small, 2 small set, 1 small topology, 21 smash product, 99 smooth morphism, 51 socle, 27 split, 165, 187 split highest weight coalgebra, 158 weak, 158 stable, 102, 139 stable point, 140 straightening formula, 255 strictly Henselian, 30 su-acyclic comodule, 217 complex, 216 su-good, 218 subbialgebra, 83
Index subcoalgebra, 83 subcomodule, 75 subvariety, xv svelte, 2 Sweedler's notation, 76 T t-closed, 235 thick subcategory, 7 tilting module, 115, 204, 223 top, 27 torus, 105 split, 105 transpose, 225 U u-acyclic comodule, 184 complex, 179 u-good, 196 unipotent, 106 unit of adjunction, 3 unit map, 72 universal family, 113 universal functor, 113 universal map, 114 universally dense, 88 universe, 1 V variety, xv very thick, 7 W Wakamatsu's lemma, 30 weight, 106 Weyl group, 109 Weyl module, 111, 159, 187 Weyl's character formula, 112 X X-approximation, 34 minimal, 34 <¥-coresolution, 28
281 A"-projective dimension, 29 •f-projective object, 29 ^"-resolution dimension, 28 Y iV-hull, 34 minimal, 34 Yoneda product, 14, 15, 127 Yoneda's lemma, 2
