Ring Theory and Algebraic Geometry: Proceedings of the Fifth International Conference (Saga V) in Leon, Spain (Lecture Notes in Pure and Applied Mathematics)

(7)

proving that is left A-linear. It is also right A-linear because ^(aOc)6

=

^d*(S)a^e(df

® c)b

i

(17)

=

(7)

=

3dt®
Notice that the dual basis for C satisfies the following equality (the proof is left to the reader):

Doi-Hopf Modules and Entwined Modules

15

Using this equality one computes

(31)

=

(34)

=

(7)

=

(9)

-

(18)

=

V d* * d* Z—/ J

This proves that ^ is^right C"-colinear. Conversely, given E.V2, first one needs to show that 9 = aJ~ 1 ((/>) € Vi. It is now more convenient to work with Horn. (C, A) rather than C* A. For / e Horn (C, A), 6,6' e /I, (30) can be rewritten as

(bfb')(c) = btf(c*)b'.

(35)

Take any c, d 6 C, a G /I and compute

(35)

=

is right A-linear)

=

^ is left A-linear)

=

(35)

a^4>(l ® d*) (c)

=

This proves that ^ satisfies (17). Before proving (18), we look at the right C-coaction pr o n / = c*(g>a S Horn (C, A) ^ C*(g)A Write pr(f) = /[0](8>/[i] € Horn (C,A)®C. Using (31), we find, for all c e C,

This means that for all / e Horn (C, A) )).

(36)

16

Brzezinski et al.

This can be used to show that 9 satisfies (18). Explicitly,

6(c(2)®d}^®c^l}

(36) (4> is right C-colinear)

=

V(c(i) <8> 0(c (2) d))

=

0(1 ® d)[0](c) ®^(1

=

<j>(\ ® d^)(c) iS> d(2)

It remains to be shown that a\ and a^1 are inverses of each other. First take 0 6 Vi. Then for all c, d <E C,

=

6(d®c).

Finally, for 4> eV2, a e A and c, d 6 C:

® c)

a Now we give an alternative description for W%.

PROPOSITION 4.8 Let C be finitely generated and protective as a k-module. Then

W = Wl ^ W2 =Romk£A(C*®A,A®C). The isomorphism j3\ : W\ —> W2 is given by j3i(z) =

and the inverse of (3i is given by fcl(4>) = 4>(e®\}.

(37)

PROOF.- We have to show that @\(z) — is left and right ^.-linear and right Ccolinear. For all c* 6 C* and a, 6 6 A,

l (7)

=

53 a/ a^, 6*
c* (gi a)6,

Doi-Hopf Modules and Entwined Modules

17

proving that <£ is right A-linear. The proof of left A-linearity goes as follows:

*, cl(2j)cf(l} (7)

=

=

(9)

=

(23)

=

=

Next one needs to show that <£> is right C-colinear. Using (36), one finds

E i (c*, i

(9)

=

^aza^,® (c*,C( ( 2 ) )(cf) ( 1 ) ® (cf) ( 2 )

Conversely, let e W2 and put z = 0(e ® 1) = ^^ a; ® c;. Using (30), we see that a(£l) = (e l)a = za, and z 6 Wi. Take z = '£llai®ci€Wl. Then

Finally, take £ W2, and write j3l l($) = is right A-linear, right C-colinear and left C*-linear,

i

18

Brzeziriski et al.

and it follows that

)), as required.

D

Suppose that C is finitely generated and projective as a /c-module. From Proposition 4.7, it follows that F is separable if and only if there exists a map (f> € V^ = Horn ^ A(A <8> C, C* A) such that (/>(! C( 2 ))(c(i)) = e(c)l, for all c e C. In the Doi-Hopf case, this implies the Maschke Theorem in [12]. Now we apply the same procedure to determine when (F, G) is a Frobenius pair.

THEOREM 4.9 Consider an entwining structure (A, C, ijj), and assume that C is finitely generated projective as a k-module. Let F : A / t('0)5 ~* MA be the functor forgetting the C-coaction, and G — • ® C be its right adjoint. Then the following statements are equivalent: 1) (F, G) is a Frobenius pair. 2) There exist z = "^ ai ® ci £ W\ and 9 € Vj such that the maps

A®C and <j6: A ® C —> C* <S> A, given by

l i

are inverses of each other. 3) C* A and A Cg> C are isomorphic as objects in AA / t(V')5PROOF.- 1) => 2). Let z e Wl and d e Vi be as in Theorem 4.4. Then <j) = (3i(z) and (f> = a i (8) are morphisms in A-Mfy)1^, and

(27)

The fact that and are right A-linear and left C*-linear implies that 4>o4> = IC*®ASimilarly, for all c e C,

(18) (26)

Doi-Hopf Modules and Entwined Modules

19

Since and <j> are left A-linear, 0 o = 2) =» 3). Obvious, since c/> and (j> are in 3) =» 1). Let (f> : C* ® A —> A <S> C be the connecting isomorphism, and put

z = (e ® 1) = E; ai ®ci e Wi, 6 = a^-1) e Vi. Applying (32) and (37), one finds e i g i l = <^)-1 ((£(£ 01)) = Evaluating this equality at c € C, one obtains (27). For all c £ C,

1 0 c = <j)(4>~l(l <£> c)) = ^az6»(cj 0 c (1) ) 0 c (2 ). i Applying e to the second factor, one finds (26). Theorem 4.4 implies that (F, G) is a Frobenius pair. D REMARK 4.10 Recently M. Takeuchi observed that entwined modules can be viewed as comodules over certain corings. This observation has been exploited in [5] to derive some properties of coring counterparts of functors F and G. It is quite clear that the procedure applied in Section 2 to extension and restriction of scalars can be adapted to functors associated to corings, leading to a generalization of the results in this Section. This will be the subject of a future publication.

5

THE FUNCTOR FORGETTING THE A-ACTION

Again, let (A,C,il>) be a right-right entwining structure. The functor

G' : forgetting the A-action has a left adjoint F' . The unit p, and the counit 77 of the adjunction are given at the end of Section 3.

LEMMA 5.1 Let M e AM(^}% N e c' M(if)}cA.

Then F'G'(M) e A.MWOS

and

C

G'F' 6 .A/1 (•) ^. The left structures are given by

a(m 0 6 ) = am 0 b and pl(n 06) = /or a// a, 6 6 ^4, m £ M, n £ AT. Furthermore HM is left A-linear, and VN is left C-colinear.

Now write V = ^(G'-F', IA^ C ). W7' = MM(lc,^'G')- Following the philosophy of the previous Sections, we give more explicit descriptions of V and W . We do not give detailed proofs, however, since the arguments are dual to the ones in the previous Section. Let

V{ = {$ £ (C ® A)*

^(c (1) ®a^)c* 2) =tf(c ( 2 ) 0a)c ( 1 ) , for all c e C,aeA}. (40)

PROPOSITION 5.2 The map a : V -> V{, a(v'} =eovc is an isomorphism.

Brzeziriski et al.

20

PROOF.- Details are left to the reader. Given $ € V, for N & M.c, the natural map v'N : N A —> IV is

n For any /c-linear map e : C —> A <8> A and c 6 C, we use the notation e(c) = e 1 (c) ®e 2 (c) (summation understood). Let W[ be the fc-submodule of Horn (C, A <8> A) consisting of maps e satisfying e 1 (c ( 1 ) )(g>e 2 (c ( 1 ) )(8)c ( 2 )

e1

=

"(I)'

(41) (42)

PROPOSITION 5.3 The map 8 : W -> W{ given by

is an isomorphism. Given e £ W[, C,1 = /3~^(e) is recovered from e as follows: for

M

PROOF.- We show that 8 is well-defined, leaving other details to the reader. Consider a commutative diagram

A-

C-

•~A <8> C J^^..^ ® C ® A

3ft and right C-colinear. Write A(c) The map A = CC^A ° (7c C o,- ® a!i • Then C(i) <8> A(c( 2 )) = ^ ci(1) ig) c i(2 ) ® a, i

Applying e to the second factor, one finds C ( i ) ® e ( c ( 2 ) ) = A(c).

The right C-colinearity of A implies that a

(gi c *

Doi-Hopf Modules and Entwined Modules

21

and hence proves (41). To prove (42) note that A = CA®C ° (nA <S> IG) is left right A-linear, hence e 1 (c)®e 2 (c)a

=

(IA ® £ ® /A)(Cii®c(( 1 ® c ) a ))

D

PROPOSITION 5.4 Let (A,C,ip) be a right-right entwining structure. 1 ) F ' = • ® A : Jv[c —> .M (VOS is separable if and only if there exists "9 G V[ such that for all c G C, i?(c&l)=e(c). (43)

2) G' : M^)^ —> M° is separable if and only if there exists e G W[ such that for all c<=C, e1(c)e2(c)=e(c)l. (44) 3) (F',G') is a Frobenius pair if and only if there exist •& G V{ and e & W( such that e(c)l

=

79( C ( 1 ) ®e 1 ( C ( 2 ) ))e 2 (c ( 2 ) )

(45) (46)

PROOF.- We only prove 3). If (F' , G') is a Frobenius pair, then there exist i/ G V and C' e W such that (1) and (2) hold. Take $ e V{ and e € W{ corresponding to i/ and ^' and write down (1) applied to n ® 1 with n G N e Mc , (47)

Taking N = C, n = c, and applying £c to the first factor, one obtains (45). Conversely, if -& e V{ and e e W^i satisfy (45), then (47) is satisfied for all AT e Mc , and (1) follows since v'N ® I A and C/v®/l are right A-linear. Now write down (2) applied to m e M e A1(V>)2i (48)

Take M = C ® A, m = c (g> 1, and apply £c to the first factor. This gives (46). Conversely, if $ € V{ and e e VFj satisfy (46), then application of (46) to the second and third factors in TOpi ® mli} ® 1> an<^ then EC to the second factor shows that (48) holds for all M e M(t/>)%. Finally note that (48) is equivalent to (2). D Inspired by the results in the previous Section, we ask the following question: assuming (F',G!) is a Frobenius pair, when is A finitely generated projective as a fc-module. We give a partial answer in the next Proposition. We assume that ip is bijective (cf. [2, Section 6]). In the Doi-Hopf case, this is true if the underlying Hopf algebra H has a twisted antipode. The inverse of ijj is then given by the formula

ip~l(a c) = c5(a(!)) (g> a (0 ).

22

Brzeziriski et al.

PROPOSITION 5.5 Let (A,C,ip) be a right-right entwining structure. With notation as above, assume that (F', G') is a Frobenius pair. If there exists c & C such that e(c) = I , and if tjj is invertible, with inverse C —> C (x) A, then A is finitely generated and projective as a k-module. PROOF.- Observe first that (A, C,
that (7-10) hold, but with A and C replaced by A°p and Ccop. In particular, £(c

= e(c)a,

(49)

This can be seen as follows: rewrite (8) and (9) as commutative diagrams, reverse the arrows, and replace i/j by
a = e(c)a

(42)

= e(cv)av

=

0(c* <

Write (7®e)A(c) = Y.T=\ci®bi®ai £ C®A®A. For i = l , - - - , m , define a* e A* by (a*, a) = •&(cf <8> a^bij. Then {ctj, a* i = I , • • • , m} is a finite dual basis of A as a k-module.

D

From now on we assume that A is finitely generated and projective with finite dual basis {a,, a* | i = 1, • • • , m}. The proof of the next Lemma is straightforward, and therefore left to the reader.

LEMMA 5.6 Let (A, C, i]j) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. Then A* (g> C € cM.($)CA. The structure is given by the formulae

pr(a*®c]

=

I/ * ,~~ \ ^ _

p \Q> 09 Cj

a* <S>c (1) <8>C( 2 ),

(52)

/ *

iooi

\ y^ /o * /o\ T^

\d , &iih )C( -\\ 09 a^ Qv C / r j \ .

/CCQ°\

We now give alternative descriptions of V and W.

PROPOSITION 5.7 Let (A,C,il>) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. Then there is an isomorphism Pi '• W[ —> W!} = Horn kA(A* ®C,C® A),

Doi-Hopf Modules and Entwined Modules

23

Pie = f l , with

The inverse of f3\ is given by J3l l (fi) = e with i

PROOF.- We first prove that (3i is well-defined. a) 0i(e) = ft is right A-linear: for all a* e A*, c e C and 6 6 A, we have

\^/

(42)

—

\

5 ^"ibib

=

(a*,e 1 (c ( 2 ) )*}cf 1 ) ® e 2 (c ( 2 ) )fe

b) /?i(e) = fi is right C-colinear: for all a* 6 A* and c e C1, we have

(9) (41)

, =

(

^^,

( l )

(3)$

(a*,e 1 (c ( 2 ) )^)cj' 1 ' ) ®e 2 (c ( 2 ) )®c ( 3 ) ) <8>C( 2 ).

c

e

) /?i( ) = fi is left C-colinear: for all a* e A* and c e C, we have

(9)

The proof that /3f ^Jl) = e satisfies (41) and (42) is left to the reader. The maps /?i and /9f 1 are inverses of each other since

(a*,e 1 (c))a l (g)e 2 (c) = el(c) ® e 2 (c),

24

Brzeziriski et al.

) ((a*, (ai

At the last step, we used that for all c 6 C and a £ A, ®a) = c®a. D

PROPOSITION 5.8 Let (A,C,^) be a right-right entwining structure If A is a finitely generated projective k -module, then the map

<*i • V{ -» V2' - Horn g£(C7 ® A, A* ® C1), defined by a\ ($} = SI, with fi(c ® a) = (i9, C(!) (g) a^ai)a* (g) c*2) is on isomorphism. The inverse of ai is given by aj~ 1 (fi) = i? with

PROOF.- We first show that ai is well-defined. Take •& e V{, and let ai(i?) a) fi is right A-linear since for all a, b G A and c € C,

(2)

b) 0 is right C*-colinear since for all a G A and c G C, /3 r (Q(c®a)) (9)

= = =

c) f2 is left C'-colinear since for all a € A and c € C1,

(9)

=

*

Doi-Hopf Modules and Entwined Modules \ ^ f\ f j

!

„

25 \ li)' ^ „

*

)a -

(7) (40)

Conversely, given fi, we have to show that a^^fi) — $ satisfies (40). Take any c ig> a e C ® ^. and write Q(c ® a) = ^ 6* ®dL £ A* ®C. Since O is right and left C-colinear, we have

Therefore we can compute a),

Thus (40) follows. Finally, we show that c*i and aj"1 are inverses of each other. c

},

1

We know that ai(a^ 1 (O)) is right A-linear. Hence suffices it to show that

for all c € C. From (51), we compute ((a* <S> c)b, IA 0 £c) = (a*, 6}e(c) = (a* g) c, 6 ® e c ). Now write fi(c ® 1) = X^r- a* ® cr

anc

^ compute

26

Brzezinski et al.

THEOREM 5.9 Let (A,C,t{j) be a right-right entwining structure, and assume that A is finitely generated and protective as a k-module. With notation as above, we have the following properties: 1) F' is separable if and only if there exists SI 6 V^ such that for all c £ C, A®£c)

=£c(c).

2) G' is separable if and only if there exists 51 £ W^ such that for all c 6 C, ]T ai(ec ® lA)tt(a*i <E> c) = ec(c}l. i

3) The following assertions are equivalent: a) (F',G') is a Frobenius pair. b) There exist e £ W[, •& € V{ such that Q, = /?i(e) and O = cti(i)) are inverses of each other. c) A* C and C ® A are isomorphic objects in

PROOF.- We only prove a) => 6) in 3). First we show that £7 is a left inverse of fi. Since fi o Q is right ^4-linear, it suffices to show that

(40)

=

^(c ( 2 ) ®e 1 (c( 3 ))^)cj' 1 ) ®e 2 (c ( 3 ) )

(45)

=

c®l.

To show that f2 is a right inverse of O we use that fl o Jl is right (7-colinear and conclude that it suffices to show that for all c e C and a* G A* , (I A' ® e c )(n(n(a* ® c))) = £ C (c)a*. Both sides of the equation are in A*, so the proof is completed if we show that both sides are equal when evaluated at an arbitrary a & A. Observe that ))(i) ® e 2 (c (2) )*a l )a* ® (cf 1} )f 2)

Doi-Hopf Modules and Entwined Modules

27

hence (I A- <8>e c )(n(n(a*®c)))(a)

(42)

=

(a

(7)

-

{^a^e^VKcJf ®e 2 (c* 2) ))

(9)

- (a*,a^ 1 (( C *) ( 2 ) V)^((^)f 1 ' ) ® e 2 ((^) ( 2 ) ))

(46)

=

(a* ) a^)e( C *) =
as required.

6

D

THE SMASH PRODUCT

Let (B, A, R) be a factorization structure (sometimes also called a smash or twisted tensor product structure, cf. [27] [22, pp. 299-300] [16]). This means that A and B are /c-algebras and that R : A®B—+B(><)Aisa fc-linear map such that for all

a,c<E A, b, de B, =

bRr®arcR,

(54)

R(a®bd)

=

bRdr®aRr}

(55)

R(a®lB) R(lA®b)

- IB® a, = b®lA.

(56) (57)

We use the notation R(a ® b) = bR aR. B$RA is the fc-module B A with new multiplication = bdR#aRc. is an associative algebra with unit IB#^A if and only if (54-57) hold. In this Section we want to examine when B#RA/A and B$RA/B are separable or Frobenius. This will be a direct application of the results in the second part of Section 2. In Section 2, take R = A, S = B#RA. For v € Vi = Horn #^(5, R), define K : B -> A by Then 17 can be recovered form K, since z/(6#a) = «(6)a. Furthermore

a«(6) = al/(6#l) = v(bR#aR) = K,(bR)aR and we find that

V ^ Vl =* V3 = {K : B-+A o/e(6) = K,(bR)aR}.

(58)

Now we simplify the description of W = Wi C (B#RA) ®A (B#RA). Note that there is a /c-module isomorphism

28

Brzeziriski et al.

defined by

Let W3 = j(Wi) C B ® B ® A. Take e = bl <8> 62 ® a2 & B ® B ® A (summation implicitely understood). Then e e W$ if and only if (4) holds, for all s = 6#1 and s = l#a with 6 e B and a e A, if and only if bb1 ®b2 ®a2 1

2

(6 )R ® (6 2 ) r (8) aRra

=

b1 626B ® a^, 1

2

2

= b ® 6 a a,

(59)

(60)

for all a € yl, fe € S. This implies isomorphisms W7 ^ M/i ^ VK3 = {e = b1 62 ® a2 e B ® B & A | (59) and (60) hold}.

(61)

Using these descriptions of V and W, we find immediately that Theorem 2.2 takes the following form.

THEOREM 6.1 Let (B, A, R) be a factorization structure over a commutative ring k. 1) B#RA/A is separable (i.e. the restriction of scalars functor G : A/ls#RA -^ MA is separable) if and only if there exists e = b1 ® 62 <S> a2 G Ws such that

6 1 6 2
(62)

is split (i.e. the induction functor F : MA ~* MB#RA *s separable)

if and only if there exists K 6 V-j suc/i i/ia^

K(!B) = IA-

(63)

5j B^nA/A is Frobenius (i.e. (F,G) is Frobenius pair) if and only if there exist K e Vs, e e W3 such that

(b2)R ® K^^jja 2 = 61 ® «(6 2 )a 2 = 1B ® 1A.

(64)

Theorem 2.7 can be reformulated in the same style. Notice that

Hom fl (5',^) = RomA(B#RA,A') ^Eom(B,A). Rom(B,A) has the following (A,B#rtA)-l>imodule structure (cf. (6)):

(cf(b#a))(d)=cf(db)a, for all a,c G C and b, d e B. From Proposition 2.5, we deduce that

V ^ V2 - 1/4 = Horn ^^(SfeA, Horn (£,>!)).

(65)

If B is finitely generated and projective as a /c-module, then we find using Proposition 2.6

W ^ W2 <* W4 = KomA>B#RA(Rom(B,A),B#RA). Theorem 2.7 now takes the following form:

(66)

Doi-Hopf Modules and Entwined Modules

THEOREM 6.2 Let (B,A,R) be a ring k, and assume that B is finitely {bi, b* | i — 1, • • • , 771} be a finite dual 1) B^RA/A is separable if and only

29

factorization structure over a commutative generated and projective as a k-module. Let basis for B. if there exists an (A, B#RA}-bimodule map

(j>: Horn (B, A) ^ B* A -> B#RA such that

2) B^RA/A is split if and only if there exists an (A, B^RA)-bimodule map B#RA -» Horn (B, A) such that

3) B#RA/A is Frobenius if and only if B* ® A and B#RA are isomorphic as (A,B#RA)-bimodules.

This is also equivalent to the existence of K € ¥3, e =

b1 ig> b2 <8> a2 £ W% such that the maps

and

!>: B#RA^Kom(B,A),

$(b#a)(d) = K(bdR)aR

are inverses of each other. The same method can be applied to the extension B#RA/B. There are two ways to proceed: as above, but applying the left-handed version of Theorem 2.7 (left and right separable (resp. Frobenius) extension coincide). Another possibility is to use "op" -arguments. If R : A®B —> B®A makes (B, A, R) into a factorization structure, then

R: Bop® A°p -> A°p <S> B°p makes (A°P,B°P,R) into a factorization structure. It is not hard to see that there is an algebra isomorphism

(A0v#ABop)op * B#RA. Using the left-right symmetry again, we find that B#RA/ B is Frobenius if and only if (A 0 P#^B°P)°P/B is Frobenius if and only if (Aop#RBop)/Bop is Frobenius, and we can apply Theorems 6.1 and 6.2. We invite the reader to write down explicit results. Our final aim is to link the results in this Section to the ones in Section 4, at least in the case of finitely generated, projective B. Let (A,C,if)) be a right-right entwining structure, with C finitely generated and projective, and put B = (C*)°p. Let {ci,c* i = 1, . . . , n} be a dual basis for C. There is a bijective correspondence between right-right entwining structures (A, C, tj}} and smash product structures (C*op, A,R). R and ip can be recovered from each other using the formulae

J?(a®c*) =

(c*,cf)cJ(gia V ) ,

tp(c® a) =

((c*)fl,c) Ci aR.

30

Brzeziriski et al.

Moreover, there are isomorphisms of categories

and In particular, B^^A can be made into an object of ^.A/f ('!/') 5, and this explains the structure on C* ® A used in Section 4. Combining Theorems 4.9 and 6.2, we find that the forgetful functor M^}1^ —» MA and its adjoint form a Frobenius pair if and only if C* ® A and A <8> C are isomorphic as (A, (C*) op #fl J 4)-bimodules, if and only if the extension (C*)op#RA/A is Frobenius.

REFERENCES [1] K.I. Beidar, Y. Fong and A. Stolin, On Frobenius algebras and the Yang-Baxter equation, Trans. Amer. Math. Soc. 349 (1997), 3823-3836.

[2] T. Brzezinski, On modules associated to coalgebra-Galois extensions, J. Algebra 215 (1999), 290-317. [3] T. Brzezinski, Frobenius properties and Maschke-type theorems for entwined modules, Proc. Amer. Math. Soc., to appear. [4] T. Brzeziriski, Coalgebra-Galois extensions from the extension point of view, in "Hopf algebras and quantum groups", S. Caenepeel and F. Van Oystaeyen (Eds.), Lee. Notes Pure Appl. Math. 209, Marcel Dekker, New York, 2000.

[5] T. Brzezinski, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, preprint math.RA/0002105. [6] T. Brzeziriski and P. M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367.

[7] T. Brzeziriski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492. [8] S. Caenepeel, B. Ion and G. Militaru, The structure of Frobenius algebras and separable algebras, K-theory, to appear. [9] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Smash biproducts of algebras and coalgebras, Algebras and Representation Theory 3 (2000), 19-42. [10] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules, Applications, Adv. Math. 145 (1999), 239-290. [11] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules II, in "Hopf algebras and quantum groups" , S. Caenepeel and F. Van Oystaeyen (Eds.), Lect. Notes Puere Appl. Math. 209 Marcel Dekker, New York, 2000.

[12] S. Caenepeel, G. Militaru, and S. Zhu, A Maschke type theorem for Doi-Hopf modules, J. Algebra 187 (1997), 388-412. [13] S. Caenepeel, G. Militaru, and S. Zhu, Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997), 4311-4342.

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[14] S. Caenepeel, G. Militaru, and S. Zhu, Crossed modules and Doi-Hopf modules, Israel J. Math. 100 (1997), 221-247. [15] S. Caenepeel and §. Raianu, Induction functors for the Doi-Koppinen unified

Hopf modules, in "Abelian groups and Modules", A. Facchini and C. Menini (Eds.), Kluwer Academic Publishers, Dordrecht, 1995, p. 73-94.

[16] A. Cap, H. Schichl and J. Vanzura. On twisted tensor product of algebras. Common. Algebra 23 (1995), 4701-4735.

[17] Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373-385. [18] L. Kadison, The Jones polynomial and certain separable Frobenius extensions, J. Algebra 186 (1996), 461-475.

[19] L. Kadison, Separability and the twisted Frobenius bimodules, Algebras and Representation Theory 2 (1999), 397-414.

[20] L. Kadison, "New examples of Frobenius extensions", University Led. Series 14, Amer. Math. Soc., Providence, 1999. [21] M. Koppinen, Variations on the smash product with applications to groupgraded rings, J. Pure Appl. Algebra 104 (1995), 61-80.

[22] S. Majid. Foundation of Quantum Group Theory. Cambridge University Press 1995. [23] C. Nastasescu, M. van den Bergh, and F. van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413.

[24] R. Pierce, Associative algebras, Grad. Text in Math. 88, Springer Verlag, Berlin, 1982. [25] M. D. Rafael, Separable functors revisited, Comm. in Algebra 18 (1990), 14451459. [26] A. del Rio, Categorical methods in graded ring theory, Publ. Math. 72 (1990), 489-531.

[27] D. Tambara. The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 37 (1990), 425-456.

Computing the Gelfand-Kirillov Dimension II J. L. BUESO, J. GOMEZ-TORRECILLAS and F.J. LOBILLO, Departamento de Algebra, Universidad de Granada. E18071-Granada. Spain. E-mail: [email protected]

1

INTRODUCTION

This paper has a twofold goal and a mixed nature. We propose an algorithm to compute the Gelfand-Kirillov dimension for finitely generated modules over solvable polynomial algebras and, from the theoretical point of view, we characterize these algebras within the class of all filtered algebras. When looking for a notion of dimension for modules over a non-commutative algebra A which allows its effective computation, it is quite natural to look at the existing algorithms in Commutative Algebra. It seems reasonable to try first the computation of the dimension for the algebra A itself. In the case of commutative finitely generated algebras over a field k, the problem is equivalent to the computation of the degree of the Hilbert polynomial of k [ x j , . . . , xn]/I, where / is an ideal of the commutative multivariable polynomial algebra k [ x j , . . . ,xn], and this can be done effectively by means of the computation of a Grobner basis for / (see [2, Section 9.3] and its references). These ideas can be exported from the case of cyclic k xi,... ,xn -modules to finitely generated ones (see [15, Section 4]). From this point of view, the notion of dimension for modules which extends properly to noncommutative algebras is the Gelfand-Kirillov dimension. The techniques from the commutative case can be easily adapted to compute the Gelfand-Kirillov dimension in the case of quantum affine spaces, namely, finitely generated k-algebras defined by relations of the type XjXi = qjiXiXj, for some nonzero scalars QJJ € k (see e.g., [17], where quantum affine spaces are called homogeneous solvable polynomial kalgebras). When dealing with more general non-commutative k-algebras, a fruitful idea is to consider R to be a finitely generated k-algebra, with finite-dimensional generating vector subspace V, in such a way that the graded algebra gr(R) associated to the standard filtration is a multivariable commutative polynomial algebra (see [6]) or a quantum affine space (see [12]). 33

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However, a very simple example like the Jordan plane defined by two generators x,y, subject to the relation yx — xy + x2 shows that the former techniques, based on standard filtrations with (semi)commutative associated algebras, had to be improved. In the Spring of 1996, the first author presented in the SAGA4 held in Antwerp-Brussels an algorithm to compute the Gelfand-Kirillov dimension for cyclic modules over solvable polynomials algebras ([18]) with respect to the degree lexicographical order (see [4, 5]), which is applicable to some quantum groups whose generators are subject to quadratic relations, like quantum matrices, quantum symplectic space or quantum euclidean space). A similar approach appeared in the later paper [17, Section 6]. However, there is a mistake in the basic result there [17, Lemma 6.1] which is carried on the whole section. An easy counterexample to [17, Lemma 6.1] is the aforementioned Jordan plane. The algorithm given in [4, 5] was extended to finitely generated modules over solvable polynomial algebras (called there PBW algebras) with respect to a weighted graded lexicographical order in [23, 7], which allows to handle algebras with nonquadratic relations (the simplest example is given by the commutation relations yx = xy + x3). Here we show that this algorithm can be used to compute the Gelfand-Kirillov dimension of any finitely generated module over any solvable polynomial algebra, without restrictions on the given term order. During our search for that algorithm we have discovered some interesting results on filtered algebras which, in particular, locate the solvable polynomial algebras (or, equivalently, the PBW algebras) in the theory of noetherian rings: they are precisely those algebras having a filtration (in most cases, non standard) whose associated graded algebras are graded quantum spaces. We also show that these algebras can always be re-filtered by finite-dimensional vector subspaces keeping a quantum space as associated graded algebra. This allows to use the well-known graded-filtered techniques to obtain that any solvable polynomial algebra R is an Auslander-regular noetherian algebra with exact and finitely partitive Gelfand-Kirillov dimension. Moreover, R is a Cohen-Macaulay algebra which satisfies the Nullstellensatz.

2

ADMISSIBLE ORDERS IN MONOIDEALS AND STABLE SUBSETS

Let N denote the additive monoid of all positive integers (including the neutral element 0). Let n be a strictly positive integer. We consider Nra as additive monoid with the sum defined componentwise. Let 61,... ,en be the canonical basis of this free abelian monoid.

DEFINITION 2.1 An admissible order < on (N™, +) is a total order such that, (a) 0 = ( 0 , . . . , 0) ^ a for every a e N".

(b) For all a, (3,7 e N™ with a ^ f3 it follows 0 + 7 ^ / 3 + 7. REMARK 2.2 By Dickson's Lemma (see, e.g., [2, Corollary 4.48]), admissible orders on Nn are good orders (i.e., any non-empty subset of N™ has a first element). By ^iex we denote the lexicographical order with € j ~
Computing the Gelfand-Kirillov Dimension

35

define \a w = aiWi + - • • anwn, i.e., a\w is the dot product (a, w). The w-weighted degree lexicographical order ^w is defined by

and a ~( m ) = n N x {!,...,m} by a + (/3,i) = (a + /3,i). It is clear that any (a,i) e N n '( m ) may be written as (a, z) = a + (0, z) for any 1 < z < m and ct € N™. In the case m = 1,

we shall identify N"'(1) with N".

DEFINITION 2.3 A non-empty subset E of N"'( m ) is said to be stable if E = E + N™. The stable subsets of Nn are called monoideals of N n . The following lemma suggests that the study of some aspects of stable subsets of N™1*-"1-1 can be reduced to monoideals. This is the case of the notion of dimension that we will consider later.

LEMMA 2.4 Let E be a stable subset o/N™' ( m ) . For every i = 1,... ,m, the set Ei = {a e N"; (a,z) 6 E} is a monoideal o/N™. Moreover, E may be written as a disjoint union

where Ei = (0,i) + Ei. PROOF.

Straightforward.

D

The following result was pointed out in [15].

PROPOSITION 2.5 Let E be a stable subset o/N">( m ). There is a set (a^ii), . . . , (as,is) of elements of E such that s

^)+Nn)

(1)

Every admissible order X on Nn induces a total order on N n > ^ m \ which will also be denoted by ;< This order is defined by putting (2)

for any (a,i) and ((3,j) in N™' ( m ) . Dickson's Lemma extends to N re ' (m) , thus X is a well-ordering on N™1'"1-1. Moreover, it enjoys the following properties (a) (a,i) ^ 7 + (a,i) for every (a,i) € N"'W and every 7 € N";

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(b) for all (a,i), ((3,j) with (a,i) ^ (/3, j ) , and every 7 6 N n , it follows 7 + (c*,z) ^ 7+C/3J)

For any function / : N —> N, consider its dimension or growth degree denned as (3)

logra

which, of course, need not be finite. Some basic properties of this invariant can be found in [24], [27, Chapter 8] or [20]. DEFINITION 2.6 For any stable subset E C N">( m ) and any weight vector w with strictly positive integer components, we define the Hilbert function HF^ : N —» N of E relative to w by putting HF£(s) = card{(a, i) e N n >( m > \E

\a w ^ s}

for every s € N. In the case that w = (1, . . . , 1) we shall use the notation HF# for the Hilbert function.

In the proof of the next Lemma we shall use the notation |(o;,z)| w = |a w. By N™ we denote the subset of N" consisting of those vectors with all their components strictly positive.

LEMMA 2.7 Let E C N"'(m' be a stable subset and let w € N™ . Then d(HF£) = d(HFE). PROOF. Consider w = max{wj, . . . ,u>n}. If |(a,i)| < s, then |(a,z)|a, < ws and, thus, HFE(S) < HF^(ws). Moreover, it follows from

that \(a,i)\u < s implies |(a,i)| < s, whence HF%(s) < HFE(s).

D

By Lemma 2.7, the following definition makes sense.

DEFINITION 2.8 The dimension of a non-empty stable subset E of N n '( m ) is denned as dim(£) = d(HF^), for any w e N£. The study of the Hilbert function of a stable subset can be reduced to monoideals, as the following result shows.

PROPOSITION 2.9 Let E be a stable subset o/N™'( m ) and consider the decomposition

given in Lemma 2.4- Then

HF£ = HF£ + • - • + HF^m

(4)

dim(E) = max{dim(B1), . . . , dim(£;m)}

(5)

and

Computing the Gelfand-Kirillov Dimension

37

PROOF. The formula (4) is clear, since (a,i) ^ E if and only if a <£ E{. The equality (5) follows now from [27, Lemma 8.1.7]. D It follows from Proposition 2.9 that the computation of the dimension of stable

subsets of N n '( m ' reduces to the case of monoideals. Let E be a monoideal of N™. In the case that E is not proper, the dimension can be computed easily as ,. ,„ / 0 dim(.E) = < . „ v ' \ n iff E = rt0

For proper ideals, we shall need the notion of support of a vector 0 ^ ex. 6 N n , defined as supp(a) = {z € {!,..., n}

at ^ 0}

Consider the set V(E) = {cr C {1, . . . , n} a n supp(a) ^ 0 Va e E} which can be computed from a set of generators ai, . . . ,ccs of E since, as can be checked,

V(E) = {ff C {!,..., n}

a n supp(a fc ) ^ 0 V/c = 1, . . . , s}

The following result allows the effective computation of the dimension of any monoideal, and hence, of any stable subset. A proof, inspired on the material of [2, Section 9.3], can be found in [5, Section 4].

THEOREM 2.10 Let E be a proper monoideal o/N™. (1) dim(£) = n - min{card(cr)

a 6 V(E)}

(2) If m is the maximum of the entries of all vectors in a finite set of generators for E, then there is an unique polynomial h(x] G Q[z] such that HF£(s) = h(s)

for every s ^ mn. REMARKS 2.11 Equivalent descriptions for the dimension dim(B) can be deduced from [11, Theorem 3.1]. The fact that HFB(S) coincides with a polynomial for s big enough was proved in [19, Lemma 16, p. 51].

3

PBW ALGEBRAS, QUANTUM RELATIONS AND FILTRATIONS

Let R be an algebra over a commutative field k, and let xj, . . . ,xn be elements in R. A standard monomial in xi, . . . ,xn is an expression xa = x"1 . . .x^™, where a = (ai, . . . , an) & N". Assume that an element r € R can be written in the form

r = Y, r<*xa

(ra e k)

(6)

aSN"

The expression (6) is called a standard representation of r. Of course, ra ^ 0 only for a finite subset A/"(r) of N n . We will often refer as polynomials to the elements

of R having a standard representation.

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DEFINITION 3.1 Let ;< be an admissible order on N™, and consider an element r of R having a standard representation (6). The exponent exp(r) of (the standard representation of) r is defined as the maximum with respect to ;< of the finite set JV'(r). We note that exp(r) depends on the given order ^ and on the standard representation (6) of r. The type of relations that are satisfied by many interesting algebras suggests the following definition.

DEFINITION 3.2 Let R be a k-algebra generated by elements xi,...,xn. If there are nonzero scalars qji e k and polynomials pji (1 sC i < j ^ n) such that the relations Q - {xjXi = qjiXiXj +pji, 1 ^ i < j ^ n} hold in R, then we will say that the generators x\ , . . . , xn satisfy a set Q of quantum relations. If in addition, for every 1 ^ i < j ^ n for some admissible order ^ on N n , then the quantum relations Q are said to be ^-bounded.

DEFINITION 3.3 An algebra R over a field k is said to be a Poincare-BirkhoffWitt algebra if R is generated by finitely many elements x\, . . . , xn such that (PBW1) The standard monomials xa with a € N™ form a basis of R as a k-vector space. (PBW2) There is an admissible order X on Nn such that the generators xi, . . . , xn satisfy a set Q of ^-bounded quantum relations. Notice that R can then be thought of as the algebra generated by x\ , . . . , xn subject to the relations Q = XjXi = q^XiXj +PJI, 1 ^ i < j ^ n, such that no further latent relations appear. We will then say that R — k{xi, . . . ,xn; Q, :<} is a PBW algebra. Here, some pertinent comments about Definition 3.3 arise. The notion of PBW algebra is a restatement of the concept of solvable polynomial algebra introduced by Kandri-Rody and Weispfenning in [18]. It follows from [10, Theorem 1.2] that the condition (PBW2) is equivalent to

exp(/sO = exp(/) + exp(g)

(7)

for every /, g £ R. Finally, if R = k{xi , . . , , xn; Q, ^} is a PBW algebra and •<' is a different admissible order on N™, we cannot expect in general that R is a PBW with respect the new order -<' .

DEFINITION 3.4 ([26, Definition 3.7]). We will say that two elements x, y of the algebra R are semi- commuting if there is 0 ^ q & k such that yx = qxy. A graded k-algebra A is called semi-commutative if R is generated as k-algebra by a finite set of homogeneous semi-commuting elements.

Computing the Gelfand-Kirillov Dimension

39

If A is a semi-commutative graded algebra with homogeneous generators j/i, . . . , yn, then there are non zero scalars q^ G k such that y^y, = qjiUiyj, 1 ^ i < j ^ n. Clearly, the standard monomials ya = y"1 .. .y^n with a € N™ span A as a kvector space. Therefore, every element of A has at least a standard representation. A special interesting case occurs when the standard representation of every element in A is unique. This leads to the following definition.

DEFINITION 3.5 A semi-commutative graded algebra is said to be a graded quantum affine space if the monomials ya are k-linearly independent (and hence, form a k-basis for A). Of course, every graded quantum affine space is a PBW algebra with respect to any admissible order. Our aim is to show that the PBW algebras are, precisely, those filtered algebras which have a quantum affine space as associated graded algebra. The first step will be Proposition 3.6. First, let us recall that an algebra R over k is (positively) filtered, if it is endowed with an ascending chain FR = {FnR \ n ^> 0} of vector subspaces, the filtration of R, satisfying for all n,m ^ 0 f. 1 e FQR,

2. FnR C Fn+\R, 3. (FnR)(FmR) C Fn+mR, 4

- R=Un>0FnR-

It obviously follows from the definition that FoR is a subalgebra of R. For any filtered algebra R with filtration FR, let us introduce the graded vector space

where F-iR = {0}. If r € FiR \ Fi-iR, then r + Fi^\R has degree i and a(r) = r + Fi-iR is the principal symbol of r. The multiplication on gr(R) is based, via the distributive laws, on the rule

f

cr(rs) 0

i rs £ i+j_i otherwise

where r G Fj/? \ Fi-iR and s € K,-J? \ Fj_iR. This well-defined multiplication makes gr(jf?) into a k-algebra, called the associated graded algebra.

PROPOSITION 3.6 Let R be a filtered k-algebra with filtration FR such that gr(.R) is semi-commutative generated by homogeneous elements y\,. .. ,yn with deg(j/i) = Ui ^ 0 for I ^ i ^ n. Put u = (MI, . . . , un) e Nn and, for 1 ^. i < j ^ n,

let 0 ^ <jjj G k such that j/jj/j = qjiyiyj(1) If Xi,... ,xn g J? are swc/i i/iat Xj = j/j + FUi-\R for 1 ^. i ^ n then

F,R = , therefore, R satisfies (PBW2) with respect to ^
with exp(pjj) -
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for some polynomials pji G R.

(2) If gf(R) is a graded quantum affine space, then R = k{xi, . . . , x n ; Q, ^u} is a PBW algebra.

PROOF.

1. We will prove that FSR = J^iai <« ^a?" by induction on s. Let Z

be the set of indices 1 <J i <J n such that Ui = 0. So, assume s = 0. If Z = 0, then Fo-R — k and there is nothing to prove. If Z ^ 0, then we can assume that FoR = gr(.R)o and, therefore, j/j = a;, for every i & Z. Since every element in gr(.R) has a standard representation in the monomials y a , we get

F0R= Now, assume that s > 0 and let r G -FS.R \ Fs_iR. Since r + Fs-iR € gr(/?)s and the elements yi, . . . , yn are homogeneous generators for gr(/?) we obtain that

|a|u =s

|a|u =s

By the induction hypothesis, f/

\7

T T*'~* £^ Pg _ -i1 -*/-?^ — I QtJL> tz X —

\ /

ITT*^ .IS.**'

And, therefore,

H U ^n

The equality follows from the fact that FR = {FsR}s^o is a filtration and Xi 6 -FUi-R for z = 1 , . . . , s. In particular, we have proved that xi,..., xn generate R as an algebra over k. For i,j such that 1 ^ i < j ^ s, there is a non-zero scalar qji such that yjj/i = qjiyiyj. Therefore,

x-Xi + Fu.+u.-iR = (x- + Fu.-iR)(xi + Fu.-iR) = which implies that ir jit

J

M j *•

'

j

"i i "'J

^

/

^

In particular, pji has a standard representation such that exp(pjj) ^ u e^ + e.,-. 2. We just need to prove that the monomials xa are linearly independent over k. Assume that the monomials xa are not k-linearly independent. We derive a contradiction as follows. There is a non trivial expression 5ZQewn caXa = 0, that can be chosen of minimal exponent (3. Let s = \/3\u, then 0= |a|u =s

|a|u <s

Computing the Gelfand-Kirillov Dimension

41

which gives in gr(_R) s the following non trivial relation

E

v——^

caxa + Fs_lR=

\a\a=s

2_,

c

aya

\a\a=s

which contradicts the linear independence of the monomials ya.

D

REMARK 3.7 If the filtration FR in Proposition 3.6 is assumed to be finite dimensional then, necessarily, the vector u has all its components strictly positive.

The following result, although not explicitly stated there, was first proved by V. Weispfenning in [33, proof of Theorem 1]. We include a slightly different proof here.

PROPOSITION 3.8 Let R be a k-algebra finitely generated by x i , . . . , x n and assume that there are non zero qji € k, polynomials pji € R and an admissible order ^ on N™ such that the following ^-bounded quantum relations are satisfied for 1 ^ i < j ^ n with exp(pjj) -< €j + €j

Then there is w = (wi, . . . , wn) & N™ such that \exp(pji)\w < Wi+Wj. In particular, if R = k{zi, . . . , xn; Q, ^} is a PBW algebra, then R = k{xi, . . . , xn; Q, ^w} is a PBW algebra. PROOF. Let i,j be such that 1 ^ i < j
— Pji t ^/

Then

K.X

Write

where Cji — €j + €j — Aji. Then C is a subset of Z" such that (the unique extension to Z™ of) ^ is positive on C. By [29, Corollary 2.2] or [33, Proposition 2.1], there is some vector w = (wi, . . . , wn) £ N™ with w^ > 0 such that (w, /3} > 0 for every (3 € C. But this means that, for every 1
for every a € Aji

n After Proposition 3.8, it is possible to decide effectively if a given set of quantum relations Q is bounded by some admissible order on N" and, in the case of affirmative answer, to compute explicitly a bounding order of the form ^.w for some vector weight w = (wi, . . . , wn) £ N™ . To do this, write

c=

C

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where Cji = e, + €j — N(pji) for every 1 ^ i < j ^ n, consider the linear programming problem minimize /(to) = 101 + • • • + ws with the constraints

PROPOSITION 3.9 The set of quantum relations Q is bounded for some admissible order on N™ if and only if the linear programming problem (9) has a solution. In this case, <£> contains some vector w with integer components which gives a bounding ordering ^<w. Some explicit examples of computation of w by means of Proposition 3.9 can be found in [8].

PROPOSITION 3.10 Let R be a \i-~algebra generated by finitely many elements Xi,...,xn and assume that there are a vector w = (wi,...,wn) € N n , non-zero elements qji e k and polynomials pji e R such that the following ^<w-bounded quantum relations are satisfied for 1 ^ i < j sC n

Q EE XjXi = XiXj + pji

with exp(pij) -<w e.^ + €j

Define, for every positive integer s, F™R = ]Ciai <« kcca. Then

(1) The algebra R is filtered with filtration FWR = {F™R}. (2) The associated graded algebra grw(R) is generated by the principal symbols j/j = Xi + F^.^R forl^is^n. (3) If hji = Pji + F™i+w._1R then the following ^<w-bounded quantum relations are satisfied in grw(R) for 1 ^ i < j ^ n

Qw = yjVi = qjiViVj + hj{

with exp(%) -<w €i + 6j

(4) If R = k{o;i, . . . , xn; Q, ^<w} is a PBW algebra, then

is a PBW algebra. PROOF. 1. By [5, Proposition 1.7 (see also [18, Lemma 1.4]) we have that for every a, (3 6 N™ there exists qa^ € k such that xaxf3 = qa3Xa+P +

c

Obviously, 1 6 F^R. By (10), the monomials xa generate R as a vector space over k. This gives R = U s>0 F™R. Clearly, if a, j3 € N n are such that a\w < \(3\w then « ~<w ft- This implies that F™R C F™ R whenever s < t. For any pair of positive

Computing the Gelfand-Kirillov Dimension

43

integers s,t, let / 6 F™R and g e F?R. We want to prove that fg e F™+tR. Clearly, it suffices to prove that xax@ € F^_tR for every a.,/3 & N™ such that a\w ^ s and \/3\w ^ t. By (10), it is enough to check that \*Y\W ^ s + t for every 7 ±>«, a + /3. But this follows from the definition of ^w. 2. For each s Jj 0, the s-th homogeneous component of the graded algebra gr(jR) is generated as a k-vector space by xa + F^.1R, with |o: w = n, and we know that xa + F^R = ya, where y > = X i + F^^R for i = 1, . . . , s. Therefore, yi , . . . , yn are homogeneous generators for gr(R). 3. We have the following computation

yjyi = (Xj + F^.^RKxi + F^R) = x,x{ + F^+W]_,R = (qjiX.Xj + F^.^.^R) + (Pji + F^.^.^R) = q^y-j + %

(11)

Moreover, since N(hji) C N(pji) we get that exp(hji) ^w exp(pji) -<w e* + €j. 4. We just need to prove that the monomials ya are k-linearly independent. Since grw(R) is a graded algebra, it suffices to prove that every linear combination of homogeneous monomials of the form Xliaj =scaya = 0 is necessarily trivial. But this is a consequence of the linear independence of the monomials xa , since we have

REMARKS 3.11 1. When computing the relations Qw it is understood that we compute a standard representation of hji with respect to the standard monomials ya. This is very easy, in fact, if pj% = £QaQa:a then % = Y.\a\v =wi+Wi a «J/ Q 2. For every positive integer s it can easily be proven that

F?R={feR

|exp(/)U<s}

(12)

3. Let R — k{xi, . . . ,xn; Q, ^w} be a PBW algebra. It is easy to see that the filtration FWR is standard if and only if w = (1, . . . , 1), i.e., -^w is just the degree lexicographical order. ^ 4. There is an analog to Proposition 3. 10. (4) for the Rees algebra R associated to R = k{xi, . . . ,xn; Q, ^<w}, which was proved in [9, Teorema 2.6.3]. Thus, R = k[t,Xi, . . . ,xn;Q,^w] is a PBW algebra, where it is a new central variable, Xj = tWiXi for every 1 ^ i ^ n, w = (l,to), and the new ^-bounded quantum relations are

where — / T •TJ• — ^ - • T . ' r - 4— X j — yjz^i^j T^

\

/

,->, o^l ^A"' j

The foregoing propositions can be restated in a more compact form which is interesting from a theoretical point of view. We give two theorems. The first one explains, from the point of view of algebras given by generators and relations, when it is possible to find a filtration with semi-commutative associated graded algebra.

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THEOREM 3.12 The following conditions are equivalent for a k~algebra R. (i) R is filtered by a finite-dimensional filtration with semi-commutative associated graded algebra. (ii) R is filtered with semi-commutative associated graded algebra. (in) R satisfies a set Q of ^.-bounded quantum relations for some admissible order X. (iv) R satisfies a set Q of ^<w-bounded quantum relations for some vector w € N™.

PROOF,

(i) =^> (ii) is obvious.

(ii) => (iii) is given by Proposition 3.6.(1). (iii) => (iv). This is given by Proposition 3.8 (iv) =>• (i). By Proposition 3.8, w can be assumed to be such that \exp(pji)\w < Wi + Wj for every 1 ^ i < j <; n. This implies that, in Proposition 3.10.(3), all hjiS are zero and, hence, giw(R) is semi-commutative. D

DEFINITION 3.13 An algebra satisfying the conditions of Theorem 3.12 is said to be a somewhat semi-commutative algebra. The following Theorem characterizes the solvable polynomial algebras introduced in [18] in terms of nitrations. This gives a neat connection between the solvable polynomial algebras and the filtered structures on noncommutative rings studied in [28].

THEOREM 3.14 The following conditions are equivalent for a k-algebra R. (i) R is filtered by a finite-dimensional filtration with gr(fl) a graded affine quan-

tum space. (ii) R is filtered with gr(R) a graded affine

quantum space.

(iii) R is a PB W algebra. (iv) R is a PB W algebra with respect to some admissible order of the form ^w with w 6 NJ PROOF.

This follows from propositions 3.6, 3.8 and 3.10 in a way similar to that

of Theorem 3.12.

4

D

CONSEQUENCES AND EXAMPLES

Everyone working in the field of non-commutative algebras would greatly appreciate some way to get properties of a given algebra just checking some properties on the defining relations. This is the case for the algebras satisfying a set of quantum relations. You need just to check if the relations are bounded for some admissible order on N™ and the concluding theorems in Section 3, in conjunction with the well developed theory of filtered algebras, will take care of everything. We will recall some of these interesting properties. Let R be a noetherian k-algebra. We say that the Gelfand-Kirillov dimension is exact if for every finitely generated left or right -R-module M, GKdim(M) is an integer and

GKdim(M) = sup{GKdim(AO,GKdim(M/AO}

Computing the Gelfand-Kirillov Dimension

45

for every submodule N of M. R is left (resp. right) finitely partitive if, given given any finitely generated left (resp. right) .R-module M, there is an integer n > Q such that, for every chain M = M0 3 MI D • • • D Mm

with GKdim(Mi/Mi + i) = GKdim(M), one has m < n. R has the radical property if every element in the Jacobson radical is nilpotent. R has the endomorphism property over k if for each simple left R-module M, End(M) is algebraic over k. If both properties are satisfied by R, then R satisfies the Nullstellensatz over k. A finitely generated left or right .R-module M satisfies the Auslander condition if for every integer n and every submodule TV of Ext^(M, R), we have Ext^(Ar, R) = 0 for every i < n. A noetherian ring with finite injective dimension is said to be Auslander-Gorenstein if every finitely generated left or right .R-module satisfies the Auslander condition. If in addition R has finite global homological dimension, we say that R is Auslander regular. Finally, if M is a left or right .R-module, the grade number of M is defined as

j ( M ) = inf{i I Ext*fl(M,.R) ^ 0} e N U {+00} The noetherian k-algebra R is Cohen-Macaulay if for all non-zero finitely generated left or right .R-modules M,

j ( M ) + GKdim(M) = GKdim(R) THEOREM 4.1 Let R be a k-algebra satisfying a set of bounded quantum relations. Then the following statements hold. (1) The algebra R is left and right noetherian.

(2) The Gelfand-Kirillov dimension is exact on short exact sequences of finitely generated left or right R-modules.

(3) The algebra R is left and right finitely partitive. (4) The Krull dimension of every finitely generated R-module is bounded by its Gelfand-Kirillov dimension. (5) The algebra R satisfies the Nullstellensatz over k.

// R is any PBW algebra, then, in addition to the foregoing properties, R is an Auslander-regular and Cohen-Macaulay algebra.

PROOF. By Theorem 3.12, R has a finite-dimensional filtration with semicommutative associated graded algebra. The noetherianity of R follows from [25, 1.6.9]. The exactness of Gelfand-Kirillov dimension on short exact sequences of finitely generated left or right .R-modules follows from [32, 4.4 Theoreme] or [26, 3.8]. R is left and right finitely partitive by [16, Theorem 2.9]. The bound on the Krull dimension can be obtained from [25, 8.3.18] (see [31] for the original citation). The Nullstellensatz follows from [26, 3.8]. Let R now be a PBW algebra. Then R has a finite-dimensional filtration such that gr(.R) is an affine quantum space. Using [22,

Bueso et al.

46

3.6 Theorem] (see also [13, Theorem 4.2]), it is easy to see that gr(/?) is AuslanderGorenstein, so by [3, 3.9 Theorem] R is Auslander-Gorenstein. Moreover, by [25, 7.5.3] (see also [14]) gr(/?) has global finite dimension, so R is Auslander regular by [25, 7.6.18] (see also [30]). Finally, by [3, 3.8 Theorem] and [25, 8.6.5], R is Cohen-Macaulay if and only if gr(R) is Cohen-Macaulay, but this follows from [22, 5.10]

D

The following are examples of PBW algebras and, hence, they enjoy the properties listed in Theorem 4.1. In each case, we include an explicit vector weight w with strictly positive integer components in such a way that the filtration given in Proposition 3.10 yields a graded quantum affine space as associated graded algebra. • Enveloping algebras of finite-dimensional Lie algebras. Let 0 be a finite-dimensional Lie k-algebra with basis xi,... ,xn. Then the enveloping algebra U(Q) = k{xi,..., x n ; Q, ^.w} is a PBW algebra, where

i < j ^ n} and w = ( 1 , . . . , 1). • Quantum matrices are a special case of the algebra H(p, A) defined in [1]. H(p, A) = k{ziQ | 1 ^ i, a ^ p; Q, ^w} is a PBW algebra, where the relations are

+ (A - l)pjiXif}Xja

if j > i, /3 > a if j > z, a < j3 if j = i, (3 > a

and where the weight vector can be expressed here as a weight matrix:

w=

.

op+i

2

P+2

(2 i+Q ) 1<: .

.

...

2

2p

<

i

• Quantum Weyl algebras. Let Q = (r (k) = k{yi,...,yn,xi,...,xn;Q,^,w} is a PBW algebra, where

(i < 3) (i < 3} (i > J) and io = ( l , l , . . . , l , l , 2 , . . . , n ) .

Computing the Gelfand-Kirillov Dimension

47

• Quantum symplectic spaces. For every nonzero element q in k, the algebra Oq(s)f>k2n) = k{j/i,... ,yn,xi,... ,xn;Q,^.w} is a PBW algebra, where

j X i = q-xixj, lyi

2

- fyiXi = i-l

- i) i=\ E (?

• Iterated skew differential operators algebras. The foregoing quantized algebras and the enveloping algebras in the solvable case are instances of the following more general construction. Quantum euclidean spaces are obtained in this way. Let

be an iterated Ore extension of k. Assume that i(x\). Put w^ = max{l, 1012} + 1. Suppose we have defined wi, . . . , ifj-i for j > 2. For k = 1, . . . ,j — l, set Wfcj = max{aiifi + - • •+a,-_iuij__i; a 6 J\f(6j(xk))} and choose u>j = maxjljU;^ — Wi; 1 < i , f c < j — 1} + 1. Then /? = k{xj, ... ,xn;Q, ^w} is a PBW algebra, where Q=

iXj + 8j(xi]

I ^ i < j ^ n}

• Polynomial Ore algebras. For 1 ^ i ^ s, 1 ^ j sj r, let a,ji,bji G k with a_jj ^ 0, and let Cjj(cc) e k [ x i , . . . , z s ] be commutative polynomials. The algebra O = k{xi, . . . , x r , tti, . . . ,us; Q, ^w} is a PBW algebra, where the quantum relations are

j + bjiUj + Cji(x)

for

1

i <j ^ 5

for for

1 1

i <j ^ r i ^ s, I ^ j

The vector weight is

to = (!,...,!, wi,...,wr) where the weights w\, . . . ,wr are constructed as follows. For each 1 ^ i ^ s > 1 ^ J ^ r > let wji be the total degree of the polynomial Cji(x). Then choose Wj = max{u>ji 1
The algebra Vq(sl3(C)). Let q be a complex number such that q8 ^ 1. Consider the complex algebra given by ten generators /i2, /is, /23, &i, ^2, ' i j

48

Bueso et al.

/ 2 , 612, 613, 623 subjected to the following relations. 613612 = 9~ 2 6l2ei3

/13/12 = 9~2/12/13

623612 = 9 2 6l2623 - 9e13

/23/12 = ?2/12/23 ~ 9/13

623613 = 9~ 2 6l3623

/23/13 = 9~2/13/23

£2 _ /2

612/12 = /12612 +

612/01 = 9~ 6

k

= k

612/13 = /13612 + 9/23^

6

612/23 = /23612

6 13 fc 2 = 9~ 1 /C26l3

&2/13 =

e

23^1 = 9^1623

&1/23 =

623fc2 = q~2k2e23

/c2/23 =

1

613/12 = /12613 -
fcf fc| - l\l\ 613/13 = /13613 -

——5 2—————25—

~

13*1 = ^

,

2,

q-ll2el2

7 ,.

/2/12 = 9"1 /J2

613/23 = /236i3 + 9^|ei2

ei2,;2 =

623/12 = /12623

e ^ / j = qr/ i e i 3

/ j / ^ = q-/13^

623/13 = /13623 ~ 9~ /12^2

C 13 / 2 = g/ 2 6l3

^2/13 = 9/13^2

623/23 —

_

By solving the linear programming problem (9) we obtain that the displayed quantum relations are ^TO-bounded, where w = (3, 5,3,1,1,1,1,1,1,1,) Bergman's Diamond Lemma shows, after some large computations, that the standard monomials in /i2, /is, /23, ki, fc 2 , li, / 2 ,612,613,623 form a C-basis of Vq(sls(C)), and, thus, it is a PBW algebra. Notice that the quantized enveloping C-algebra of s[s(C) is a factor algebra of V^(sIs(C)).

5

GROBNER BASES FOR MODULES

Let R = k{Qi,..., Qn\ :<,} be a PBW algebra and consider the free left _R-module Rm with basis ei,...,em. Every finitely generated left .R-module is isomorphic to a factor module of the form Rm/K for some m and some submodule K of Rm which has to be finitely generated (R is noetherian). The computational treatment of finitely generated left /?-modules has its basic tool in the notion of Grobner basis of submodules of free modules of finite rank. For this purpose, we will introduce some notions and the corresponding notation. Our point of view is influenced by the theory developed for the commutative case in [21] and [15]. Every element / e Rm can then uniquely be expressed as Y^iLi fiei> where /; € R, for 1 < i < m. For every (a,i) g pj n .( m ) ) consider the element x^a'^ = xaei of Rm, where xa = x*1 • • -x%». Since B = {xa-} a g N"} is a k-basis of R, it follows that

Computing the Gelfand-Kirillov Dimension

49

Bm — {x(a'^; (a,i) £ N"'( m )} is a basis of Rm as a k-vector space. Therefore, every element / £ Rm has a unique standard representation

We define the Newton diagram of / to be

and the exponent of / to be exp(/) = maxA/"(/). Here, the maximum is computed with respect to the order on N"'(m) denned in (2). If (a,i) is the exponent of /, then

We will refer to c^a^ as the leading coefficient of / and we will write lc(/) = C( QJ J). The exponent of the submodule K of Rm is denned as the stable subset of N ra '^ m ^

Exp(tf) = {exp(/)

feK}

DEFINITION 5.1 A subset 9l , . . . ,gt of K is said to be a Grobner basis of K if

t

+N")

(13)

By Proposition 2.5, every submodule K of Rm posseses a Grobner basis, which is expected to be a set of generators for K. This can be deduced from the Division Algorithm, which is also the fundamental tool to compute effectively Grobner bases. By [10, Theorem 1.2], for every a, j3 £ PP, there is an unique nonzero scalar qa,p € k and a polynomial pa,/3 G R such that

xax13 = qa,0xa+f3 + pa>0

(exp(p Qj/3 ) -< a + (3)

Given / £ Rm with exp(/) = (a,k) e N™'( m ) j we define the scalar exponent of / as sexp(/) = a € N™. We will write (a,i) < n '( m ) (/3,j) if and only if i = j and (/3, i) = 7 + (a, z) for some 7 £ N™. As the reader could check by using Dickson's Lemma, the Division Algorithm given in Algorithm 1 is mathematically sound. The element r obtained in the Division Algorithm is said to be a remainder of 77* __ the division of / by the set F = {/lt . . . , fs}. We will denote by / any remainder. Of course, this notation is somewhat ambiguous, as the remainder is not in general uniquely determined. We have the following consequence of the Division Algorithm.

PROPOSITION 5.2 A finite subset G = {
elements is a Grobner basis of K if and only if f — 0 for every f £ K . In particular, every Grobner basis of K is a set of generators.

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Algorithm 1 Division Algorithm for free modules_______ _____ INPUT: / , / ! , . . . , /s € Rm with /, ^ 0 (1 < i < s) OUTPUT: hi,... ,hs,r such that / = hlf1 + ••• + hsfs + r and r = 0 or Af(r) D UI=i( ex P(/i)) + N n ) = 0 and max(exp(/i!) +exp(/ 1 ),... ,exp(/i s ) + exp(/J,exp(r)) = exp(/). INITIALIZATION: hi := 0 , . . . , hs := 0, r := 0, 5 := f WHILE g ^ 0 DO IF there exists i such that exp(/J <™'( m ) exp(g) THEN choose i minimal such that exp(/ i ) < n '( m ) exp(g)

__

O-i = l C (ff)l c (/i)'? S exp(g)-sexp(/ i ),sexp(/ i ) /H := hi + aixsexP^)-^p(f,)

g

:=g

-

aiX^xp(g)-sexp(fi)fi

ELSE r := r + lm(g) g := g - lm(gr)

The effective computation of the Gelfand-Kirillov dimension of Rm/K we are interested in needs as an important ingredient the computation of a Grobner basis for K. We present a version of Buchberger's algorithm that fits our PBW algebras (see Algorithm 2). This algorithm is a special case of the one we proposed in [10] with the purpose of the effective computation of "Ext functors" for modules over the left PBW rings introduced there. DEFINITION 5.3 Let f,g be vectors in the free left .R-module Rm, and write exp(/) = (a,j), exp(g) = ((3,k). Let 7 = (71, ...,7 n ) be defined by 7* = *i,/3j} for 1
SP(/ ) f f ) = [ g-i a a i c ( f f ) a .-y-a / _ g -i

lc(/)^-/3g

Htkk.

Algorithm 2 Left Grobner Basis Algorithm for Modules INPUT: F = { / ! , . . . , f,} C Rm with /^ 0 (1 < i < s)

OUTPUT: G = {glt... ,gt}, a Grobner basis for Rfi + --- + Rfs. INITIALIZATION: G := F, B := { { f , g } ; f ^ g e G} WHILE B ^ 0 DO Choose any {/,
Computing the Gelfand-Kirillov Dimension

6

51

HOMOGENEOUS GROBNER BASES

Let R be a filtered k-algebra with filtration FR. A filtration on a left /?-module M is a family FM = {FnM n ^ 0} of vector subspaces of M satisfying

1. FnM C Fn+iM for all n > 0, 2. (FnR)(FmM) C Fn+mM for all n,m > 0,

3- M = \Jn>QFnM. The graded vector space

is endowed with a structure of graded left gr(/?)-module in the following way. If ?7i G FjM \Fj-iM, the principal symbol of m is given by
fcr(rm) a(r)a(m) — < I0

itrm<£ FJ+i_iM otherwise

where r G ^.R \ F^fi and m G FjM \ Fj^M. Now, let # = k{:ri,...,:r n ;Q, i^} be a PBW algebra (by Proposition 3.8 any PBW algebra is of this form, after replacing, if needed, the old order ;< by a new weighted order ^w). By Proposition 3.10, grw(R) = k { x i , . . . , xn; Qw, ^w} is a PBW algebra. Let Rm be a finitely generated free left /^-module with basis BI, ..., em. This module has a canonical filtration FwRm given by F™Rm = (F™R)Rm for every positive integer s. It is immediate that F™Rm = (F™R)ei ® • • • ® (F™R)em. An immediate consequence is that grw(Rm) is a free graded left gr w (_R)-module with basis BI, ... em. Every B-submodule K of Rm inherits a filtration FWK given by F?K = K n F™Rm for every positive integer s. Also, the factor module M = Rm/K is filtered by Ff(M) = (F™Rm + K)/K. With these nitrations, the short exact sequence of left /^-modules

0 -+ K -* Rm -> Rm/K -* 0 is strict in the sense of [27], and, hence, we obtain a short exact sequence of graded left gr™(.R)-modules

0 -^ g?w(K) -* grw(Rm) -+ grw(Rm/K) -> 0 which implies, in particular, that grw(Rm/K) ^ giw(Rm)/grw(K). Therefore, a presentation of grw(Rm/K) can be constructed from a presentation of Rm/K. We will show that this construction can be done effectively.

LEMMA 6.1 For any non-zero vector f G Rm we have exp(/) = exp(<j(/)).

PROOF. First, we will prove that exp(/) = exp(<j(/)) for every / e R. So, let a = exp(/) and write |ct|u, = s. Therefore, / = caxa + 5^/3-< a C/3X^• Since R is a PBW-algebra with respect to ^w, it follows that

52

Bueso et al. *(f)=CaXa

+

V fi -<w fi

I/3|» =s

so exp(
=

'

/j

0

if |

if|exP(/J)U<S

and'/ = ( s / i , . . . , s / ™ ) . Clearly,

and, thus, exp(/) = (CM) = (exp(/ i ),i) - (exp^/O),*) = exp(
PROPOSITION 6.2 For every submodule K of the free left R-module Rm , we have

PROOF. This follows in a straightforward way from Lemma 6.1.

D

THEOREM 6.3 1. If {glt.. .,gt} is a Grobner basis for K then

is a Grobner basis for grw(K). 2. If {hi, . . . , hs} is a Grobner basis for grw(K) and g1, . . . , gt 6 K are such that a (di) = hi for every 1 ^ i ^ t, then {gl, . . . , gt} is a Grobner basis for K . PROOF.

1. Indeed, if {g1, . . . ,gt} is a Grobner basis of K, then

As gr™(.fQ) = Exp(.K') by Proposition 6.2, and exp(g i ) — e\p(a(gi)) by Lemma 6.1, clearly

so {a(gl\... ,a(gt)} is a Grobner basis of grw(K), indeed. 2. Similar to part 1.

D

Computing the Gelfand-Kirillov Dimension

7

53

THE GELFAND-KIRILLOV DIMENSION

Let us recall the definition of the Gelfand-Kirillov dimension of finitely generated kalgebras (details can be found in [27], [20], [24]). If V is a finite-dimensional vector subspace of R and n is a natural number, then Vn denotes the vector subspace of R generated by all n-fold products v\ • • -vn, where Vi & V. It is understood that V° = k. Define y(n) _ V^ yi i=0

Now, assume that R is finitely generated as k-algebra by V. The GelfandKirillov dimension of R (GKdim(R), for short) measures the asymptotic rate of polynomial growth of the 'dimension function' f ( n ) = dim^ V^n\ In fact, GKdim(.R) is the infimum of the real numbers r such that f ( n ) < nr for n 3> 0. It is known that this value is independent of the choice of the generating subspace V. The GK dimension of a finitely generated left .R-module M is given by the growth of the function /(n) = dimk(V^U), where U is any finite-dimensional vector subspace of M that generates M as left .R-module. This real number coincides with the dimension of the function f ( n ) as defined in (3). Our next goal is to show that the Gelfand-Kirillov of Rm/K can be computed from a given set of generators { f i , - - - , f s } of the submodule K. The first step is to replace the given admissible order X by some order ^<w for some vector w = ( u > i , . . . ,wn) with strictly positive integer components. This is possible due to Proposition 3.10 and the effective computation of the weight vector with strictly positive components is give by Proposition 3.9. Thus, assume that R = k{xi,... ,xn; Q, ^w} is a PBW algebra, which will be filtered by the iD-filtration FWR. Any finitely generated left R-module can be presented as M = Rm/K, where Rm is a free left .R-module with basis e i , . . . , e n , and K is a submodule of Rm. We endow M with the to-filtration FWM. The Hilbert function relative to this filtration is defined as

HF^(s) = dimk(F?M)

(14)

LEMMA 7.1 With the previous notation we have GKdim(M) =

PROOF. Let HFgrw ( M ) denote the Hilbert function of the graded left giw(R) module gr w (M), namely,

3=0

By [32, Proposition 3.2, Lemme 2.2],

GKdim(M) = GKdim(gr u '(M)) = Finally, for each positive integer s,

FW(M]

dirnk(gr-(M)J) j=0

j=0

j=0

'

54

Bueso et al.

which implies that HFgrw ( M ) = HF^.

n

LEMMA 7.2 For every positive integer s, the set

K

s, a,

is a frasis /or i/ie k -vector space F™(M). In particular,

PROOF. Let G — {g1, . . . ,gt} be a Grobner basis for K and consider f + K with |exp(/)|u, < s. By the Division Algorithm, we know that / = qig^ + • • • + qtQt + f where exp(r) ^<w exp(/) and A/"(r) n Exp(K ) = 0, i.e.,

It follows from (cc,z) ^w exp(r) for all (a,«) £ A/"(r), that KQ:,?)!^, < s, so

r =

Moreover, if (a,i) <£ Exp(/f), then a;^a'^ ^ X. We thus have proved that the set of all x^a'^ + K with the property that |(a,z)|u, < k for (a,z) ^ Exp(K) is a system of generators for FSU)(M) = {/ + K] |exp(/)| TO < s}. Let us now prove their linear independence. If

then r = E( Q ,i)^Ex P (K) c(a,i)X(a^ £ ^> w h e r e M ( r ) r \ E x p ( K ) = 0. By Algorithm 1 and Proposition 5.2, we obtain that r = Gr = Q, so C(a,i) =0. D

THEOREM 7.3 Let M be a finitely generated left R-module with a presentation M = F/K. Then GKdim(M) = dim(Exp(A")). PROOF. This is a consequence of Lemmas 7.1 and 7.2.

D

Let us now describe the algorithm to compute the Gelfand-Kirillov dimension of a given finitely generated left .R-module M = Rm/K over a PBW algebra R = k{xi,... ,xn;Q,^<}. Given a set of generators {/ 1 ; ...,/ s } for K, proceed as follows.

1. Compute a vector weight w = (wi,..., wn) € N+ such that

R =. is a PBW algebra.

Computing the Gelfand-Kirillov Dimension

55

2. Compute, by using the algorithms described in Section 5, a Grobner basis G for K.

3. Compute, from G, the stable subset Exp(K) of N"'( m ).

4. The Gelfand-Kirillov dimension of M is the dimension of 'Exp(K), which is computed by using Theorem 2.10.(1). REMARK 7.4 When w = (!,...,!), it follows that the ^-filtration of M = Rm/K is just the standard filtration of M. On the other hand, we have that the Hilbert function associated to this filtration is, precisely, WFExp(K)- By using the results of Section 2, we get that HFE X p(/f)( s ) 'ls a polynomial of known degree, for s big enough. By Theorem 2.10.(2) and Proposition 2.9, this polynomial can be explicitly computed by interpolation. Details can be found in [9]. This applies in particular to Weyl algebras and enveloping algebras of finite dimensional Lie algebras, which are PBW algebras with respect to -<degiexReferences

[1] M. Artin, W. Schelter, and J. Tate, Quantum deformations of GLn, Commun. Pur. Appl. Math. 44 (1991), 879-895.

[2] T. Becker and V. Weispfenning, Grobner bases. A computational approach to commutative algebra, Springer-Verlag, 1993. [3] J.-E. Bjork, The Auslander condition on noetherian rings, Sem. d'Algebre P. Dubreil et M.-P. Malliavin 1987-1988 (M.-P. Malliavin, ed.), Lecture Notes in Mathematics, no. 1404, Springer-Verlag, 1989, pp. 137-173. [4] J. L. Bueso, F. J. Castro, J. Gomez-Torrecillas, and F. J. Lobillo, Computing the Gelfand-Kirillov dimension., SAC Newsletter 1 (1996), 39-52, http://www.ugr.es/~torrecil/Sac.pdf/

[5] ____, An introduction to effective calculus in quantum groups, Rings, Hopf algebras and Brauer groups. (S. Caenepeel and A. Verschoren, eds.), Marcel Dekker, 1998, pp. 55-83. [6] J. L. Bueso, F. J. Castro, and P. Jara, The effective computation of the GelfandKirillov dimension, P. Edinburgh Math. Soc. (1997), 111-117. [7] J. L. Bueso, J. Gomez-Torrecillas, and F. J. Lobillo, Effective computation of the Gelfand-Kirillov-dimension for modules over Poincare-Birkhoff-Witt algebras with non quadratic relations., Proceedings of EACA-98, Universidad Complutense and Universidad de Alcala, 1998.

[8] ____, fiCudndo un Algebra finitamente generada es de tipo PoincareBirkhoff-Witt?, Proceedings of EACA-99, Universidad de La Laguna, Tenerife, 1999, pp. 195-204.

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[9] J. L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Dimension de GelfandKirillov y Polinomio de Hilbert-Samuel de Grupos Cudnticos: Metodos Algoritmicos, http://www.ugr.es/~torrecil/academia.pdf, Research Award of the Academia de Ciencias Matematicas, Fisico-Qufmicas y Naturales de Granada, 1999. [10] J.L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Homological computations in PBW modules., preprint, 1998. [11] G. Carra Ferro, Some properties of the lattice points and their applications to Differential Algebra, Comm. Algebra 15 (1987), 2625-2632. [12] F. Chyzak, Fonctions holonomes en calcul formel, Ph.D. thecnique, 1998.

thesis, Ecole Poly-

[13] E. K. Ekstrom, The Auslander condition on graded and filtered noetherian rings, Seminaire Dubreil-Malliavin 1987-1988, Lecture Notes in Mathematics, vol. 1404, Springer, 1989, pp. 220-245. [14] K. L. Fields, On the global dimension of skew polynomial rings, J. Algebra 13 (1969), 1-4. [15] A. Galligo, Algorithmes de calcul de base standards, Prepublication de 1'Universite de Nice, 1983. [16] J. Gomez-Torrecillas and T.H. Lenagan, Poincare series of multi-filtered algebras and partitivity, J. London Math. Soc., to appear. [17] Li Huishi, Hilbert Polynomial of Modules over Homogeneous Solvable Polynomial Algebras, Comm. Algebra 27 (1999), 2375-2392. [18] A. Kandri-Rody and V. Weispfenning, Non-commutative Grobner bases in algebras of solvable type, J. Symb. Comput. 9/1 (1990), 1-26. [19] E.R. Kolchin, Differential York, 1973.

Algebra and Algebraic Groups, Academic Press, New

[20] G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman Pub. Inc., London, 1985. [21] M. Lejeune-Jalabert, Effectivite D.E.A., Univ. Grenoble.

des calculs polynomiaux, 1984-85, Cours de

[22] T. Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. (1992), 277-300.

[23] F. J. Lobillo, Metodos algebraicos y efectivos en grupos cudnticos., Ph.D. Universidad de Granada, 1998.

thesis,

[24] M. Lorenz, Gelfand-Kirillov dimension and Poincare series, Cuadernos de Algebra, vol. 7, Universidad de Granada, 1988.

Computing the Gelfand-Kirillov Dimension

57

[25] J. McConnell and J. C. Robson, Noncommutative noetherian rings, Wiley Interscience, New York, 1988.

[26] J. C. McConnell, Quantum groups, filtered rings and Gelfand-Kirillov dimension, Noncommutative Ring Theory, Lecture Notes in Math., vol. 1448, Springer, 1991, pp. 139-149. [27] J.C. McConnell and J.C. Robson, Noncommutative Noetherian rings, J. Wiley and Sons, Chichester-New York, 1987.

[28] T. Mora, Seven variations on standard bases, Preprint, 1988. [29] T. Mora and L. Robbiano, The Grobner fan of an ideal, J. Symbolic Comput. 6 (1988), no. 2-3, 183-208.

[30] A. Roy, A note on filtered rings, Arch. Math. 16 (1965), 421-427.

[31] S. P. Smith, Krull dimension of the enveloping algebra o/sl(2,C), J. Algebra 71 (1981), 89-94. [32] P. Tauvel, Sur la dimension de Gelfand-Kirillov, Comm. Algebra 10 (1982), 939-963.

[33] V. Weispfenning, Constructing Universal Grobner Bases, Proceedings of AAECC 5, Springer LNCS, 356, 1987, pp. 408-417.

Some Problems About Nilpotent Lie Algebras J.M. CABEZAS and E. PASTOR, Dpto. de Matematica Aplicada, Universidad del Pais Vasco, Vitoria-Gasteiz. Spain. E-mail:[email protected] and [email protected] L. M. CAMACHO, J. R. GOMEZ and A. JIMENEZ-MERCHAN, Dpto. Matematica Aplicada I, Universidad de Sevilla, Sevilla. Spain. E-mail:[email protected], [email protected] and [email protected]

J. REYES and I. RODRIGUEZ, Dpto. de Matematicas, Universidad de Huelva, Huelva. Spain. E-m&il:[email protected] and [email protected]

Abstract

In this paper we survey the classification of some important families of nilpotent Lie algebras. We also consider some geometric problems on such families and several related computational topics. Most of the results have been obtained by at least one of the authors of this paper. So, we give an overview of the work developed on Lie theory over the last years by some members of the research group directed by J.R. Gomez.

1

INTRODUCTION

The importance of Lie algebras in different domains of mathematics and physics has become increasingly evident in recent years. For instance, in applied mathematics Lie theory remains a powerful tool for studying differential equations, special functions and perturbation theory. The study of the constructive aspects of Lie theory has become practical recently by the use of computers. Computers provide a valuable new research tool in Lie theory, which makes it possible to test (and even to formulate) various theoretical conjectures about Lie algebras. One of the first constructive problems to appear is 59

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related with the classification of such algebras. It is in this general framework, that our paper can be considered. A Lie algebra (Q,/J.) is a vector space 0 over a field IK, with a bilinear mapping M : 0 x 0 —> 0 denoted (X, Y) —> f-i(X, Y) — [X, Y] and called bracket product, verifying

1. [X,X} = 0, 2. [X, [Y, Z}\ + [Y, [Z, X}] + (Z, (X, Y]] = 0 for all the elements X, Y, Z of 0. The second condition is called Jacobi Identity. The dimension of the Lie algebra is the dimension of the vector space 0. Most of the Lie algebras which appear in this paper will be considered over the complex field C and will have finite dimension. By taking a basis (Xo,Xi,... ,Xn_i) in 0, the algebra is completely determined by its structure constants, that is, by the set of complex constants {C^}, defined by a(Xi,Xj) = X)fc=i ^tjXk- Then, we can identify the algebra 0 and its law fj,. Thus, the set Cn of laws of Lie algebras is an affine algebraic set defined by the polynomials expressions

(i)

£<

(2)

One could think of a program of classifying all Lie algebras by considering the above equations to be solved for unknown structure constants. This turns out to be a very complicated problem because of the non-linearity of (2). In fact, the general classification is an open problem. Levi's classical and famous theorem decomposes the classification of Lie algebras into the classification of semisimple and solvable Lie algebras. The classification of semisimple Lie algebra is well-know from the works of Killing and Cartan in 1914. The classification of the solvable Lie algebras can be reduced in a certain sense to obtain the classification of the nilpotent Lie algebras, which can actually be considered the unsolved problem. The descending central sequence of a Lie algebra 0 is defined by C°0 = 0, <^0 = [Q,Ct~lQ}. The first ideal CIQ = [0,0] is the derived algebra of g. IfC f c 0 = {0}, for some k, the Lie algebra is said to be nilpotent. The smallest integer k such that C fc g = {0} is called the nilindex of 0. The nilpotent Lie algebra g is filiform if dimC z g = n — i — 1 for 1 < z < n — 1. Therefore, an n-dimensional filiform Lie algebra 0 has nilindex n— 1. Hence, any filiform Lie algebra has a maximal nilindex, and filiform Lie algebras are considered to be the "less" nilpotent Lie algebras. The nilpotent Lie algebras with nilindex n — 2 will be called quasi-filiform. The first non-trivial classifications of some classes of low-dimensional nilpotent Lie algebras are due to Umlauf [57]. Some of the lists of nilpotent Lie algebras obtained by Umlauf contain errors and they are incomplete. Then, several authors have worked providing new lists (see [23], [24], [49], [52], [53], [54], [55], in the bibliography), but the complexity of the involved computations leads frequently to

errors. We remark that the first exact classification of the nilpotent complex Lie algebras in dimension n < 6 is probably due to Morosov [48]. It is also obtained in a different way by Vergne [59], who showed the important role of filiform Lie algebras in the study of variety of nilpotent Lie algebras laws.

Nilpotent Lie Algebras

61

Recently a new invariant has been introduced by Goze to study nilpotent Lie algebras. Let g be a nilpotent Lie algebra. Let gz(X) be the ordered sequence of Jordan block's dimensions of the nilpotent operator a,d(X), where X £ g. In the set of these sequences, we consider the lexicographical order. Then the characteristic sequence

gz(g) =

max {gz(X}}

is an invariant of 0. An n-dimensional Lie algebra Q is filiform if and only if the Goze invariant gz(g) is (n — 1, 1). Using this invariant, Ancochea and Goze gave the classification of nilpotent Lie algebras in dimension 7 [3] and the classification of filiform Lie algebras in dimension 8 [2]. For a presentation of this problem see [44]. The list [2] contains some errors and it has been corrected by the authors [4] and by Selley [56]. The geometric approach to study nilpotent Lie algebras is one of the main methods which has been developed over the last years. The set A/"n of nilpotent algebra laws is an affine algebraic variety; two Lie algebras are isomorphic if and only if they belong to the same orbit of the natural action of the general linear group. In this approach the notion of filiform Lie algebra appears in a natural way; the subset J-n of filiform laws is an open set in J\fnAmong the geometric aspects of the variety J\fn we point out the description of their orbits, their irreducible components, etc. It is known that J\fn is reducible when n > 7 [5] , [46] . The closure of an irreducible component of Tn is an irreducible component of j\fn. Thus, we can obtain information about J\fn, by studying the variety Tn. For example, the variety }-n has at least 3 irreducible components if n > 12 [46], and consequently the variety J\fn as well. In fact, for n sufficiently large, the number of components of J\fn is at least of order n [5] . A survey of the problem is exposed in [44] , where we can find the description of the components of Fn, n < 10. In [44], one also finds the description of some components of the variety Afn, n > 12, which are obtained by using some Lie algebras belonging to JT^. These filiform Lie algebras can be graded in a certain way as we will see in Section 6. The importance of these graded filiform Lie algebras was shown by Khakimdjanov in the study of Mn [46]. In the cohomological study of the variety of laws of nilpotent Lie algebras established by Vergne [59] , the author also considers a class of graded filiform Lie algebras that plays a fundamental role. In fact, using such graded filiform Lie algebras, Goze and Khakimdjanov give in [45] the geometric description of the characteristically nilpotent filiform Lie algebras. Thus, it is clear that knowing a family of graded algebras of a certain class of nilpotent Lie algebras we get valuable information to solve problems on such a class. We now describe how this paper is structured, and the problems studied in each section. In Section 2 we present some geometric aspects about J-n. We have also summarized the classification of the n-dimensional filiform Lie algebras with n < 11. Finally, as an approach to a general filiform family, we consider the /c-abelian filiform Lie algebras, whose classification is given in any dimension. Section 3 and Section 4 are devoted to the study of nilpotent Lie algebras with small nilindex, that is opposite to the filiform ones. Concretely, in the third section

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we study the p-filiform Lie algebras, defined by the Goze invariant, in order to generalize the filiform case to nilpotent Lie algebras with given nilindex. We show how they can be useful to know the variety J\fn when p is close to the dimension, and there is very little information. In Section 4 we study the metabelian Lie algebras (nilindex 2) and some Lie algebras with nilindex 3 in order to look at relevant algebras with small nilindex where the classification is not possible. Section 5 and Section 6 return again to the study of Lie algebras of large nilindex. In fact, we consider quasi-filiform and 3-filiform Lie algebras. However, in these sections we deal with some graded families of those nilpotent Lie algebras in any dimension n. The obtained classifications are a valuable utility to study the variety Mn • Section 5 provides an extension of the graded algebras used by Vergne to the p-filiform Lie algebras, p = 2,3. In Section 6 we consider the gradation with a larger number of subspaces that the natural gradation provides, and we study the filiform and quasi-filiform case. The obtained algebras are better than the naturally graded ones to solve some cohomological problems. Finally, in the last section we show how symbolic calculus can be used to deal with some similar problems. In particular, how to make adequate use of some of the features that the sofware package Mathematica provides. The authors thank Professor Y. Khakimdjanov for his constructive suggestions. 2

FILIFORM LIE ALGEBRAS

The first non-trivial classifications of some classes of low-dimensional nilpotent Lie algebras are due to K. Umlauf. In his thesis [57] (Leipzig, 1891) he presented the list of nilpotent Lie algebras of dimension m < 6. He also gave the list of nilpotent Lie algebras of dimension m < 9 admitting a basis (Xo,Xi,... ,Xm-i) with [Xo, Xi] = Xi+i for i = 1,..., m - 2. Now, the nilpotent Lie algebras with this property are called filiform Lie algebras. Umlauf's list of filiform Lie algebras is exact only for dimensions m < 7; in dimensions 8 and 9 his list contains errors and it is incomplete. In the last years, it has been evident to study again nilpotent Lie algebras due to the importance of them in classification problems and their applications in different domains of mathematics and physics. Unfortunately, many of these papers are based on direct computations (by hand) and the complexity of those computations leads frequently to errors. M. Vergne obtained an exact classification of the nilpotent complex Lie algebras in dimension m < 6, and showed the important role of filiform Lie algebras, terminology introduced by herself, in the study of the variety of nilpotent Lie algebras laws [59]. Indeed, the subset Fn of filiform laws is a Zariski open set in the variety of nilpotent Lie algebras Afn- In spite of the developed study of filiform Lie algebras over the last years the classification problem is still open. In this section we

summarize the results obtained in the last years. 2.1

Obtaining laws of families of filiform Lie algebras

Obtaining a complete list of Lie algebras is a problem whose complexity increases with the dimension of the algebra. Thus, an approach to the study of the variety of nilpotent Lie algebras is the description of the algebraic sets of laws of filiform Lie

Nilpotent Lie Algebras

63

algebras and their irreducible components. In [1] it is shown that the set of laws of 8dimensional nilpotent Lie algebras is the union of 8 irreducible components. These have been determined explicitly using some algorithms of symbolic computation developed ad hoc by Valeiras [58]. A modified implementation developed by himself in the language Mathematica can be found in [44]. Obtaining an initial handled set of polynomials, which determine the variety subject to study is an necessary first step in any subsequent study of such a variety. In [35] can be found how to associate a descent vector to each X G C J 0 which lets us write the brackets by using a recursive relation. Then we get an algorithm to obtain families of filiform laws by fixing an adapted basis. THEOREM 2.1 There exists a polynomially bounded algorithm which generates the set of laws of filiform Lie algebras, for each dimension n. By using the Jacobi identities we can obtain the conditions which the parameters have to verify to determine the family of filiform laws. Some of the obtained equations in the generated family could allow another more simple form and hence some of the parameters could be eliminated. In this way, and with an implementation of the algorithm stated in the above theorem, the 11-dimensional family of filiform Lie algebra laws was presented in [22]. Finally, in [35] one may find the study for dimension 12. THEOREM 2.2 The sets of the filiform Lie algebra laws over C11 and over C12 can be parametrized (up to isomorphism) respectively by the points of the affine algebraic sets J^u C C16, of Krull dimension 12, and F\-2 C C21 of Krull dimension 13. Finally, in [31], the polynomial equations which define the variety jFn let us determine and describe its two irreducible components. 2.2

Low-dimensional filiform Lie algebras

Ancochea and Goze gave the classification of nilpotent Lie algebras in dimension 7 [3] and the classification of filiform Lie algebras in dimension 8 [2],[4], by using the characteristic sequence introduced by Goze to study nilpotent Lie algebras. Using this invariant, several authors (see [26], [6]) gave the classification of filiform Lie algebras in dimension 9 and 10. The method of Ancochea and Goze, very well adapted to dimension 8, has been transformed in very complex computations in dimensions 9 and 10. The lists [6], [26] are again incomplete and many parameters can be eliminated. According to [46] filiform Lie algebras share some common properties starting in dimension 12. This suggests to study separately filiform Lie algebras for dimensions > 12 and to do a detailed study when the dimension is < 11. Using an approach of Vergne [59], the techniques developed by Khakimdjanov [46], and introducing the notion of elementary changes of bases we have finally given a complete classification up to isomorphism of all complex filiform Lie algebras of dimension m with m < 11 [33], In this way, for instance, the 130 families of the list [6] may be reduced to less than 65. Moreover, some families of Lie algebra laws obtained in [33] do not appear among those 130 families. Hence, the list [6] is not very useful due to the errors that such a list contains.

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The simplest (n + l)-dimensional filiform Lie algebra is Ln+i, defined by

where (Xo, Xi, . . . , Xn) is a basis of Ln+i (the undefined brackets being zero). We note that any (n + l)-dimensional filiform Lie algebra is isomorphic to a linear deformation of Ln+i (see [46]). Let A be the set of pairs of integers (k, r) such that I < k < n—I,2k+l < r
if 1 < i < k < j < n, and ^k,r(Xi,Xj) = 0 otherwise. Let JFm be the variety of filiform Lie algebras laws in an rn-dimensional vectorial space. Any (n + 1)dimensional filiform Lie algebra law fj, G J~n+i is isomorphic to /J,Q + fy, where JJ,Q is the law of Ln and $ is a 2-cocycle defined by

(fc,r)€A

and verifying the relation fy o $ = 0 with

* o #(x, y,z) = 3> (*(x, y), 2:) + * (#(?/, z), x) + # (*(z, x), y) . A basis (^Q; • • • , Xn) of an (n + l)-dimensional filiform Lie algebra with law fj, is called adapted, ii^X^Xj) = n0(Xi,Xj) + ^ > ( X i , X j ) , 0 < i,j < n. Denoting by [a\ the integral part of a from now on, all brackets of an (n + l)-dimensional filiform Lie algebra in an adapted basis (XQ, Xi, . . . , Xn) are determined by the brackets

+ a fcil X 2 fc+3 + • • • + ak>2k-2Xn,

l
If n is odd we have the supplementary bracket [^( n -i)/2; -X"(n+i)/2] = a (n-i)/2,n -XnThe non-uniqueness of an adapted basis for a filiform Lie algebra is clear, so we introduce adapted and elementary changes of bases. A change of basis / G GL(V) is called adapted to the law JJL G J-n+i(V) if the image of an adapted basis is an adapted basis. The most important observation here is that to study the closed subgroup of the group GL(V) composed of all adapted changes, denoted by GL a a(V), it is enough to study certain elementary changes of bases.

THEOREM 2.3 Let f be an element ofGL^V).

f(Xo)

Then f is given by

= aoXo + aiXi + • • • + anXn i)l 2
where agbi ^ 0. In fact, f is a product of the elementary changes of bases

f(X0) i/(a,6)= { f ( X l )

=

aX0 bXi

a, 6 G C * .

Nilpotent Lie Algebras

65

f(X0)

= X0

f(Xl)

=

Xi

f(X0)

=

X0,

f(Xi)

=

Although the factorization of / obtained in the previous theorem is not unique, it may be chosen in a certain order. When a parameterized family of filiform Lie algebras is given, in order to eliminate the unnecessary parameters, the key is to find a sequence of elementary changes of bases. In [33] it is shown how the elementary changes of basis act. We can remark here that if a filiform Lie algebra law is given by its matrix of coefficients {a,k,m}, then the application of the change of basis i/(a, 6) gives a new law {bk,m}, with bk,m = bak,m/am~2k. So, we can transform the two nonzero coefficients ak,m,o,r,tt with ?7i — Ik ^ t — 2r, to 1 (if such a pair does not exist, we can transform only one of the coefficients in the law). The changes of bases r(a, k) and a(b, k) are basically used in parameters elimination. In each application, some coefficients are unaffected, and, under certain restrictions, we can eliminate one of the parameters. Therefore, parameter eliminations must be ordered in such a way that eliminated parameters remain zero in the following changes of bases. In this way the classification for the filiform Lie algebra with dimension less than or equal to 11 is obtained in [33].

THEOREM 2.4 Every n- dimensional complex filiform Lie algebra law, with n < 11, is isomorphic to a law p?n of the List of Laws given in [33]. This approach develops the method of Ancochea and Goze [2] and allows to simplify the computations, but when dimension increases we cannot obtain reliable results without the help of computers. The symbolic programming language Mathematica is useful as an assistant to get the classification. This can be seen in [32]. Finally, as an important application, the classification obtained in [33] has been used to study symplectic structures on the filiform Lie algebras [34]. 2.3

/c-abelian filiform Lie algebras

In addition to know some families of filiform Lie algebra laws in concrete dimensions, we also know the classification of filiform Lie algebras g with the condition CIQ abelian, obtained by Bratzlavsky [7]. The notion of k-abelian nilpotent Lie algebra is introduced in [27] (if CkQ is abelian) to consider the most general situation. This is a synthesis of two properties in the theory of nilpotent Lie algebras: the filiformity (the nilpotent Lie algebras with this property are "less" nilpotent) and the commutativity (the nilpotent Lie algebras with this property are "most" nilpotent). The classification of 2-abelian filiform Lie algebras is obtained in [27] by using the properties of the elementary change of bases in the 2-abelian case.

THEOREM 2.5 Every (n + 1)- dimensional (n > 6) 2-abelian filiform Lie algebra law is isomorphic to one of the laws n i i S ^ ( f l ) , H2,a,t(^}> M3,s,t(^)> /-M, 3,3+2(^)1

Mis.t, Mi4,s(^), Mi5,s(fi), fj,ie,s, Mi?, which are defined in [27].

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A Lie algebra is called characteristically nilpotent if all of its derivations are nilpotent. The definition and first example of such a Lie algebra has been given by Dixmier and Lister [25]. Khakimdjanov has shown that the family of characteristically nilpotent Lie algebras, which had been considered as narrow, is in fact very large [46]. As an application of the obtained classification, and by using the criterion of characteristically nilpotence given by Goze and Khakimdjanov [47],[45], we can conclude that all Lie algebra laws described in Theorem 2.5, except the 2-abelian algebras of laws fig,Sla+2(^o), Mis.t, Mie.s, Mi?, are characteristically nilpotent.

3

THE FAMILY OF p-FILIFORM LIE ALGEBRAS

The study of the variety of nilpotent Lie algebras through the filiform ones in the previous section is centered in those algebras of maximal nilindex among the nilpotent Lie algebras having the same dimension. When we want to study that variety by considering nilpotent Lie algebras with small nilindex the difficulties increase more and more as we can see in Section 4. To study some problems about nilpotent Lie algebras with small nilindex we have chosen in each concrete nilindex a family which allows us to use it in a similar way like the filiform family is used to obtain results on the complete variety. Namely, an n-dimensional nilpotent Lie algebra g is said to be p-filiform if its Goze invariant is gz(g) = (n — p, 1, /?., 1). Thus, the family of p-filiform Lie algebras is a large family of Lie algebras, including the filiform ones as a particular class. Indeed, the filiform and quasi-filiform Lie algebras are the p-filiform ones with p — 1,2 respectively. In this section we are interested in p-filiform families with p close to the dimension of the algebras of such families, i.e., when the family of algebras considered has small nilindex and where the complete variety of nilpotent Lie algebras is not very well-known. 3.1

p-filiform Lie algebras with p > n — 3

When we consider the family of p-filiform Lie algebras with dimension equal to n, it is obvious that 1 < p < n — 1. If p = n — 1 the situation is trivial because the family is just reduced to the abelian algebra in the appropriate dimension. If 0 is a p-filiform Lie algebra, we can always choose an adapted basis of the algebra g. That is, a basis (Xo, Xi,..., Xn-p, Y j , . . . , Yp-i) such that XQ e g— [g, g] and verifying [X0,Xi] = Xi+1 l
{X0,Xn_p}

= 0

[X^Yj]

= 0

l<j
We now describe how the family of p-filiform Lie algebras can be determined for the n-dimensional nilpotent Lie algebras when p = n — 2 and p = n — 3. The results obtained in [13] show how it becomes more difficult to obtain these classications as p decreases. We have adapted techniques already applied to specific dimensions; that is, we have considered appropriate arguments about nilpotentcy (generally, adjointnilpotency), characteristic sequences and those conditions which are derived from assumption that the algebras be p-filiform. Then, with an adapted selection of

Nilpotent Lie Algebras

67

changes of bases, we have obtained the explicit classification for the p-filiform families with p > n — 3. Indeed, firstly we have considered some Jacobi identities in which XQ appears to get a preliminary expression of the brackets which define the family of algebras considered on an adapted basis. Those brackets can be dramatically simplified by considering some conditions which the structure constants have to verify in order to guarantee that Yj £ Im ad(Xo), I < j < p — I , for the vectors in the adapted basis considered, and to preserve the appropriate nilindex of the algebras. Secondly, as the Goze invariant of the algebras considered is (n — p, 1, 1, . . . , 1), there cannot exist a nonzero minor of order n — p in ad(Z), for all Z <£ [g,g]. Thus, some appropriate choices for Z (for example: XQ + AYj , Xi + AYj , AXo + Xi, . . . , A € K) with a process of finite induction, lead to some more restrictions. Finally, some adjoint-nilpotency considerations on any vector X of g (for instance, the characteristic polynomial P(A) of the adjoint matrix corresponding to such a vector has to be A"), some relations obtained from the remaining Jacobi identities, and some appropriate changes of basis (from simple change of scale to other, very complicated, ones) we obtain the goal of the classification by eliminating unnecessary parameters. In [13] we can see that the (n — 2)-filiform Lie algebras are direct sums of abelian algebras and Heisenberg algebras. THEOREM 3.1 In dimension n > 3, there are exactly [(n— 1)/2J nilpotent, real or

complex, pairwise non-isomorphic (n — 2) -filiform Lie algebras, denoted as gL 1 and whose laws in the basis (XQ, X\, . . . , Xn-p, Y\, . . . , ^p-i) are given by 4

[X0,*i]

=

^

X2

One also finds in [13] the classification of the family of (n — 3)-filiform Lie algebras, which are obtained in a similar way. We can see how the different dimensions of the center and of the derived algebras show that they are pairwise non-isomorphic algebras.

THEOREM 3.2 In dimension n > 5, there are exactly n — 1 nilpotent, real or complex, pairwise non-isomorphic (n — 3) -filiform Lie algebras, denoted as g/ 3 1 ^, and whose laws in the basis (Xo, Xi, . . . , Xn_p, Yj, . . . , Yp_i) are given by

(X0,Xi}

= Xl+1

[*1>^~4]

=

Y?k-i,Y2k

— —

l
X3

= X3

V ln-4-

,withl<

l
S

<

n

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As an application we have studied some cohomological properties of the p-filiform Lie algebras obtained above in which the determination of the algebra of derivations of each p-filiform Lie algebra is essential. They can be found in [20], [12] where it is also described how symbolic calculus can be used to solve similar problems. 3.2

(n — 4)-filiform Lie algebras

We now show the increasing difficulties involved in the problem when we consider p = n — 4. Obtaining the classification of (n — 4)-filiform algebras by using the method considered above was not successful. Thus, it has been necessary to develop new techniques. Actually, we have used central extensions of certain algebras with dimension smaller than the dimension of those algebras that we want to study. Let g be a p-filiform Lie algebra of dimension n. If we obtain its quotient by a one-dimensional central ideal, the resulting quotient algebra has dimension n — 1 and its Goze invariant is either (n — p — 1,1,..., 1) or (n — p, 1 , . . . , 1). In this second case we can obtain again the quotient algebra by a one-dimensional central ideal, obtaining a quotient algebra of dimension n — 2 and Goze invariant either (n — p — 1 , 1 , . . . , 1) or (n — p, 1 , . . . , 1). If we follow the process (that finishes because dim(Z(g)) is finite), it is obvious that we always get a quotient algebra of Goze invariant (n — p — 1,1,...,!) of dimension n — k. Conversely, we invert the process and from the algebras of characteristic sequence (n — p — 1,1,...,!) and dimension n — 1 we construct the central extensions that have Goze invariant equal to (n — p, 1,..., I ) . These p-filiform Lie algebras will be called algebras of first generation. If we apply the same process to the same algebras, but in dimension n — 1, we obtain p-filiform algebras that will be called of second generation. The process will continue until an algebra appears in some generation. In [11] we can find that all nilpotent Lie algebras of dimension n and Goze invariant (4,1, (?.~~.4,1) are obtained as central extensions of a first generation of only three algebras of dimension n — l and Goze invariant (3,1, ^r.4,1). Specifically, the "first" of each one of the two existing families with dimension of the derived algebra 2 and the only one with dimension of the derived algebra 3. Any (n — 4)filiform nilpotent Lie algebra of dimension n, that is a central extension of second generation, is a central extension of the first generation as well.

THEOREM 3.3 In dimension n > 8, there are exactly 6n — 29 pairwise nonisomorphic complex (n — 4) -filiform nilpotent Lie algebras. They are distributed in twelve families, denoted by 0/ 4 ^

j \ , 1 < i < 12, whose laws can be found in [11].

We can see in [11] how these algebras are pairwise non-isomorphic by considering some invariants as the dimension of several appropriate subspaces: ZQ, C1^, C 2 0, f)i = Cen(ClQ), etc. Finally, it is shown in [11] that the p-filiform Lie algebras can be obtained as extensions by derivations of filiform Lie algebras of dimension n — p + I for p > n — 4. The case p = n — 3 appears in [10]. 3.3

p-filiform Lie algebras with n — 6 < p < n — 5

In a way similar to the cases studied above, we can obtain a general simplified

expression for the families of p-filiform Lie algebras when n — 6 < p < n — 5, but the classification of such families is by now unsolved. However, we have studied some

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69

interesting subfamilies in each case in order to get an approach to the complete family. The generic family of laws of the (n — 5)-filiform Lie algebras is obtained in [8], and the generic family of laws of the (n — 6)-filiform Lie algebras can be found in [18]. The laws of the family of (n — 5)-filiform Lie algebras allows us to get the classification when we consider some restrictions on the derived subalgebras of the family. Indeed, we can see in [8] or [16] the algebras g obtained when dim[g,g] = 6, that is when such derived subalgebras have dimension as large as possible.

THEOREM 3.4 If Q is an (n — 5)-filiform Lie algebra of dimension n > 9 with dim(C 1 g) = 6, then it is isomorphic to one of the algebras 0/ 5 ^ ^, I < i < 5, which are defined in [8]. We have also considered the families of (n — 5)-filiform and (n — 6)-filiform Lie algebras when n = 8. The classification for these families completes the p-filiform family in that dimension, which shows more information about the variety A/S. All of the 3-filiform Lie algebras can be reduced to only three [17]. The classification of 2-filiform Lie algebras (quasi-filiform Lie algebras) is obtained in [19], where it is shown how symbolic calculus was used to get the goal. Concretely, by using algorithms implemented in the Mathematica programming language [60].

4

LIE ALGEBRAS WITH SMALL NILINDEX

We consider here some type of nilpotent Lie algebras with small nilindex. Only the abelian algebra is obtained in each dimension having nilindex equal to 1. However, when the nilindex is greater than 1 the difficulties to determine all pairwise non-isomorphic algebras increase more and more. Indeed, the classification of the nilpotent Lie algebras with nilindex 2 is a very hard problem. The classification of those (2p + l)-dimensional algebras which have Goze invariant ( 2 , . . . , 2,1) is equivalent to the classification of the bilinear forms on Cp with values in Cp, which include all Lie algebras of dimension p. Thus, it is impossible to obtain a complete classification of such nilpotent Lie algebras [44], Therefore, to know examples of families of such Lie algebras with small nilindex is a valuable utility to study the variety of nilpotent Lie algebras J\fm.

A nilpotent Lie algebra of nilindex 2 is called a metabelian Lie algebra. Now, we are interested in certain types of metabelian Lie algebras and certain types of algebras with nilindex 3 determined by their Goze invariants. Goze's invariant allows to separate a family of nilpotent Lie algebras with a fixed nilindex into some families which we can study more easily. In this way, we can separate the family of the n-dimensional metabelian Lie algebras by considering those having Goze invariant equal to (2,2, . P A,2,1 n7?P\l), where p goes from 1 to [(n — 1)/2J. The simplest case to be considered is the family determined by ( 2 , 1 , . . . , 1). These are the (n — 2)-filiform Lie algebras and they have been studied in [13]. The classification can be found in Section 3, where we have surveyed the p-filiform Lie algebras. In the same way, the nilpotent Lie algebras with nilindex 3 which have Goze invariant equal to ( 3 , 1 , . . . , 1) are the (n — 3)-filiform Lie algebras, and these are also classified for arbitrary dimension in [13].

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We present in this section some general results about the metabelian Lie algebras, including the classification for a general family of such algebras under certain assumptions. We also study some concrete families of nilindex 2 and 3 in order to demonstrate how and where the difficulty increases in these classification problems. 4.1

Metabelian Lie algebras

To study the metabelian Lie algebras we separate them into families with Goze invariant (2,2,. P A,2, 1 n7??\ I). Then, if Q is an n-dimensional Lie algebra with nilindex 2 there is p 6 TL such that 1 < p < [(n - l)/2j and gz(g) = (2,2,. p) . ,2,l n r. 2 p \l). When p = 1, that is gz(g) = (2, 1, . . . , 1), the algebra Q is one of the (n — 2)filiform Lie algebras determined in [13]. In the general case, the difficulty to obtain the classification lies in the fact that there are too many parameters with very few restrictions. Let Q be a metabelian Lie algebra. A basis

is called adapted if

[Xo,X2i-i] = Xx, l
by n-2p-l

k=l

k=l

Let s denote the rank of the matrix (of), where af is the appropriate coefficient a 2t-i 2j-i fr°m the vector Yk in the bracket [ X z i - i ^ X ^ j - i ] above. Then, the derived algebra C J g has no more than s different elements X^i, 1 < i < p from the adapted basis considered. It follows that s < ( p ), and then we can bound the dimension of the algebra C1Q. THEOREM 4.1 Let Q be an n-dimensional metabelian Lie algebra and let p € N verify n > 2p + 1. Then p
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71

difficult as dim(C 1 0) decreases. In fact, when we consider dim(C 1 0) to have the largest possible value, the classification of such a family of Lie algebras can be obtained in any dimension. Indeed, let g^ p denote the algebras defined in the basis

(Xo,Xi, X2, • • • , X-2P, YI, . . . , Yn_2p-i)) by

where n > 2p + 1 > 5. For every p, the algebra g^ p

has Goze invariant equal to

n

gz(0) = (2, 2, . )., 2,1 7??\ 1) and its derived algebra verifies dim(C10jll)p) = p+ (%). Then, we can see in [43] how any n-dimensional metabelian Lie algebra g with Goze invariant gz(g) = (2, 2, p)., 2,1 n7??\ 1) and dim (C x g), having the largest possible value, is a trivial extension of an appropriate Lie algebra g^ . THEOREM 4.2 Let g be an n-dimensional metabelian Lie algebra with Goze invariant gz(g) = (2, P)., 2,1 n72?\ l},n>2p+l + (P) and dim(C J 0) =P+(%). Then the algebra is g = g^p ® Cn~m, where m = ( p f 2 ). It remains an unsolved problem, to obtain the classification when dim(C 1 g) decreases. Indeed, if the derived algebra has the smallest possible value we always find a family of algebras in which every algebra is a linear deformation of one of the model algebras For instance, a metabelian Lie algebra g with Goze invariant gz(g) = (2, 2 , 1 , . . . , 1) and dim(C 1 g) = 2 belongs to the family

[X0,X2i-i]=X2i, [ X i , Yi] =6jj^2 + &ii^4>

1 < i < 2, 1 < z < n — 5,

[Yi, Yj]

1 < i < j < n - 5.

=0^X2 + 4X4,

But we cannot separate non-isomorphic algebras from the family so far. However, a cohomology study about the algebras g° p was introduced in [41], and it can be found in [51], in order to illustrate the dimension of the orbits of such algebras. Moreover, we can also see in [51] how the number of metabelian Lie algebras obtained increases, when we consider dim(C 1 g) ^ p. THEOREM 4.3 Let g be an n-dimensional metabelian Lie algebra, n > 10, with Goze invariant g = ( 2 , 2 , 2 , 1 , . . . , ! ) and dim(C 1 g) > 3. Then g is isomorphic to a trivial extension of one of the following algebras defined in [51]: 0^ 3, g^ 3, g^, 3; 0n,3> 3n,3> 8n,3> 8n',3> ^ —T — l~T~ \ > Sra',3> ^ —r — {.^~2~ \ ' 3rt'3' ^ —r — l^2~ \ '

0^,3, 0 < r < l1^^; Qn^, 0 < r < L11^12]; g^f 3 , 0 < r < L2^11] • We remark that when we consider a metabelian Lie algebra g with Goze invariant ( 2 , 2 , 1 . . . , 1) and dim(C 1 g) > 2, then g has to be a trivial extension of gjj 2 -

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4.2

Lie algebras with nilindex 3

The study of the Lie algebras with nilindex 3 has been started in [51], in order to check if the results about metabelian Lie algebra could be generalized to other algebras with small nilindex. In this case, Goze's invariant could be one of the following (3, .p)., 3, 2, .q)., 2, 1, n -^T 2
defined in [51], we can separate the study of these algebras from the possible values (p,q). For example, the pair (1,0) corresponds to the (n — 3)-filiform Lie algebras which have been studied in [13]. Although for these algebras we have no relation to bound dim(C 1 g) as in Theorem 4.1 for the metabelian ones, we can see how the dimension of the derived algebra plays an important role in those algebras with small nilindex. Thus, we can express the family determined by (p, q) = (1, 1) in an adapted basis [51], and hence we can obtain the classification of those families of algebras with dim(C 1 0) > 3. For the non-split algebras mentioned in Theorem 4.4 below, we can also find some cohomological results in [51].

THEOREM 4.4 LetQ be an n- dimensional Lie algebra, n>8, with Goze invariant (3, 2, 1, . . . , 1) and dim(C 1 g) > 3. Then g is isomorphic to a trivial extension of one of the algebras Qln iti, 1 < i < 4n — 24, defined in [51]. To classify the Lie algebras having Goze invariant (3, . P A,3, 1, n7??\ 1) seems a much harder problem than its "similar" (2, . P A,2, 1,n7??\ 1) in metabelian Lie algebras. Indeed, in the simplest case (p,q) = (2,0) we have to consider for a Lie algebra 0 its derived algebra verifying 4 < dim(C 1 g) < 8. However, when we consider the largest value dim(C 1 0) = 8 the algebra Q just can be a trivial extension of one determined Lie algebra [42].

THEOREM 4.5 Let Q be an n- dimensional Lie algebra, n > 11, with Goze invariant (3,3, 1 n.~.6}, 1) and dim(C 1 0) = 8. Then Q is isomorphic to a trivial extension of the algebra Q^ 2 0 , defined in the basis (Xo, Xi,Xz, • • • , XQ, YI, . . . , Yn-?) by [Xo,Xi\=Xi+i, i €{1,2,4,5}, [X1)x2]=y1,

[X1,X4}=Y2, [XltX5=Y3,

To know if the algebras with Goze invariant (3, .PA, 3, l,n7?P\ 1) and having derived algebra as largest as possible can be determined as extensions of an appropriate family g^ p 0 is a very interesting question which should be a generalization of Theorem 4.2.' '

5

NATURALLY GRADED NILPOTENT LIE ALGEBRAS

In the cohomological study of the variety of laws of nilpotent Lie algebras established by Vergne [59] the classification of a class of graded filiform Lie algebras plays a

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73

fundamental role. The achieved classification allows for an easy expression of a filiform Lie algebra. The gradation considered by Vergne is provided in a natural way from the descending central sequence of any nilpotent Lie algebra. Now, the algebras obtained in this way are called naturally graded Lie algebras. Vergne proves that, up to isomorphisms, there is only one naturally graded filiform Lie algebra in odd dimensions and two of them when the dimension is even. This fact also allows other authors to deal with different aspects of the theory. For example, using such graded filiform Lie algebras, Goze and Khakimdjanov give in [45] the geometric description of the characteristically nilpotent filiform Lie algebras. Thus, it is clear that knowing the graded algebras of a certain class of nilpotent algebras we get valuable information about the structure of such a class. This may facilitate later the study of several problems that can appear within the whole of the class. In this section we study the naturally graded Lie algebras for the nilpotent Lie algebras with nilindex near their dimension. Thus, we concretely are interested in p-filiform Lie algebras for whose values of p the methods used in the sections above fail. 5.1

Naturally Graded filiform and Quasi-filiform Lie Algebras

If g is a nilpotent Lie algebra of dimension n and nilindex k, it is naturally filtered by the descending central sequence of g, (Cl&)0 k. Associated to g there exists a graded Lie algebra grg = ©iezgi, taking g; = 5j/5i+1. Thus, we have

When grg and g are isomorphic, denoted by grg = g, we will say that the algebra is naturally graded. So, if g is a naturally graded Lie algebra, such a natural gradation is finite, that is grg = gi © g 2 © • • • © g^, with [g^, QJ] C Qi+j, for i + 3 £ k, and the number of subspaces jjj is as large as the nilindex of g. The filiform case, characterized by its maximum nilindex fc = n — 1, in each dimension was studied by Vergne [59], who introduced the notation, proving that there are only two algebras of this type, Ln and Qn, when the dimension of the algebra is even, and there is only one, Ln, if n is odd. The algebras Ln and Qn are those defined on the basis (Xo, Xi, . . . , Xn-i), by

[X0,Xi]=Xi+1, (The undefined brackets, except for those expressing anti-symmetry, are supposed to vanish). As we have seen in Section 2, any n-dimensional filiform Lie algebra is a linear deformation of the algebra Ln [44]. Now, we want to study the naturally graded quasi-filiform Lie algebras, that is we will consider the n-dimensional Lie algebras with nilindex equal to n — 2. In

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order to make easy the structure of such a family we have to find a basis in which the brackets of the algebras of the family are suitable. We can see in [29] how to obtain an homogeneous adapted basis of the algebra 0, i.e., a basis whose properties are stated in the theorem below. THEOREM 5.1 Let Q be an n- dimensional naturally graded quasi-filiform Lie algebra. Then, there exists a basis (Xo, Xi, . . . , Xn_2, Y) of the algebra Q such that XQ, X\ G gi, Xi
z
Thus, we can use the laws of the family to reduce the unnecessary parameters. When g is an n-dimensional naturally graded filiform Lie algebra such that dim(C 1 0) = n — 3, we can consider a natural gradation where gi =< Xo,Xi,Y >. Then, the algebras obtained are reduced to a trivial extension of the naturally graded filiform Lie algebras. In fact, g = Ln-\ © C, if n is even, and either g = Ln_i © C or g = Qn-i © C, if n is odd. These graded quasi-filiform Lie algebras will be called split, and their only interest is to emphasize the natural gradation underlying the filiform subalgebra of the extension. We will denote by Q(n,r} an n-dimensional quasi-filiform Lie algebra in which we can choose an adapted basis so that the natural gradation verifies Y & Qr (note that dim(g r ) = 2, r ^ 1). Thus, we can view the pair of integers (n,r), I < r < n — 2, as an invariant for any naturally graded quasi-filiform Lie algebra. Then, the result in the previous paragraph says that if g = g( n ,i), then the algebra g is a split algebra. Moreover, if g = Q(n,r) with r even we can see in [30] that dim(C 1 g) = n — 3, so g is also a split algebra. Then, we can describe the family of the n-dimensional non-split naturally graded quasi-filiform Lie algebras. LEMMA 5.2 Let g = Q(n,r) be a non-split naturally graded quasi-filiform Lie algebra.

Then r is odd and g belongs to the parameterized family defined in an adapted

basis (X0,Xi,...,Xn-2,Y)

by

[Xi,Y]=aXi+r

where a £ C if 3 < r < n — 3, and a = 0 if r = n — 2. We can see how the quasi-filiform Lie algebras L( n>r ) and Q( n ,r) defined in the basis If n > 5, and 3 < r < 2 {(n - 1)/2J - 1, r odd,

If n > 7, n odd, and 3 < r < n — 4, r odd,

Nilpotent Lie Algebras

75

[X0,Xi} = I

(n,r)

belongs to the family in the lemma above. So, these algebras are naturally graded algebras. They play an almost similar role as the algebras Ln and Qn in the filiform family. In each dimension n, for each acceptable (n,r), 3 < r < n — 5 we obtain that the naturally graded quasi-filiform Lie algebras are just L( n>r ) when n is even, and I/( n]r ) or <5( n>r ) when n is odd. Those cases in which r is close to the dimension n are special. When r = n — 2 (n odd), then g = £( n ) n _2)- When r = n — 3 (n even) or when r — n — 4 (n odd), we obtain also one more algebra noted T( n)7 .), which is called terminal and is defined in [30], where the algebras £(7,3), £}g 5 \, and £?g 5^ are defined as well.

THEOREM 5.3 Every n-dimensional naturally graded quasi-filiform Lie algebra is isomorphic to one of the following algebras: If n is even to Ln-\ ® C, T(n,n-3) or ^(n,r)i with r odd and 3 < r < n — 3. If n is odd to Ln—\ ® C, Qn-i ® C, L(n,n-2)> r (n,n-4)i ^(n,r) or Q(n,r)> with r odd and 3 < r < n — 4. When n = 7 or n = 9 we add the algebras £(jtz)> £}g $\i £?g 5)As an application, we refer to [40], [21] for a characterization of the space of derivations of the naturally graded quasi-filiform Lie algebras and a description of how symbolic calculus can be used to solve similar problems [28]. 5.2

Naturally Graded 3-filiform Lie Algebras

The complete n-dimensional family of nilpotent Lie algebras with nilindex n — 3 can be separated in the subfamilies characterized by the Goze invariants (n — 3,1,1,1), and (n — 3,2,1). The first one is the family of the 3-filiform Lie algebras. We are now interested in the naturally graded Lie algebras for such a family in order to look for a family of graded algebras which extends the results in the section above. In fact, the situation for naturally graded 3-filiform Lie algebras is a generalization of the filiform and quasi-filiform Lie algebras. There is one locally finite family of terminal algebras (depending on one parameter) for n odd or n even; furthermore, there are one or two locally finite families (depending on two parameters) for n odd or n even, respectively. Precisely, for n > 11, there are O(n 2 ) non-split Lie algebras.

THEOREM 5.4 Let Q be a naturally graded 3-filiform Lie algebra of dimension n, grg = ®igs0i and let (Xo, Xi,... ,Xn^3, YI, Yj) be an adapted basis of Q . Then

gi Qi

D D

< X0,Xi > <Xi>

Bi

=

{0}

2 n - 2.

More precisely, the decomposition obtained is g = gi © 32 © • • • © fln-s with [0i,0j] C Qi+j, for?+j < n-3, verifying that 2 < dim(gi) < 4 and 1 < dim(flj) < 3, 2 < z < n-3.

Cabezas et al.

76

The vectors YI, Y% are special. The triple (n, 7-1,7-2) will represent the case of naturally graded 3-filiform Lie algebras of dimension n, where the integers r± and r-2 indicate the positions of the coordinates of (dim(gi),dim(02), • • • , dim(g n _3)) different from those of (2, !,...,!). And 0( ra ,n,r 2 ) represents an algebra of the appropriate family determined by (71,7-1,7-2), 1 < r\ < r% < n — 3. Some general

facts about (71,7-1,7-2) can be found in [14], [15]. • The cases (71,7-1,7-2), where either r\ or r-2 is even are not admissible. Moreover, when r\ = 7-2, then r\ is odd. • The admissible cases (71,1,7-2) produce split Lie algebras: either a direct sum or a direct sum g n _2 ©C 2 , where Qn-i and Qn-2 correspond to a naturally 0 ra _i graded 2- filiform and 1-filiform Lie algebra of dimension n—l and n — 2, respectively. • The case (4, 1, 1) corresponds to the abelian Lie algebra. Moreover, in the case (5, 1, 1) two algebras are obtained: one split and the Heisenberg algebra. Of course, the general structure of the family of naturally graded 3-filiform Lie algebras is more complicated than that of the quasi-filiform ones.

LEMMA 5.5 Any naturally graded 3-filiform Lie algebra of dimension n is isomorphic to one whose law can be expressed, with (Xo, X\, . . . ,Xn^s, Yj, Yjj) an adapted basis, by 1 < i < n-4,

+ (-

[Xi,Y2] = fXi+r^ [Yi , Y2] = hXn_3

1
l
where 3 < ri < r 2 < TI — 3, ri,r2 od
by If n > 8, and 3 < r\ < r<2 < n — 3, n, r-2 odd, r

;l = X,A

If n > 10, n even, and 3 < r\ < r^ < n — 5, r\,r-2 odd, 1 < i
Nilpotent Lie Algebras

77

The naturally graded 3-filiform Lie algebras of dimension n and type (n,ri,r2) when r-2 is "terminal", i.e., when r% is close to n — 3 (so Y? € Qi, Xn-s G QJ in a gradation of the algebra, where jjj is near QJ) has been studied carefully in [14]. The terminal families r(n,r, n — 5) (n even, n > 12, r odd, 3 < r < n — 7) and r(n,r,n — 4) (n odd, n > 9, r odd, 3 < r < n — 6) can also be found in [14]. As in the quasi-filiform case, we have to consider some exceptional algebras in low dimensions (n < 10): one for each dimensions 8 and 9 and two infinite families for n = 10 [9]. In the general case (n > 11), we will always suppose to be working with Lie algebras g with n > 11 and 3 < r\ < r%, ri,r2 both odd. We can see in [14] that, for odd dimension, there is one bi-parameter locally finite family of algebras, i( n ,r 1 ,r 2 ) (parameters n, r ^ ) , and one terminal one-parameter locally finite family, T( n ; r i ] n _ 4 ). For even dimension another bi-parameter family, <3( ra ,r-i,r 2 ) and °ne terminal oneparameter family, T(n;r.1]TJ_5) appear, but T(,i ]T . lin _4) disappears (r-2 = n — 4 is even if n is even and this is impossible). In order to obtain the desired classification, it is essential that any algebra g should belong to the family is an extension for ideals of g' = 0/C l ™~ 4 (g), which is in a natural way a graded 3-filiform Lie algebra of dimension n — 1 and type (n — I,ri,r2). Thus, the general classification is obtained using induction on the dimension of the algebras of such a family. The following theorem shows that there are no terminal algebras when r^ < n — 7.

THEOREM 5.6 Any naturally graded 3-filiform Lie algebra Q of type (n,TI,^) with n > 11, 7*1, TI both odd, 3 < TI < TI < n — 7, is isomorphic to an algebra of the family L(n, TI,^) if n is odd or to an algebra of the families J^( n , r i i r 2 ) or Q( ra ,rj,r 2 )

if n is even. Thus, the results obtained for the naturally graded 3-filiform family are a generalization of the filiform and quasi-filiform families. This allows to conjecture about the structure of the naturally graded p-filiform Lie algebras when p is greater than 3.

6

LENGTH OF NILPOTENT LIE ALGEBRAS

The difficulties to obtain the classification of a class of nilpotent Lie algebras lead to study the algebras which can give useful information about such a class. In this way, the graded Lie algebras play an important role. We have considered in Section 5 the naturally graded algebras for some families of p-filiform Lie algebras. In that natural gradation on nilpotent Lie algebras, the subspaces in the gradation and the existence of an appropriate homogeneous basis (necessary to obtain the classification) are a natural consequence of the central descending sequence of the Lie algebras considered. However, it introduces a restriction by fixing the number of subspaces in the gradation by means of the nilindex. Several authors, for instance Goze, Khakimdjanov, and some of the authors of this paper, have considered graded Lie algebras which possess not just one natural gradation, but a gradation with a large number of subspaces, because this condition facilitates the study of some cohomological properties for such algebras (see [13], [44]). For such a "length" of the gradation, the main interest is in algebras whose length is as large as possible.

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In this section we are interested in the study of the filiform and quasi-filiform Lie algebras with length larger than their nilindex. Thus we have obtained other graded Lie algebras different from those in the section above. These graded Lie algebras with a greater number of subspaces than the naturally graded ones can be useful examples of filiform and quasi-filiform algebras to approach some problems about the variety of nilpotent Lie algebras. 6.1

Connected gradations

We suppose in this section that all Lie algebras are defined over the field of complex numbers. We also consider Z-graded Lie algebras; that is, admitting a decomposition Q = ®j 6 g Si> where the subspaces Qi satisfy [0i,gj] C Qi+j for all i,j 6 Z. We will say that a nilpotent Lie algebra Q admits a connected gradation 0 = Qni © • • • ® Qn2, when each Qi, with n\ < i < 77-2, is nonzero. The number of subspaces l(($Q) = n^ — n\ + 1 will be called the length of the gradation. DEFINITION 6.1 The length /(g) of a Lie algebra g is defined as I(Q) = max {/(®g) = n2 - n\ + 1 : g = g ni ® • • • © g na is a connected gradation} . This means that /(g) is the greatest number of subspaces from the connected gradation which can be obtained in g. Thus, every Lie algebra g has at least length equal to 1, because we can consider the connected trivial gradation g = go- On the other hand, an algebra cannot have length greater than its dimension. Thus, for every nilpotent Lie algebra g we have 1 < /(g) < dimg. For example, the filiform Lie algebra Ln admits the gradation Ln = Si ® • • -®Qn, where the subspace Qi = < X j _ i >, with 1 < i < n, is generated by the element Xi-\ of the basis (Xo, Xi,..., Xn-i) over which the algebra Ln is defined. Although such a connected gradation on Ln is not natural (QI ^ Q/Cls), it could be more useful to be considered because each subspace Qi has dimension equal to 1. Moreover, this gradation has the greatest possible number of nonzero subspaces, and this shows that the length of such an algebra is l(Ln) = n, If we consider a family of nilpotent Lie algebras and we look for the graded algebras with a given length, then we cannot suppose an adequate basis to express the algebras in a easy way (as Ln in an adapted basis). This is a first problem to solve in order to obtain the classification of such a graded family of Lie algebras. Thus, firstly we will obtain an homogeneous adapted basis that allows us to determine the structure of the family of Lie algebras considered. Secondly, we will separate that family in subfamilies determined by the gradations which we must consider. Finally, we will obtain the non-isomorphic algebras. 6.2

Filiform Lie Algebra of maximum Length

In addition to Ln, there are other filiform Lie algebras that admit this gradation. The algebras Rn and Wn are defined in a homogeneous basis (Xo,..., Xn-i), by n

~

f {X0,Xi} =Xi+1

1
=X2+J

2<j
Nilpotent Lie Algebras

79

[X0,Xi]=Xi+l

l
r v- y i _ 6 (i-i)! (j-i)! (j-i)

- ———

——— i < j < n — 2 — i.

As for L n , we can check that l(Rn) = n and l(Wn) = n. Therefore, Rn and Wn are also maximum length filiform Lie algebras. The importance of these algebras was shown in the study of reducibility of the variety of laws of the nilpotent Lie algebras A/"ra. Khakimdjanov [46] describes in the variety j\fn, with n > 12, two irreducible components which contain an Rn and Wn respectively. Thus, the determination, up to isomorphism, of the filiform Lie algebras with length equal to their dimension can provide us, then, with new information about the variety J\fn. To obtain the naturally graded algebras, Vergne [59] uses the fact that if g is a filiform Lie algebra of dimension n, there exists an adaptedbasis (Xo, Xi,..., Xn_i), such that

[A-0,AVi] = 0 is verified. It can be observed that the basis considered in the definitions of Ln, Rn and Wn are adapted bases. When g is an n-dimensional filiform Lie algebra with 1(0) = n, we can see in [37] how to get an adapted basis by choosing homogeneous vectors from the subspaces of any decomposition 0 = ffig such that l(ffig) = n. This allows us to obtain the structure of the graded filiform Lie algebras considered.

THEOREM 6.2 If Q is an n-dimensional filiform Lie algebra of maximum length I(Q) = n that admits the decomposition g = g ni ffi • • • © Sni+n-i, then there exists

an adapted and homogeneous basis of g. Of course, we cannot obtain different Lie algebras from the gradations considered by just reordering the indices. In fact, we can see in [37] that g admits the decomposition g = g _ n + 2 f f i - • -ffigi if and only if g = g_i ffi- • -ffig n _2, and g admits

the decomposition g = gi ffi • • • ffi g n if and only if g = g_ n ffl • • • ffi g_i. In the first case the only algebra of the family is g = Ln, but the algebras Kn and Q'n verify 0 = 81 ® • • • ffi 0m where Kn (n > 8) and Q'n (n > 7, n odd) are defined in the basis (X0,Xi,...,Xn-.i) as follows: F

V l V [Ao,VA,] = Ai+i,

fj, X

r

• X

j n-2 | _ . l

1 ^ " ^--' ,«

1 < z < n - O2,

= (-1)*"1 ( I ^^ I - Z) X

I n-2 I

•] = ( — ! Y ('~ 1 )("~ 3 ~') a v

,

I

<J

<\ "~ 4 I ,

o < 7 < "~3

with a = 0, if n is even and a = 1, if n odd.

I [Xo,Xi] = Xi+i,

l
Moreover, the algebras Ln, Rn, Wn, Kn and Q'n, which admit the decomposition 0 = 0i © • • • ® Qni determine the filiform Lie algebras of maximum length.

80

Cabezas et al.

THEOREM 6.3 Let g be an n- dimensional filiform Lie algebra of maximum length /(g) = n and n > 12. Then, either Q = Ln, g = Rn, Q - Wn, g = Kn or g = Q'n (n odd). In addition to these filiform Lie algebras, we must consider the algebras with dimension less than 12 and maximum length, obtained in [36]. 6.3

Quasi-filiform Lie algebras of length greater than their nilindex

The number of subspaces for the natural gradation in the quasi- filiform case is n— 2, where n is the dimension of the algebra [29]. So, in addition to the Lie algebras with length n, the algebras with length equal to n — 1 can also provide useful information to know better the complete quasi-filiform class. As in the filiform case, to obtain the n-dimensional naturally graded quasifiliform Lie algebras we can see in [29] how to get an homogeneous adapted basis (X0,Xi, . . . ,Xn-2,Y), such that

[X0,Xi}=Xi+1,

i
In [38], we have shown that the adapted basis can also be chosen homogeneous for any possible gradation with the maximum length on quasi-filiform Lie algebras.

THEOREM 6.4 Let g be an n-dimensional quasi-filiform Lie algebra of maximum length l(g) = n that admits a decomposition g = g ni © • • • © Qm+n-i- Then there exists an adapted and homogeneous basis (Xo, Xi, . . . , Xn_2i Y) °f 3 such that: (a) If the decomposition o/g is g = g _ n + 2 ® - • - ® g o ® 0 i , then g =< Y > © < X\ > © < X2 > © • • • ffi < ^o > or Q =< Xi>®<X2>®---®®<X0>. (b) If the decomposition of Q is g = Q-n+3 © • • • © 01 © 32, then g =< Xi > © < X-2 > © • • • © < ^0 > © < ^ > •

(c) If the decomposition of Q is g = g0 ©gi © • • • ©0 n -i, then g =< Y > © < X0 >

© < Xi > © • • • © < Xn_2 > • (d) If the decomposition o/g is g = gi©g2©- • -®Qn, then g =< X0>®®< Xi > © • • • © < Xn-2 > orQ=<X0>®<X1>®---®< Xn^2 > ® . Obviously, every trivial algebraic extension g = g'©C of a filiform Lie algebra g' of maximum length is also a quasi-filiform Lie algebra of maximum length, because we can join the subspace C at the end of the gradation of g' to get the appropriate gradation for g. Thus, the algebras Ln_i © C, Rn~i © C, W n _i © C, Kn_i © C and Q'n_i © C are quasi-filiform Lie algebras of maximum length equal to n [37]. All these algebras are examples of split Lie algebras g © C, whose decompositions lie actually in g'. The non-split quasi-filiform Lie algebras of maximum length have been determined in [38], where we can also find the study of some geometric problems about the graded quasi-filiform Lie algebras obtained. THEOREM 6.5 (Quasi-filiform Lie algebras of maximum length) Let Q be an n-dimensional non-split quasi-filiform Lie algebra of maximum length l(g) = n, with n > 13 . Then the algebra g is either g = gL 2_n-. (n odd), g = g? ^ or Q = s f n i ) ; where the algebras 0 L 2 _ r a ) > 0 L i ) ; 0L, i)

(Xo, Xi, . . . Xn-2, Y) as follows:

are

defined in the basis

Nilpotent Lie Algebras

81

// n > 5 and n is odd,

Ifn>5, 2 0(n 1}

'

' j\ V i , yv Vi II — _ yv V j_|_3)

9 _
Now we consider an n-dimensional quasi-filiform Lie algebra g of length l(g) = n — 1. We remark that the natural gradation for this algebra has n — 2 subspaces. Moreover, we do not know if such an algebra is naturally graded. If g does, then the algebra is one of the algebras obtained in Section 5 with another appropriate gradation different from the natural one. Thus, the most interesting point is to study families with few naturally graded Lie algebras. Then we could obtain other graded algebras to study the chosen family. When the derived algebra of g has the minimal dimension dim(C 1 g) = n — 3, then g = Ln-\ © C or g = <5 n _i © C [29], so we first consider the family of quasi-filiform Lie algebras verifying such a condition about their derived algebras. In [39] we have studied the possible gradations to be considered, and we can see that if g is an n-dimensional quasi-filiform Lie algebra of length n — 1 and dim(C 1 g) = n — 3, then it is always possible to choose an homogeneous adapted basis of the algebra (Xo,Xi,... ,Xn-2,Y) for any gradation g = ©g such that /(0g) = n — 1. This allows us to get the structure of the family, and finally to determine the algebras of the family considered. The theorem below shows how hard it is to obtain a classification.

THEOREM 6.6 Let g be an n-dimensional quasi-filiform Lie algebra with length = n— 1, dim(C ll g) = n — 3 and n > 13. Then either g = Qn-\ ffi C, g = gL -,, Q

Q

/

lT~l QllfTL

_ J _ J _ J

CLTLQi T) Odd

A 1

OT T) ~~ TL — 4 /

4

0 —- Q/

/

\

r

'

^

^

^

x

O 1

ITL — t3 "^ T) "^ TL — o /

'ra •. (n — 5 < p < n — 3) for 3 < p < n — 3, which are defined in [39]. The low-dimensional cases (n < 13) and an approach to the study of the ndimensional quasi-filiform Lie family of algebras g with length l(g) = n — 1, verifying dim(C 1 g) ^ n — 3, can be found in [50].

7

SYMBOLIC CALCULUS ON LIE ALGEBRAS

Some of the problems studied in this survey require a lot of computations in order to reach the goal we are aiming at. A family of Lie algebras is usually determined by a set of polynomial equations involving the structure constants of the family, and it is obtained from some restrictions which have to be considered about such algebras arising from Jacobi's relations, nilindex, Goze's invariant, etc. Then, to

82

Cabezas et al.

determine a family of Lie algebras in a reliable way is a very complicated problem. Moreover, the basic object to study is an n-dimensional Lie algebra, whose law is usually given in an adequate basis. Often, we have to change the considered basis and we have to determine how the family is modified. Obviously, a general change of basis in arbitrary dimension cannot be done. So we have to "guess" the change of basis which simplifies the law of the family. When a family is determined in an easy way, for instance when we know the different algebras up to isomorphism, we are often interested in several problems to approach the complete variety of nilpotent Lie algebras by the family that is considered. Thus, we usually want to know the dimension of the orbits of those representative algebras in the family, their spaces of derivations, etc. In some cases the theorems come from the algebras whose laws are very simple, but in many cases we must make a lot of computations to get examples which allow us to find out the right way to the solution of the considered problem. Thus, we have developed many algorithms to assist us where a computation "by hand" could be either dangerous or almost impossible. Most of them have been implemented in the symbolic programming language that the package Mathematica provides [60]. The software package Mathematica has been very useful. It has been used for its programming capacities and as a powerful symbolic calculator. It has been considered always as an assistant to obtaining examples from concrete algebras in which we are interested for high dimensions. Moreover, we have used the developed Mathematica packages to check some conjectures that could be made in order to modify the packages when a general result was stated. Thus, we have used Mathematica in an interactive way, and we often had to solve some problems different from the initial ones. Mathematica has become a well-know symbolic software tool to perform computations in a reliable way. We have shown in several sections of this survey how some of the results obtained have been presented in Computational Algebras Conferences or Mathematical Symposiums. The programs we developed can be found in the appropriate references or directly through the authors of this paper.

REFERENCES

[1] J. M. Ancochea, J.R. Gomez, M. Goze, G. Valeiras, Sur les composantes irreductibles de la variete des lois nilpotentes, J. of Pure and Appl. Algebra, 106(1996), 11-22. [2] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie filiformes de dimension 8, Arch. Math., 50(1988), 511-525. [3] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie nilpotentes de dimension 7, Arch. Math., 52:2(1989), 157-185. [4] J. M. Ancochea-Bermudez, M. Goze, On the varietes of nilpotent He algebras of dimension 7 and 8, J. of Pure and Appl. Algebra, 77(1992), 131-140.

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[5] J. M. Ancochea-Bermudez, M. Goze, You. B. Hakimjanov (Khakimdjanov), Sur la reductibility de la variete des algebres de Lie nilpotentes, C. R. A. S. Paris, 313:1 (1991), 59-62.

[6] L. Boza, F. J. Echarte, J. Nunez, Classification of complex Filiform Lie Algebras of dimension 10, Algebra, Groups and Geometries, 11:3(1994), 253-276.

[7] F. Bratzlavsky, Classification des algebres de Lie nilpotentes de dimension n, de classe n — I , dont I'ideal derive est commutatif, Acad. Roy. Belg. Bull. Cl. Sci., 560 (1974), 858-865. [8] J.M. Cabezas, L.M. Camacho, J.R. Gomez, R.M. Navarro, A class of nilpotent Lie algebras, to appear in Communications in Algebra, 2000. [9] J.M. Cabezas, L.M. Camacho, J.R. Gomez, R.M. Navarro, E. Pastor, Lowdimensional naturally graded 3-filiform Lie algebras, Preprint MA1-01-OOVI, Universidad de Sevilla, 2000.

[10] J.M. Cabezas, J.R. Gomez, Las algebras de Lie (n — 3)-filiformes como extensiones por derivaciones, Extracta Mathematicae, 13(3) (1998) 383-391. [11] J.M. Cabezas, J.R. Gomez, (n — 4)-filiform Algebra, 27(10) (1999) 4803-4819.

Lie algebras, Communications in

[12] J.M. Cabezas, J.R. Gomez, Cohomological properties ofp-filiform Submitted to Journal of Lie Theory.

Lie algebras,

[13] J.M. Cabezas, J.R. Gomez, A. Jimenez-Merchan, Family of p-filiform Lie algebras, In Algebra and Operator Theory, 1997. Ed. Y.Khakimdjanov, M. Goze, Sh. Ayupov, 93-102, Kluwer Academic Publishers, 1998.

[14] J.M. Cabezas, J.R. Gomez, E. Pastor, Structure theorem for naturally graded 3-filiform Lie algebras, I Colloquium on Lie Theory and Applications, Vigo (Spain), 2000. [15] J.M. Cabezas, J.R. Gomez, E. Pastor, Naturally graded 3-filiform Lie algebras, Submitted to Proceedings of Edinburgh Mathematical Society, 2000. [16] L.M. Camacho, Algebras de Lie p-filiformes PhD thesis, Universidad de Sevilla, 2000.

[17] L.M. Camacho, J.R. Gomez, R.M. Navarro, 3-filiform Lie algebras of dimension 8, Ann. Math. Blaise Pascal, 6(2) (1999) 1-13. [18] L.M. Camacho, J.R. Gomez, R.M. Navarro, Family of laws of (n — 6)-filiform Lie algebras, Preprint MA1-02-OOVI, Universidad de Sevilla, 2000. [19] L.M. Camacho, J.R. Gomez, R.M. Navarro, The use of Mathematica for the classification of some nilpotent Lie algebras, IMACS-ACA'99, Meeting El Escorial (Spain), 1999.

[20] L.M. Camacho, J.R. Gomez, R.M. Navarro, Algebra of derivations of (n — 3)filiform Lie algebras, ILAS'99 Meeting Barcelona, 1999.

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[21] L.M. Camacho, J.R. Gomez, R.M. Navarro, Cohomology of some nilpotent Lie algebras, Extracta Mathematicae, 15 (2000). [22] F.J. Castro, J.R. Gomez, A. Jimenez-Merchan, J. Nufiez, How to obtain families of laws Lie algebras, Proc. EACA'95, 29-37. Serv. Publ. U. Cantabria, 1995. [23] A. Cerezo, Les algebres de Lie nilpotentes reelles et complexes de dimension 6, preprint N. 27, Nice, 1983.

[24] J. Dixmier, Sur les representations unitaires des groupes de Lie nilpotents III. Canad. J. Math., 10(1958), 321-348. [25] J. Dixmier, W.G. Lister, Derivations of nilpotent Lie algebras, Proc. AMS, 8(1957), 155-158.

[26] J. R. Gomez, F. J. Echarte, Classification of complex filiform nilpotent Lie algebras of dimension 9, Rend. Sem. Fac. Sc. Univ. Cagliari, 61(1) (1991), 21-29. [27] J.R. Gomez, M. Goze, Y. Khakimdjanov, On the k-abelian Filiform Lie Algebras, Communications in Algebra, 25(2) (1997) 431-450. [28] J. R. Gomez, A. Jimenez-Merchan, The graded algebras of a class of Lie algebras in Mathematics with Vision (ed. V. Keranen, P. Mitic), 151-158, Computational Mechanics Publications, 1995.

[29] J. R. Gomez, A. Jimenez-Merchan. Algebras de Lie graduadas y cdlculo simbolico. Actas del Primer Encuentro de Algebra Computacional y Aplicaciones, EACA'95, 57-65, 1995.

[30] J.R. Gomez, A. Jimenez-Merchan, Naturally Graded Quasi-Filiform Lie Algebras, Submitted to Journal of Algebra. [31] J. R. Gomez, A. Jimenez-Merchan, You. B. Hakimjanov (Khakimdjanov), On the Variety of Nilpotent Lie Algebra Laws of Dimension 11, Rendiconti Cagliari

66 (2) (1996) 137-142. [32] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimjanov, Classifying Filiform Lie Algebras with Mathematica in Innovation in Mathematics, (ed. A. Hietamaki, V. Keranen, P. Mitic), 169-176, Computational Mechanics Publications, 1997.

[33] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Low-Dimensional Filiform Lie Algebras. Journal of Pure and Applied Algebra 130 (1998), 133-158. [34] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Symplectic Structures on the Filiform Lie Algebras. To appear in Journal of Pure and Applied Algebra. [35] J. R. Gomez, A. Jimenez-Merchan, J. Nufiez-Valdez, An algorithm to obtain laws of families of filiform Lie algebras, Linear Algebra and its Aplications 279 (1998), 1-12.

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[36] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Tratamiento simbolico de algebras de Lie filiformes graduadas conexas. Actas del Segundo Encuentro de Algebra

Computational y Aplicaciones, EACA'97, 136-142, 1996.

[37] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Filiform Lie Algebras of Maximum Length. Submitted to Acta Mathematica Hungarica. [38] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras of Maximum Length. Submitted to Linear Algebra and Applications. [39] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras of Length dim[g, g] + 2. ILAS'99 Meeting Barcelona, 1999. [40] J.R. Gomez, R.M. Navarro, Espacios de derivaciones de algebras de Lie con Mathematica, Proc. of IV Encuentro de Algebra Computational y Alicaciones, (EACA'98). Sigiienza, 1998. [41] J.R. Gomez, I. Rodriguez. Geometrical properties of model metabelian Lie algebras, Submitted to Algebras, Groups, and Geometries. [42] J.R. Gomez, I. Rodriguez. A special case of family of Lie algebras with nilindex 3. ILAS'99 Meeting Barcelona, 1999. [43] J.R. Gomez, I. Rodriguez. Metabelian Lie algebras with maximal derived. ILAS'99 Meeting Barcelona, 1999. [44] M. Goze, You. B. Hakimjanov (Khakimdjanov), Nilpotent Lie algebras, Kluwer Academics Publishers, 1996.

[45] M. Goze, You. B. Hakimjanov (Khakimdjanov), Sur les algebres de Lie nilpotentes admettant un tore de derivations, Manuscripta Math. 84 (1994), 115224. [46] You. B. Hakimjanov (Khakimdjanov), Variete des lois d'algebres de Lie nilpotentes, Geometriae Dedicata, 40(1991), 229-295. [47] Yu. Hakimjanov (Khakimdjanov), Characteristically nilpotent Lie algebras, Mat. Sbornik, 181:5 (1990), 642-655 (russian). English transl. in Math. USSR Sbornik, 70:1(1991). [48] V. Morozov, Classification of nilpotent Lie algebras of sixth order. Izv. Vysch. U. Zaved. Mat., 4:5(1958), 161-171. [49] O. A. Nielsen, Unitary representations and coadjoint orbits of low-dimensional nilpotent Lie groups, Queen's Papers in Pure and Appl. Math., 63, 1983. [50] J. Reyes. Algebras de Lie casifiliformes

graduadas de longitud maximal. PhD

Thesis, Universidad de Sevilla, 1998.

[51] I. Rodriguez. Algebras de Lie con invariante de Goze dado. PhD Thesis, Universidad de Sevilla, 2000. [52] M. Romdhani, Classification of real and complex nilpotent Lie algebras of di-

mension 7, Linear and Multilinear Algebra, 24, 167-189, 1989.

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[53]

Saffiullina, On the classification of nilpotent Lie algebras of dimension 7, Kazan, 1976 (in Russian).

[54]

C. Seeley, Seven-dimensional nilpotent Lie algebras over the complex numbers, PhD. Thesis, Chicago, 1988.

[55]

C. Seeley, Degenerations of 6-dimensional nilpotent Lie algebras on C, Commun. Algebra, 18:10(1990), 3493-3505.

[56]

C. Seeley, Some nilpotent Lie algebras of even dimension, Bull. Austral. Math. Soc. 45(1992), 71-77

[57]

K. A. Umlauf, Uber die Zusammensetzung der endlichen continuierlichen Transformationsgruppen insbesondere der Gruppen vom Range null, Thesis, Leipzig, 1891.

[58]

G. Valeiras, Sobre las componentes irreducibles de la variedad de leyes de algebras de Lie nilpotentes complejas de dimension 8, PhD. Thesis, Univ. Sevilla, 1992.

[59]

M. Vergne, Cohomologie des algebres de Lie nilpotentes. Application a I'etude de la variete des algebres de Lie nilpotentes, Bull. Soc. Math. France 98(1970), 81-116.

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S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, 1991.

On L*-Triples and Jordan #*-Pairs A. J. CALDERON-MARTIN, Departamento de Matematicas. Cadiz. 11510-Puerto Real. Cadiz, Spain. E-mail: [email protected]

Universidad de

C. MARTIN-GONZALEZ, Departamento de Algebra, Geometria y Topologfa. Universidad de Malaga. Apartado 59, 29080-Malaga, Spain.

Abstract In [12], Lister introduced the concept of Lie triple system. He classified the finite-dimensional simple Lie triple system over an algebraically closed field of characteristic zero. Neher also studies Lie triple systems and their relations with Jordan triple systems in [13]. In order to study infinite-dimensional Lie triple systems, we introduced the notion of //-triple and obtained a classification of //-triples admitting a two-graded //-algebra envelope in [3]. However, the problem on the existence of L*-algebra envelopes is still open. We prove in this paper, using Jordan //"-pairs techniques, that every infinite-dimensional topologically simple //-triple, verifying a purely algebraic additional property, has a two-graded L*-algebra envelope and then we classify them.

1

PREVIOUS RESULTS ON //-TRIPLES

Let A be a C-algebra and * : A —» A a conjugate-linear map, for which (x*)* = x and (xy)* = y*x* hold for any x, y e A. Then * is called an involution of the algebra A. We recall that an H*- algebra A over C is a nonassociative C-algebra, which is also a Hilbert space over C with inner product (• •), endowed with an involution * such that (xy\z) = (x\zy*) = (y\x*z) for all x,y, z € A. A two-graded H*-algebra, is an //"-algebra which is a two-graded algebra whose even and odd part are selfadjoint closed orthogonal subspaces. We call the two-graded //'"'-algebra A topologically simple if A2 ^ 0 and it has no nontrivial closed two-graded ideals. In the sequel an //-algebra will mean a Lie //*-algebra. The classification of topologically simple //-algebras is given in the separable case by Schue (see [14], [15]) and later in the 87

88

Calderon-Martin and Martin-Gonzalez

general case in [8], [1] and [13]. If HI and H-2 are Hilbert spaces with scalar products ( • | - ) i , z = l,2 and / : H i —> H-2 a linear map such that

for any x, y € H I , then we will say that / is a k-isogenic map. We can define an H*-triple as a complex triple system (T, < -, •, • >) provided with: 1. A conjugate-linear map * : T —> T such that (x*)* = x and <x,y,z >* = < z * , y * , z * >

for any x, y, z G T. 2. A complex Hilbert space structure whose inner product is denoted by (•[•) and satisfies

(<x,y,z>\t) = (x ) = (y| ) = (z ) for any x, y, z,t €. T.

From [5] it follows that the triple product < - , - , - > of T is continuous.

We define the annihilator of an //"-triple (T, < •, •, • >) as

Ann(T) = {x e T :< x,T,T >=< T,x,T >=< T,T,x >= 0}. This set Ann(T) turns out to be a closed ideal of T. The definition of topologically simple H "-triple is similar to that used in the case of //"-algebras. The structure theorems for //"-triples given in [5] reduce the interest on H *-triples to the topologically simple case. Essentially, these theorems claim that under suitable conditions, one can split any //"-triple as the orthogonal direct sum of its annihilator plus the closure of an orthogonal direct sum of topologically simple //"-triples. We define an L* -triple as a Lie //""-triple. If L is an L*-algebra, then it can be viewed as an L*-triple by defining the triple product [x,j/,zj = [[x,y],z] for any x,y,z <E L, this L*-triple will be denoted by LT . Furthermore, for any two-graded L*-algebra, L = Lo-LLi, its odd part LI is an L*-subtriple of LT . If T is an //"-triple isometrically *-isomorphic to the L*-triple LI for some two-graded L*-algebra L, we shall say that L is a two-graded L*-algebra envelope of T iff LQ := [Li, L I ] . It is

also easy to check that if T is an L*-triple with two-graded L*-algebra envelope L, then T is topologically simple if and only if L is topologically simple in the graded sense. We refer to [3, Theorem 1] for the following classification of topologically simple L*-triples that admit a two-graded L*-algebra envelope.

THEOREM 1.1 Let T be a topologically simple L* -triple admitting a two-graded L* -algebra envelope. Then T is one of the following: 1. The L* -triple associated to an L*- algebra L by defining the triple product [a, 6, c] := [[a, 6],c] and the same involution and inner product of L.

L*-Triples and Jordan tf*-Pairs

89

2. Skw(A,r) with A a topologically simple associative H* -algebra, T an involutive *- automorphism of A, the involution and inner products induced by the ones in A, and the triple product as in the previous case. 3. Sym(A,a)

with A as in the previous case but now a is an involutive *-

antiautomorphism, and the triple product, involution and inner product induced by the ones in A. 4- Skw(A,T)r\Skw(A,a) with A as before, T an involutive *- antiautomorphism, a an involutive ^-automorphism such that ar = ra , and the involution and inner product induced by the ones in A.

The problem on the existence of L*-algebra envelopes is still open. However, if T is an L*-subtriple of A~ for a topologically simple ternary //"-algebra A, that is, for any x, y, z & T one has

[x, y, z] =< x, y, z > - < y, x, z > - < z, x, y > + < z, y, x > where < •, •, • > is the triple product of A, then it can be proved that it has a twograded L*-algebra envelope. Indeed, from the classification of topologically simple ternary //"-algebras (in the complex case) of [7, Main Theorem, p. 226], one may see that there is an associative topologically simple two-graded //"-algebra B = Bo-LBi (see [6] for classification theorems) such that A is the ternary //"-algebra associated to BI (with triple product < x,y,z >= xyz for all x,y,z 6 BI). Let // = Lo-L^i be the two-graded L*-subalgebra of B" generated by T. It is easy to prove that LI = T and LQ = [T, T], hence the topological simpleness of T implies that of L in the graded sense.

2

PREVIOUS RESULTS ON JORDAN //"-PAIRS

Let A = (A+ , A~) be a pair of modules over a commutative unitary ring K, and < - , - , - >: Aa x A~a x A" —» A",

two trilinear maps such that

(x,y,z) H->< x,y,z> for a g {+,—}• Then A is called an associative pair if the following identities are satisfied: « x,y,z >,u,v >=< x, < u,z,y >,v >—< x,y,< z,u,v » for x,z,v € A" and y, u 6 A~a .

Let A = (A+,A ) be a pair of /iT-modules and Q° : A° —> homK(A~CT,A'7) two quadratic operators for a € {+,—}• We define the trilinear operators

and the bilinear operators

Da : Aa x A~a —> End(Aa)

90

Calderon-Martm and Martin-Gonzalez

as {x,y,z}° = D"(x,y)z := Q°' (x + z)y - Qa' (x)y - Q°' (z)y , for z, z £ Aa , y € A~a and cr 6 {+,—}. We shall say that A = (A+,A~) is a Jordan pair if the next identities and its linearizations are true:

Q°(Q°(x)y) = Q°(x)Q for x, z e A° , y <E A~a and cr e {+, -}. One can find a classification of prime, nondegenerate Jordan pairs with nonzero socle in [10, Teorema 6] or [9, Theorem 7]. If A is an associative pair, then AJ will denote the symmetrized Jordan pair of A, that is, the Jordan pair whose underlying /('-module agrees with that of A, and whose quadratic operators are given by Qa (x)y =< x,y,x >a .

Let A = (A+,A~) be a complex pair and * = (* + ,*~) a couple of conjugatelinear mappings *°" : Aa —> A~a for which *CT o *~CT = Id and

for xa , za 6 Aa and y a G A C, endowed with an involution * = (* + ,*~) such that

t-cr for xa,za,ter e ^ and y~a e A" We also recall that an H*-psdr A is said to be tope/logically simple when

< Aa,A~a,A° > ^ 0 and its only closed ideals are {0} and A. If A is an associative H*-pair, then AJ can be canonically equipped with a Jordan H*-pair structure. Moreover, if £ : A —» Aop is an involutive *-antiautomorphism, then both Sym(A, £) and Skw(A, ^) can also be structured as Jordan //"-pairs in obvious ways. Given an infinite dimensional and topologically simple Jordan //*-pair, we can always describe it from a Jordan H*algebra or from an associative //*-pair. Indeed, every topologically simple Jordan //"-pair J is prime, non-degenerate and with non-zero socle (see [2, Proposition 1]), then, the classification of Jordan pairs given in [10, Teorema 6] implies that forgetting the H*-structure, the underlying Jordan pair J is one of the following: Type (i). J = (V, V) with (V, {-, •}) a quadratic Jordan algebra, being the triple products {x, y, z}a = {x, { y , z}} - {y, {x, z}} + {z, {x, y}} for any x, y, z e V. Type (ii). J is a subpair of (L(X,Y),L(Y,X))J containing (F(X, Y),F(Y,X))J a with the triple products {x,y,z} := xyz + zyx , where (X,X') and (Y,Y') are dual pairs over an associative C-division algebra A. Type (iii). J is a subpair of

(Sym(L(X, Y), |), Sym(L(Y, X), H))

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91

containing (Sym(F(X,Y),$),Sym(F(Y,X),W where (X,YJ), (X,YJ°r) are a dual pair and its opposite over an associative C -division algebra (A,r) with involution T, where the triple products are {x, y, z}" = xyz + zyx and with ft the adjoint operator. Type (iv). J is a subpair of

(Skw(L(X, 7), fl), Skw(L(Y, X ) , «)) containing (Skw(F(X,Y),$),Skw(F(Y,X),$)) where ( X , Y , f ) , ( X , Y , f ° P ) are a dual pair and its opposite over C, where the triple products are {x, y, z}a = xyz + zyx and with ft the adjoint operator. If J is a topologically Jordan //*-pair of Type (i), then one can define an involution and inner product on V so as to obtain that V is a Jordan //"-algebra. We prove, in [2], that any topologically Jordan //"-pair of Type (ii) is of the form J = AJ , where A is a topologically simple associative //*-pair. Similar arguments apply to a topologically simple Jordan //*-pair J of types (iii) or (iv), give us that J is of the form J = Sym(A, £) or J = Skw(A, £) respectively, with A as above, and £ an involutive *-antiautomorphism from A to Aop, (see [4] for more details). The concept of polarized triple system of a pair will also be used with the same meaning as in [13].

3

MAIN RESULTS

Neher finds in [13] the relation between simple polarized Lie triple systems, simple polarized Jordan triple systems and simple Jordan pairs. We can obtain similar results in an H "-context so as to give the next

THEOREM 3.1 (a) Let L = L+ J_ L~ be a topologically simple polarized L* -triple. Then J(L) : — L with the product

is a topologically simple polarized Jordan H* -triple. (b) Let J = J+ _L J~ be a topologically simple polarized Jordan H* -triple. Then L ( J ) :— J with the product x,y,z := {x,y,z} — {y,x,z} is a topologically simple polarized L* -triple. (c) The operations L —> J ( L ) and J —> L(J) described in (a) and (b) are inverses of each other.

PROOF.- (a) and (b). According to [13] we have that J(L) := L (resp. L(J) := J) is a Jordan triple system (resp. Lie). It is easy to check that the involution and inner product of L (resp. J) make J(L) := L (resp. L ( J ) := J) an //"-triple. Since any ideal of J(L) is an ideal of L, we conclude that J(L) is a topologically simple Jordan //"-triple. Our next claim is that L(J) is topologically simple. We first observe that AnnL(J) = 0. Indeed, from [13], AnnL(J) is a IT-invariant ideal of L(J), being II the automorphism of L ( J ) defined as H((x+,x~)) : — (x+, -x~), hence [13] shows that AnnL(J) is an ideal of J and therefore AnnL(J) = 0. Finally, let J be a closed ideal of L(J), as [ I , L ( J ) , L ( J ) } is an ideal of J ([13]), either [7, L( J), L(J)\ = 0 and then / = 0, or [I, L ( J ) , L ( J ) ] = J which implies / = L( J ) .

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The item (c) is immediate to check.

It follows, easily, that a Jordan H*-pair is topologically simple if and only if its associated polarized Jordan _ff*-triple is also topologically simple. The next result is clear by [13].

PROPOSITION 3.2 Let J; = (J+, Jr) (i e {1,2}) be topologically simple Jordan H*-pairs, and let L(Ji) be their associated polarized L*-triples. Then, if J\ and J% are *-isometrically isomorphic, L(J\) and L(J^) are also *-isometrically isomorphic. If L(Ji) and £(1/2) are *-isometrically isomorphic then J\ is *-isometrically isomorphic with J% or J%p.

THEOREM 3.3 Let (T, [ - , - , • ] ) be an L*-triple. Then there exists an associative algebra U such that [x, y, z] = xyz — yxz — zxy + zyx for any x,y,z € T. PROOF.- The proof consists in the construction of algebra envelope of T and then, denning U as the L. Define LQ to be the span of ad(Li,Li), where product operator. We have that L = LQ ® LI with

L = LQ ® LI, a two graded Lie universal enveloping algebra of a d ( x , y ) ( z ) = [ x , y , z ] is the left the product defined by

[(ad(x,y),z),(ad(u,v),w)}

=

= ( a d ( [ u , v , y ] , x ) - ad([u, v , x ] , y ) + ad(z,w), [z,y, w] - [ u , v , z ] ) is a two graded Lie algebra envelope of T, that is, {x, y, z] — [[x, y]z] for any x, y, z € T. (In fact, one can construct an involution *, preserving the grading, and a nondegenerate hermitian form /, in such a way that LI is isometrically *-isomorphic to T considered as an L*-triple with the restriction of * and of /, however, it does not seem easy to complete this construction so as to produce on L an i*-algebra structure). The universal enveloping algebra of L, see [11, Chapter V] is the algebra U we are looking for.

THEOREM 3.4 (MAIN THEOREM) Let (T, [•, -, •]) be an infinite dimensional topologically simple L*-triple system. Let U be an associative algebra such that [x, y, z] = xyz —yxz — zxy + zyx for any x,y,z e T. If xyx G T for every x,y e T, then T has a two-graded L*-algebra envelope and therefore is one of the L*-triples described in Theorem 1.1. PROOF.- By section 1, we only have to prove that T is an L*-subtriple of A~, being A a topologically simple ternary H ""-algebra. It is straightforward to prove that T ® T with the product [(a, 6), (c, d), (u, v)} := (adu + uda — cbu — ubc, bcv + vcb — dav — vad),

involution (a, 6)* := (6*, a*) and inner product < ( a , b ) \ ( c , d ) >=< a\c > + < b\d > is a polarized L*-triple and that

T

—>

T®T

x

H->

(x,x)

is a fc-isogenic *-monomorphism of L*-triples. Now, the structure theorems for polarized L*-triples (analogous to the ones in [5]), allow us to assert that T is

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93

*-isomorphically /c-isogenic to an L*-subtriple of a topologically simple polarized L*-tripleT+ LT~. Let (J = J+ _L J~, {-,-,•}) be the polarized Jordan H*-triple associated to + T _L T~ (see Theorem 3.1). Following the description of topologically simple Jordan #*-pairs given in Section 2 we have to study the following four possibilities: (a) If J = (V, V), where V is a quadratic Jordan //*-algebra, then we have

4> : T+ ±T~

—»

L(V ± I/)

an *-isometric isomorphism of //-triples. One easily shows the existence of

+ : and 4>~ : T —> also check that

V

T

—»

V

such that 4>(x,y] = (4>+ (x) , <j>~ (y)) for x,y € T. We can

for any x + , y ~ , 2 + £ T. Therefore ~ (4>+}~l , and a suitable involution. However, (V, [•, •, •]') turns out to be a nontopologically simple L*-triple (moreover, Ann(V) ^ 0 since [I, a, b]' = 0 for a, 6 £ V), which is impossible. Thus, J has to be one of the other types of topologically simple polarized Jordan //""-triples, that is: (b) J = (A+ _L A~)J with (A+ A. A~ , < • , - , - > ) a topologically simple ternary ff*-algebra. Then, for any (x+,x~), (y + , y~), (z+,z~) £ T, we have that

<

that is, T is a subtriple of (^4+ ± A~)~, with A+ X A~ a topologically simple ternary ff* -algebra, as we wanted to prove. (c) and (d). If J = Sym(A+ _L A~,£) or J = Skw(A+ ± A~,£), with (>1+ 1 ^4"", < - , - , • > ) a topologically simple ternary _ff*-algebra, we can prove as in case (b) that T is also a subtriple of a topologically simple ternary H* -algebra. This completes the proof. Note that, following the proof of the last theorem, we have that every nonquadratic topologically simple L*-triple system that embeds in a polarized i*-triple system (particularly topologically simple polarized L*-triple system) has a twograded L*-algebra envelope.

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REFERENCES [1] M. Cabrera, A. El Marrakchi, J. Martmez and A. Rodrfguez Palacios, An Allison-Kantor-Koecher-Tits construction for Lie #*-algebras. Journal of Algebra. 164 (1994), no.2, 361-408.

[2] A. J. Calderon and C. Martin, Dual pairs techniques in H* -theories, Journal of Pure and Applied Algebra 133 (1998) 59-63.

[3] A.J. Calderon Martin, C. Martin, Two graded L*-algebras, preprint, Universidad de Cadiz (Spain), 1999. [4] A.J. Calderon Martin, C. Martin, On associative ff*-structures induced by Jordan JY*-pairs, preprint, Universidad de Cadiz (Spain), 1999. [5] A. Castellon, J. A. Cuenca, Associative #*-triple Systems. In "Workshop on Nonassociative Algebraic Models'. Nova Science Publishers (eds. Gonzalez S. and Myung H. C.). New York, 1992. pp. 45-67. [6] A. Castellon, J. A. Cuenca, and C. Martin, Applications of ternary #*-algebras to associative /f*-superalgebras, Algebras, Groups and Geometries, 10, (1993), 181-190.

[7] A. Castellon, J. A. Cuenca, and C. Martin, Ternary H*-algebras. Bolletino U.M.I. (7) 6-B (1992), 217-228. [8] J. A. Cuenca, A. Garci'a and C. Martin, Structure theory for L*-algebras. Math-Proc. Camb. Phil. Soc. (1990), 361-365. [9] A. Fernandez Lopez, E. Garcia Rus and E. Sanchez Campos. Prime Nondegenerate Jordan Triple Systems with Minimal Inner Ideals. Nova Science Publishers Inc. (1992) 143-166. [10] A. Garcia, "Nuevas aportaciones en Estructuras Alternativas", Tesis Doctorales/Microficha num. 157. Secretariado de Publicaciones de la Universidad de Malaga. 1995. [11] N. Jacobson, Lie algebras. Interscience 1962.

[12] W.G. Lister, A structure theory of Lie triple systems. Trans. Amer. Math. Soc. 72 (1952), 217-242. [13] E. Neher, On the classification of Lie and Jordan triple systems. Comm. in Algebra. Vol. no. 13 (12), 2615-2667.

[14] J. R. Schue, Hilbert Space methods in the theory of Lie algebras. Trans. Amer. Math. Soc. vol 95 (1960), 69-80.

[15] J. R. Schue, Cartan decompositions for I/*-algebras. Trans. Amer. Math. Soc. vol 98 (1961), 334-349.

Toric Mathematics from Semigroup Viewpoint A. CAMPILLO, Departamento de Algebra, Geometrfa y Topologfa, Universidad de Valladolid. 47005-Valladolid. Spain. E-mail: campillo @agt. uva. es

P. PISON, Departamento de Algebra, Computacion, Geometrfa y Topologfa, Universidad de Sevilla. Aptdo. 1160. 41080-Sevilla. Spain. E-mail:ppison@cica. es

I

INTRODUCTION

Toric geometry is a subject of increasing activity. Toric varieties are objects on which one usually can check explicitly properties and compute invariants from algebraic geometry. This happens for the so-called normal toric varieties, i.e. algebraic varieties which are constructed from rational fans in a euclidean space. In the last 10 years the theory of non normal toric varieties has also been developed providing a very different and new scope as well as interesting and beautiful new applications. Normal toric geometry mainly uses techniques from convex geometry, as it is technically founded on the concepts of fan and cone. Fans are sets of polyhedral cones in such a way that each cone provides an affine chart of the toric variety. Namely, those charts have, as coordinate algebra, the algebra of the semigroup of lattice points lying inside the dual cone of the corresponding cone of the fan. To study non normal toric geometry one needs to be more precise than to consider only cones. In fact, what one needs is to consider affine charts where coordinate algebras are semigroup ones for more general classes of semigroups. Thus, convex geometry should be used only as a tool by taking into account that nice semigroups generate concrete polyhedral cones. The purpose of this paper is to show how mathematics in toric geometry can be understood as the theory of appropriate classes of commutative semigroups with given generators. This viewpoint involves the description of various kinds of derived objects as abelian groups and lattices, algebras and binomial ideals, cones and fans, 95

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affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics. Our approach consists in showing the mathematical relations among the above objects and clarifying their possibilities for future developments in the area. For that purpose, we will survey some recent results and concrete applications.

2

SEMIGROUP AND GENERATORS OF TORIC GEOMETRY

The central object we will consider along the paper should be finitely generated cancellative commutative semigroups with a specified system of generators. By a commutative semigroup we understand here a set S endowed with an internal commutative operation denoted by + having a zero element denoted 0. Semigroup homomorphisms are maps preserving the operation + and the element 0. Thus, one has the category of semigroups. Cancellative for 5 means that S is isomorphic to a subsemigroup of an abelian group, or in other words that the semigroup homomorphism S —> G(S), where G(S) is the abelian group generated by S, is injective. Here G(S) is the abelian group of classes of pairs (m,n) G 5 x 5 for the relationship (m,n) ~ (m',n') iff m + n! — m' + n. Thus, our central object should be the data of a semigroup S as above plus a surjective semigroup homomorphism

TTO : Nh -> 5, where N is the semigroup of nonnegative integers. Notice that TTQ is just the same data than the choice of a generator system of the semigroup S, namely the generator system ni,... ,nh, where rij = TTQ(CJ), ej being the /i-uple with j-coordinate 1 and other coordinates 0. Since toric geometry is a subject providing explicit computations and results, one can think that toric mathematics essentially consists in the detailed study of maps TTo of the above type. Along the paper it will be shown how the above statement stands when dealing with affine or projective toric objects. For studying such a map TTQ one needs to understand the structure and behavior of its fibers 7r^"1(?7i) for m & S. This is an elementary and difficult problem which,

for many purposes, becomes the key problem of toric geometry. A first remark is that one should consider some kind of finiteness hypothesis, namely requiring that the fibers TT^^TO) be finite for every m. The following result gives some distinct characterizations of that hypothesis.

PROPOSITION 2.1 (see [4]) Let TTQ : N'1 —> 5 be a surjective semigroup homomorphism where S is a cancellative commutative semigroup. Then, the following conditions are equivalent: 1. KQ (TO) is finite for every m.

2. There is no infinite sequence m € S, TOI, . . . , m j , . . . € S — {Q}, such that m — TOI — • • • — mi 6 S for every i. 3. Sn(-S) = {0}.

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4- There exists a semigroup hornomorphism A : 5 m = 0.

N such that A (TO) = 0 iff

Semigroups satisfying conditions in Proposition 2.1 are called, in the literature, of different forms according to the property one wants to emphasize. Thus, they are said to be combinatorially finite in view of (1) (see [5]), Nakayama in view of (2) (see [18] and [24]), strongly convex in view of (3) (see [11]), or positive in view of (4). The terminology which will be used along this paper is that of "positive". The description of the fibers is related to the study of relations among the chosen generators of S. Since the "kernel" of TTQ does not exist in the category of semigroups, to describe the relations one needs a different object, the congruence F of TTo, to define those relations. The congruence F is the binary relation on N^ consisting of those pairs (u, v) g N'1 x N'1 such that u, v belong to the same fiber TTQl(rn) for some m e S. Congruences are binary equivalence relations on semigroups allowing to give a semigroup structure on the quotient, i.e. with the property that (u, v) 6 F and w is in the semigroup (i.e. Nh in our case) then (u + w,v + w) e F. Since 5 is a finitely generated semigroup, by [13, 1.6], one has that the congruence F is finitely generated, i.e. that F is the least congruence containing one finite set of elements in it. In other words, one can say that 5 is a finitely presented semigroup. In the rest of the paper we will show how to treat and exploit the information in a semigroup with their generators and relations. This will involve several fields of mathematics on each of which one will derive concrete perspectives and consequences. The figure below shows the scheme of the spirit of our discussions.

\

T Semigroups (Generators and relations)

I Arithmetics!

3

ABELIAN GROUPS AND LATTICES

Consider a map TTQ : N'1 —> 5 as in section 1. Since the assignment to a semigroup of the group generated by the semigroup is functorial, one has an induced exact sequence of abelian groups given by

0 -> L -» G(Nh) = Zh -> G(S) -» 0, where L is a subgroup of Zh, so L is finitely generated and torsion free. We will refer to L as the lattice associated to the data TTQ. Notice that L is nothing but the kernel of surjective induced group homomorphism -K : Zh —> G(S), so that L is the object keeping the information of the group theoretical relations among the semigroup generators n\,..., n^.

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The relation between the congruence F and the lattice L can be described easily. In fact, if (u, v) e F then there is a unique element w € N'1 such that (u — w, v — w) 6 F and the supports of u — w and v — w are disjoint. Here by support of an element of Zh we mean the set of indices whose coordinates are nonzero for such an element. Notice that if < denotes the componentwise product ordering on Zh, then w is nothing but the infimum of u, v for that ordering. Thus, one has a well defined map b : F -> N'1 x L,

given by 6(u, v) = (w, u — v). If (w,l) € N'1 x L, set 1 = 1+ - I", where 1+ = sup(l,Q), l~ = sup(-l,0) and sup(,) stands for the supremum relative to the ordering <. Then the assignment to the element (w, 1) of the couple (1+ + w, 1~ + w) is a map N h x L —> F which is, by construction, inverse to 6. So one has the following result. PROPOSITION 3.1 The map b is a bijection. It follows from Proposition 3.1 that the information in F is just the same as in L and how one can get one from the other. Moreover, from free abelian groups and their sublattices one can study the semigroups we are interested in. In fact, if a lattice L C Zh is given, then from the obvious exact sequence

0-> L-+Zh -^Zh/L->0, one can consider the subsemigroup S of the group Zh/L given by the image of Nh and generators given by the images of the elements ei,... ,e^. Also notice that the condition on S to be positive is equivalent to the condition L D Nh = (0). Notice that, in general, the abelian group G(S) = Zh/L can have torsion, so the semigroup S can also have torsion in the sense that it can contain elements ?7i, n € S, m ^ n and integers a G N such that am = an. If T is the torsion subgroup of G(5), the image of S in G(S)/T is a new semigroup S of the same kind than 5. Notice that S is positive iff S is so. This follows from the fact that LnN' 1 = (0) iff LnN' 1 = (0), L being the lattice for the induced map 7f0 : Nh -> 5. Finally, we remark that S is not only the image of N'1 by TT, but S is also the image of other subsets of Zh, in particular of the set N^ + L. This new set is also a semigroup which has an obvious structure of N^-module. As semigroup it is not positive (except in the trivial case L = 0 for which 5 = (0)), however if S is positive then N^ + L has the property analogous to (2) in Proposition 2.1, i.e. there is no infinite sequence of elements m = TOO > TOJ > . . . > TOJ > . . . in N'1 + L. In other words, if 5 is positive then N ft + L is generated by its minimal elements for the ordering <. Note that such minimal elements are nothing but the primitive elements of the set N'1 + L, i.e. those elements which are not a sum of a nonzero element of Nh with another element of N'1 + L.

4 SEMIGROUP IDEALS AND ALGEBRAS In this section we will fix a commutative field k. Then, one has a functor from the category of semigroups to that of /c-algebras taking each semigroup to its semigroup /c-algebra. Notice that, for any semigroup S, the semigroup /c-algebra k[S] consists

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of the vector space generated (freely) by the symbols x m > one f°r each m e 5, endowed with a multiplication given on symbols by the rule xm ' Xn = X m + n > f°r m,n € S. Now, consider a map TTQ : N'1 —> 5 as in section 1, and apply the above functor to it. One gets an exact sequence

0 -» 7 -> A = k[Nh ?$R = k[S] -» 0, where / is the kernel of the fc-algebra homomorphism • • •) x" h i so that / can be understood as the ideal of polynomial relations of such symbols. The ideal 7 is binomial as it is generated by the binomials X" — X" for (u, v) ranging over the congruence F. Using Proposition 3.1 one sees that it is also generated by X' — X' where 1 ranges over the lattice L. Anyway, notice that to generate / it is enough to take a finite number of binomials X" — X w , where the couples (u, v) generate the congruence F. Now, assume that S is positive. Then nice properties occur. First, by 3) in Proposition 2.1, one has that the irrelevant MR = ®TOJO kxm and MA = ©m-*;o A™ are ideals of R and A respectively. Second, by 1) one has that each Am is a finitely dimensional vector space. Third, by 2), Nakayama's lemma holds for Sgraded modules; in particular one can speak about minimal systems of homogeneous generators for / which are nothing but those inducing a basis of the vector space I/MA!- It is clear that one can consider minimal sets of binomial generators for /. In fact, one can consider S'-graded free homogeneous resolutions of R as an A-module. If 5 is positive, Nakayama's lemma shows that one can consider the minimal free resolution (which is unique up to isomorphism) which is one of the type n ^ pfp^ —> p ••• ^ lp p = AA —> fS. riH —> ^. U, r\ U —> —> p 1*2^—> f\ ^—>l -TQ where each Fi is a free S-graded finite A-module, the ^ are graded of degree 0 and p is the projective dimension of R as A-module, i.e. the least integer p such that Fp ^ 0. The Auslander-Buchsbaum theorem shows the relation p + r — h, where r is the depth of R. The integer r ranges over the values 0 < r < d, where d is the Krull dimension of R. Notice that the Krull dimension of R coincides with the rank of the abelian group G(S). The last statement follows from the computation of dimensions in terms of transcendence degrees. It implies, in particular that the dimension of the /c-algebra k[S] does not depend on the field k. This is not the case

for the integer r which could depend on k.

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Commutative algebra provides interesting particular cases. First, when r = d, the ring k[S] is said to be Cohen-Macaulay. This property depends on S and k but not on the map TTQ. If k[S] is Cohen-Macaulay and, moreover, Fp has rank 1 as an A-module, then k[S] is said to be Gorenstein. This is a case in which the minimal resolution is self-dual, i.e., by applying the functor Hom( — ,A) and considering the natural grading, the induced exact sequence

0 -> Eom(FQ, A) -> Hom(Fi, A) -> • • • • • • -* Eom(Fp, A) -» Coker(<4) -> 0, is S-graded isomorphic to the minimal resolution of R. Again the Gorenstein property depends on S and k, not on TTQ. Finally, k[S\ is said to be a complete intersection if / can be generated by h — d homogeneous elements (in fact binomials). Equivalently, complete intersection means that the congruence F can be generated by h — d pairs. The complete intersection property only depends on 5 and implies the Gorenstein property.

5

CONES AND FANS

With assumptions as above, the next object one can associate to a semigroup 5 is the cone C(S) generated by S, i.e. the cone generated by the image of S in the Q-vector space VQ := G(S) (8>z Q. Since the base ring extension from Z to Q kills the torsion, the cone C(S) obviously coincides with that of its image 5 in G(S)/T. If 5 is not positive, then C(5) is equal to the whole VQ, so it contains trivial information. Thus, the interesting case turns out to be the case in which S is positive. Note that S is positive if and only if C(S) is a strongly convex cone (i.e., if one has C(S) n —C(S) = 0). This fact justifies the terminology in (3) section 1. Now, if one takes into account the generators of 5, then one has that the cone C(S) is the rational polyhedral one generated by (i.e. it is the convex hull of) the images in VQ of the generators n\,..., n^. Thus, convex geometry occurs as a useful technique of toric mathematics. There is a very important case, in which the cone C(S) determines the semigroup 5. In fact, a semigroup is said to be normal if it is torsion free and if, moreover, one has 5 = C(S) n G(S). It is well known that 5 is a normal semigroup if and only if the semigroup fc-algebra A;[5] is an integrally closed domain, i.e. a normal ring. Hochster's theorem [14] shows that if S is normal then, in fact, k[S] is CohenMacaulay. A trivial example of normal semigroups are the free semigroups, i.e., those which are isomorphic to N* for some integer t. In fact, free semigroups are the only ones such that the fc-algebra k[S] is a regular ring. The terminology "regular" is coherently used also in convex geometry, being applied to a cone on Q* which is generated by a basis of the lattice Z*. Notice that a semigroup is free if and only if it is normal and if the cone it generates is regular. Toric geometry appears initially as the study of normal toric varieties. Thus, the development of normal toric geometry is based on convex geometry and, therefore, one can say that normal toric mathematics is convex geometry mathematics. Coming back to the general case, the convex cone C(S) provides a new interesting invariant for a semigroup 5, namely the number of edges e of C(S). Comparing

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with the dimension, one has e > d, and the equality holds whenever the cone C(S) is simplicial. Thus, semigroups for which e = d will be called simplicial along the paper. Free semigroups are a very special case of simplicial semigroups. Toric varieties also include non affine ones. Affine toric varieties are nothing but the affine varieties X with coordinate fc-algebra equal to a semigroup fc-algebra with the assumptions of section 1. General toric varietes are algebraic varieties which can be covered by affine toric varieties with overlappings which are also affine toric. Normal toric varieties are usually given in terms of convex geometry. The data consists in a fan $ of rational polyhedral cones in Q™, i.e, a set {crj^g^, where <& is a finite set, each a a strongly convex polyhedral cone in Q n , the faces of each a in $ are also in $, and the intersection of every couple of two cones in $ is a common face of both of them. The variety is constructed in the following way. For each a in <&, consider the semigroup 5CT of integer coordinates points which lie inside the dual cone of a, and let Xa be the affine toric variety given by Sa. Then the toric variety X is the join of the affine varieties Xff, the intersection of any two Xa, XT

of those affine charts being the toric variety Xanr. Thus, for a normal toric variety, the fan <& not only determines the variety but it represents and exhibits its geometry. In fact, cones in the fan correspond to affine charts in such a way that intersection of cones correspond to the overlapings of the corresponding charts. For non normal toric varieties one can proceed in a similar way, but taking a further precision on the semigroups. Thus, one needs a fan 4> as above plus, for each cone a, a subsemigroup S'a of Sa generating the same cone as Sa and in such a way that the intersection of two charts with respective coordinate algebras k[S'a] and k[S'T] is the affine chart with coordinate algebra /c[S^.nr]. Thus, one sees that, also in the global case, toric mathematics is not only convex geometry but again involves finitely generated cancellative semigroups. The support of a fan <£> is defined to be the union of the supports of the cones in the fan. The fan is said to be complete if its support is Q n . Toric varieties built from complete fans are complete algebraic varieties. The next section is devoted to the particular case of projective varieties, a subclass of complete toric varieties.

6

AFFINE AND PROJECTIVE TORIC VARIETIES

Toric varieties are algebraic ones, so algebraic geometry is naturally related to toric mathematics. Particularly interesting algebraic varieties are the affine and projective ones. When some data TTQ : N^ —-> S, is given, the semigroup 5 gives rise to the (abstract) affine toric variety X = Spec(k[S]), whereas the choice of generators provided by TTQ gives rise to an embedding of X into the affine space A.h. The dimension of X is just the rank d of the abelian group G(S). Below, we discuss and emphasize how abstract and embedded projective toric varieties can also be described in nice terms. Let S be a finitely generated cancellative commutative semigroup. Assume that 5 is endowed with a semigroup map A : 5 —> N such that the semigroup is generated by the elements in the set 5j = A - 1 (l). Then, for any choice of the field k, the couple (S, A) gives rise to an (abstract) (d — l)-dimensional projective algebraic scheme, namely Z = Proj(k[S}), where k[S] is now viewed as an Ngraded algebra by relaxing its natural 5-grading via the map A (in other words,

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degree i 6 N homogeneous elements are the sum of homogeneous elements of Sdegrees in A~ 1 (z)). Along the paper, couples (S, A) as above will be referred to as polarized semigroups. For a polarized semigroup (5, A), one has the property that m G 5 is a sum of i > 0 elements of Si if and only if one has A(m) = i. This property has two immediate consequences. First, the set Si (and hence any fiber X ~ 1 ( i ) ) is finite, as 5*1 is nothing but the set of irreducible elements in S. Second, the semigroup S is, a fortiori, positive. For the last statement, notice that to prove positiveness, when a map A as above already exists, one only needs to check A~ x (0) = 0 which follows from the afore mentioned property. Now, assume that 5 is torsion free. Then, since k[S] is a domain, the projective algebraic scheme Z is, in fact, a projective algebraic variety.

PROPOSITION 6.1 Let (S, A) be a polarized semigroup such that S is torsion free. Then Z = Proj(k[S}) is a projective toric variety. In fact, since S is torsion free, it can be viewed as a subset of VQ. On the other hand, the map A extends to a group homomorphism AZ : G(S) —> Z and to an R-linear map AR : VR —» R, where VR = G(S) ®z R» Now, let QI be the convex hull of the set Si in VR, and let S° C Si be the vertex set of S x . Notice, that S°,Si and QI lie in the affine hyperplane in VR given by A^^l). Fix m° € S°. Then, the semigroup S(m°) generated by the set of elements of type TO — TO° with m G Si is a new positive finitely generated semigroup whose associated group is A^^O). In particular, it follows that the dimension of the affine toric variety X(m°) given by S(m°) is d— 1 where d = rankG(S), i.e. the dimension of the projective variety Z. Moreover, X — Spec(k[S}) being the projecting cone of Z, the construction shows that the affine toric varieties X(m°), when TO° ranges over Sf, form a covering of Z as affine charts, making Z into a projective toric variety. This shows the proposition. For projective normal varieties it is possible to describe which Cartier divisors are ample and very ample ones. By a polarization of a projective variety one means picking a very ample Cartier divisor class. It provides an embedding of the variety in a projective space. When the variety is toric, one sees that the polarization produces a polarized semigroup (S, A) in such a way that the variety is isomorphic to the one given by the couple (S, A). See [11] for details. Thus, it is equivalent to give an embedded projective toric variety and to give a polarized semigroup. Notice that, for a given polarized semigroup (S, A), the set Si is the set of irreducible elements of S, so it is the only generator set contained in Si which gives the embedding of the affine toric variety X = Spec(k[S\) which is the projecting cone of Z. Thus, a polarized semigroup provides a canonical embedding of the projective toric variety into ph~l where h is the cardinality of Si. We remark that the fan giving rise to the projective variety Z lies in the dual space of the hyperplane Aq^O). Namely, the cones of the fan are exactly the duals of the cones generated by the semigroups S(m°). By construction, it is easy to see that such a fan is a complete one which corresponds to the algebraic geometric fact that any projective variety is complete. Finally, as it occurs for affine toric varieties, the main algebraic geometric characteristics of projective toric varieties are recognized in terms of the polarized semigroup (S, A). Thus, Z = Proj(k[S}) is said to be arithmetically Cohen-Macaulay

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(resp, Gorenstein) if and only if the algebra k[S] is Cohen-Macaulay (resp. Gorenstein). In the same way, Z is projectively normal if and only if k[S] is normal, i.e., if the semigroup 5 is normal. Finally the variety Z is normal (resp. regular) if and only if each semigroup S(m°) is normal (resp. free). Notice that to be projectively normal means that Si = 5,, where Si = A~ 1 (z) and Si = C(S) n X ~ 1 ( i ) , i.e. if every element in 5, is a sum of i elements of Si for all i's. Normalness can be characterized in rather similar terms, using the Ehrhart and Hilbert functions. The Ehrhart (resp. Hilbert) function is the map E (resp. H): N —> N given by E(i) = card(Si) (resp. H(i) = card(Si)), which coincides with a polynomial map of degree d—1 with coefficients in Q for i big enough. Then, under the most general conditions, the leading terms of the polynomials for E and H are equal, and the variety Z is normal exactly when both polynomials are equal. Obviously, in those terms, projective normalness is characterized by the property E = H.

7

POLYTOPES, SIMPLICIAL AND CELLULAR COMPLEXES

Once one has an embedded affine or projective toric variety, one looks at describing and computing, when possible, equations and syzygies for the embedding. Most results in this direction are recent and they use combinatorial objects such as simplicial and cellular complexes or polytopes. Note, from section 5, that the projective case is reduced to the affine one, as for a given polarized semigroup, the equations (and syzygies) of the embedded projective variety it defines are the same as the equations (and syzygies) for its projecting cone affine variety. Such an affine variety is nothing but the (affine) toric variety given by the semigroup S of the polarization with Si as chosen system of generators. Along this section, we will assume that a map TTQ : Nh —> S, as in section 1, is fixed, and that 5 is a positive semigroup. Denote by A the generator system of S given by TTQ, by II the set of primitive elements of the N^-module M = N'1 + L, and, for every m £ S, by Tm the set of monomials of 5-degree equal to m. Notice that the set Tm can be identified with the fiber iiQl(m). Recall that the fact that S is positive implies that M is generated by II and that each TTO is finite. Then, there are several combinatorial objects with vertex set one of A, II or Tm which are naturally associated to TTQ as described below. Associated to any fixed element TO in S one has the simplicial complexes A TO , Qm and the polytope Qm defined, respectively as follows. First, Am is the simplicial subcomplex of parts F of A such that m — np € 5", where np = X^eF n- Second, @ m is the simplicial subcomplex of parts G of Tm such that all the monomials of G have a non unit greatest common divisor (i.e. those monomials share at least one variable). Third, fim is the polytope in VR = Zh ®z R- given by the convex hull of the set Tm = 7r^ 1 (rn). Notice that on the set S one can consider an ordering ^ defined by TO' ^ m if and only if TO — m' € 51, and that, if TO' ;< TO then one has A m / C A m , 0m/ times a monomial of degree TO — m! is a subcomplex of Qm and, finally, the translation of fl m / by any vector in the fiber i^Ql(m — m!) is a subset of f2 m . Associated to the whole of 5, one has two useful regular cellular subcomplexes of parts of II. Namely, on one side one has the so-called Taylor complex H which

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is nothing but the (simplicial) complex of all parts of II, and, on the other hand, the hull complex S which is the subcomplex whose faces are the subsets of II which correspond with some unbounded face of the convex hull of the set of points of VR. of type ta = (tai,...,tah) for a = (ai,...,a^) € M where t is any big enough real number. The mentioned correspondence is the obvious one taking into account that any vertex of the above convex hull is necessarily one of type tb with b e II. See [2] for details on the construction and properties of the hull cellular complex. Sometimes, a subcomplex of E, the so-called Scarf complex is considered. It is, in fact, a simplicial complex which is defined to be the set of parts H of II satisfying the property a# ^ a/// for every H' ^ H where, a# stands for the supremum of the elements in H for the ordering < of section 2. The hull and the Scarf complexes coincide when the data TTQ is generic, i.e. when the congruence F can be generated by couples (u, v) such that the unions of the supports of u and v is the set {1, 2 , . . . , h}. In the sequel, we will often use reduced homology with values in the field k for simplicial and cellular complexes. The corresponding z'-th reduced homology vector spaces will be denoted by Hi. The description of equations has to do, in practice, with the determination of sets of binomial generators of the semigroup ideal / (section 3) which are either a minimal set of generators or a Grobner basis. For each monomial ordering (i.e. a total order on the set of monomials for which the monomial 1 is the minimum and which is closed under multiplication by constant monomials) one has a well denned reduced Grobner basis with respect to such an ordering, which also happens to be generated by binomials (see [22] for details). Thus, each such reduced Grobner basis can be understood either as a subset of the congruence F or of the lattice L (sections 1 and 2). The union of reduced Grobner bases for all the possible monomial orderings is called the universal Grobner basis, and it has the property of being, simultaneously, a Grobner basis for all monomial orderings. Again the universal Grobner basis can be seen as a subset of F or L. A reduced Grobner basis with respect to a concrete ordering can be computed from any other generator system by means of the well known Buchberger algorithm. The description of the universal Grobner basis becomes more difficult and it will be stated precisely just below. To find the universal Grobner basis, consider the subset U of S consisting of those elements m G S such that the polytope fi m has an edge which is not parallel to some edge of some fi m / for some TO' -< m. Then, for each m 6 U consider the binomials of type Xu — Xv, where the coordinates of u — v are relatively prime and the segment [u,v] is an edge of £2m. A result by Sturmfels, Weismantel and Ziegler [21] shows that the set of all binomials one obtains in this way when m ranges over U is exactly the universal Grobner basis of /. Such universal basis is finite as one can see that it is contained into the so-called Graver basis which is itself finite. The Graver basis consists of the binomials corresponding to the primitive elements of the lattice L, i.e. those elements 1 = 1+ — 1~ in L for which there are no other 1' = 1'+ - I'" in L such that 1 ^ 1' and 1'+ < 1+ and l'~ < 1~. To find minimal sets of homogeneous generators of / one can proceed as follows. Consider the set C of elements m £ S such that Ho(Qm) ^ 0, i.e. those elements for which the complex Qm is not connected. The set C is finite. For each TO 6 C pick a monomial Xu in each connected component of Qm and distinguish the monomial Xv picked for one concrete component. Then the binomials Xu — Xv, where Xu ranges over the picked monomials for the other components, are the degree m terms

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of a minimal system of homogeneous generators of /. Thus, when TO ranges over the set C, the whole set of obtained binomials Xu — Xv is a minimal set of homogeneous generators for the ideal. A different way to find homogeneous generators for /, which involves the complexes A m , is also available for higher order syzygies, and it will be discussed next in this higher order context. We do not know if such discussion could also be reasonably done in terms of the complexes 0m as well as of the complexes Am or whether Qm could be used to describe the universal Grobner basis. The description of syzygies consists in obtaining either the minimal 5-graded resolution (section 3) or concrete resolutions with other special properties, for example, the property of preserving the symmetries relative to the action of the lattice L. With notations as in section 3, the z-th order syzygy module is the 5-graded module A^ = ker((fi). Notice that one has A^o = /. For each degree TO G S, the number of generators of degree m in any minimal set of generators for Ni is, by Nakayama's lemma, the dimension of the /c-vector space Vi(m) = (Ni}m/(MANi)m. A first and key connection between syzygies and toric geometry is a result due initially to Hochster, [15], and considered again by several authors in [7], [1], [5], which asserts that one has an explicit and natural vector space identification of type

V-(m) = ffi(A m ), where Hi stands for the reduced simplicial homology with coefficients in the field k. Moreover, computations of direct and inverse images by the isomorphisms giving rise to the above identification are available. This result illustrates how combinatorics play a natural role also for describing syzygies, and, therefore, how one has many reasons to include combinatorics among toric mathematics. The first direct applications of the above result are given by Briales, Campillo, Marijuan and Pison in [4] to give an effective algorithm to compute minimal systems of binomial generators of the ideal /. To apply for i > I the above natural isomorphisms, the main difficulties which arise are first to compute those values of m such that ffj(A m ) is nonzero, and, second, to determine the homology. If one is able to avoid these two difficulties in concrete cases, then from the fact that the isomorphisms are explicit, one can derive successive methodic constructions of minimal sets of generators for the syzygy modules in the minimal resolution of R (see [5] for details). To approach the first difficulty, we will mention that, recently, Briales, Pison and Vigneron [6] ( [19] for the case i = 1) determine appropriate finite subsets Cj of S with the property that m $. Ci implies //j(A m ) ^ 0. As a consequence, they obtain an algorithm for computing the minimal resolution, (see [6] for details), because the second difficulty is quite well understood from a computational viewpoint, as concrete homologies can be calculated by means of linear algebra and integer linear programming as pointed out in [18], [19] and [6]. However, integer programming being also a technique to whose development toric geometry also is contributing (as we will show at the end of the paper), it is convenient to try to better understand the explicit structure of the homologies H_(Am). This is treated by Campillo and Gimenez in [8]. For it, one considers a partition A = £ U C, where £ is a subset of generators whose image in Vq generates minimally the cone C(S), in the sense that, for each edge of C(S), £ contains exactly

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one element whose image generates such an edge. Notice, that one has e = card(£), e being nothing but the invariant of S in section 4. Algebraically, one deduces that k[S] becomes a finite extension of k[£]. Thus, the minimal graded resolution of k[S] as an ^4-module can be compared with its minimal resolution as a B-module, where, now, B = fc[N e ] corresponds, as in section 1, to the semigroup generated by the set £. This situation puts in evidence two kinds of objects. First, one has the Apery set relative to £, which is nothing but the set Q of elements q 6 S such that q — n $_ S for every n E £. In other words, the Apery set is nothing but the set of exponents whose corresponding symbols generate minimally k[S] as a fc[£]-module, and, therefore, it is a finite set. Second, for each m € S one has the analog of Am for this relative situation, namely the simplicial subcomplex Tm of parts J of £ such that m — nj e S. Thus, one can see that the dimension of Hi(Tm) is exactly the number of degree m elements in a minimal set of £ -homogeneous generators of the z-th-order syzygy module in the above minimal resolution of the S-module k[S]. Now, for a fixed m € S, one has a key long exact sequence of type

. . . -> Hi+l(Qm) -^Ki^ ffi(A m ) -> Hi(Qm) -+#<_! ^ ... where H.(Qm) and K, are appropriated vector spaces of the following nature. First, H.(Qm) is the homology of a complex associated to the vertex m of a graph QQ with coloured edges constructed from the knowledge of Q, which has C as colour set and which is called the Apery graph. The vertex set of QQ consists of the elements m of type q + nj where q & Q and I C C. Edges of colour n £ C join a vertex

m1 to another m whenever m — m' = n. The complex associated to m has as z'-th chain space the one freely generated by the subsets / C C of cardinality i + I such that TO — n/ E. Q, the boundary map being the projection of the usual simplicial boundary. Second, the spaces K. are much more difficult to describe and we avoid the details. However, they can be computed in successive steps in two different and complementary ways. One, in terms of new graphs of exactly the same type than QQ but with other concrete sets instead of Q. Another in terms of homologies of type H.(Tmi) where the elements m' are of type TO — n/ with / C C. See [8] for the details and some applications. An extra consequence of the construction of the complexes Tm is that one has a way to characterize the depth r of the ring k[S}. Recall that the three integers r, d and e associated to a positive semigroup are such that r < d < e. The integers d and e are easily obtained from 5. To obtain r, in [8] it is proved that if TO is an integer with 1 < r$ < d, then the inequality r > TO is equivalent to the fact He-ro(Tm) = 0 for every m & S. In particular, for ro = d one gets a characterization of the Cohen-Macaulay property by the property

He-d(Tm) = 0 for every m € S, which, for the simplicial case e = d means that all the complexes Tm are connected. From this, it is easy to recover the well known characterization due to Goto [12] with asserts that, for simplicial semigroups, the Cohen-Macaulay property is equivalent to the property

TO € G(S),n, n' e £,n ^ ri ,m + n e S,m + n' 6 S1 => TO e S.

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Other characterizations of the Cohen-Macaulay property for nonsimplicial cases are given in [23] and [20]. A standard application which illustrates the use of the technique of the above long exact sequences is to the case of simplicial Cohen-Macaulay semigroups (i.e. those for which r = d = e). Since, in that case, the complexes Tm are connected, one can deduce that Ki = 0 for every i, so that one gets

for every m and i. Thus, the minimal resolution for simplicial Cohen-Macaulay semigroups can be derived from a unique combinatorial object, the Apery graph. For the general case, there are other ways to derive free resolutions for R from a unique combinatorial object. Namely, as shown in [2] this can be done either from the Taylor or from the hull complexes, H and E respectively. Let us explain how this works. For it, consider for each of the above cellular complexes an associated complex of ^-modules given as follows. The z-th order chains are the elements of the free A-module generated by the i-dimensional faces of the considered cellular complex, and the boundary map is given on any such face

# by

y^e(H,H') —— H' a V "' where the sum ranges over all the faces H' of the considered cellular complex, e(H,H') € {0,1,—!} denotes the incidence index for the cellular complex, and &H •, a/H are the elements defined above. Recall that, from the definition of regular cellular complexes, the incidence index satisfies the properties e(H, H') — 0 unless H' is a facet of H, so the above sum is extended only to facets of H in the cellular complex. Because of the properties of the Taylor and hull complexes, one has that what one actually gets are free A-module resolutions of k[M] = k[Nh + L}. Moreover, the resolution, which is Z^-graded by construction, is in fact invariant by the action of the lattice L induced from its action on II. This means, that each one of the chain of A-modules is in fact also a free .AfLj-module, where A[L] is the algebra of the group L on the coefficient ring A. Notice that one has A[L] = k[Nh x L], so the first projection N ft x L —> N^, seen as an N^-module homomorphism, gives rise to a surjective A-linear map A[L] —> A. Now, by extending scalars via the map A [L] -—> A and taking into account that

k[M] ®A[L] A = k[S] = R, one gets a complex which is in fact S-graded and exact. Thus, according to the considered cellular complex, one gets two S'-graded resolutions of R, which are respectively called the Taylor and the hull resolution.

Both resolutions are, in general, far from being minimal; however, if the data TTQ is generic, then the hull resolution is (isomorphic) to the minimal one. However, they are interesting and useful since they keep the action of the lattice L. Notice that, as commented before, the hull complex is equal to the Scarf complex for the generic case, so, in that situation the Scarf complex can be directly used instead of the hull one for constructing the above resolution, which is minimal besides. Nevertheless, we remark that the generic case is combinatorially characterized by the fact that

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the simplicial complexes Am which are not connected have connected components which are full simplices. This is a strong assumption from the combinatorial point of view, so, in general, if one wants to know about the minimal resolution the only available description (by the moment) is that discussed before, based on the study of the simplicial complexes A m . In the general (non generic) situation the hull resolution has, nevertheless, another nice property, as in fact it is a finite one, i.e. the free ^-modules involved are or finite rank and the number of them is finite. This is a non obvious statement which follows from the fact that the Graver basis is finite. See [2] for details.

8

MULTINUMERICAL SEMIGROUPS

In practice, the toric data TTQ (of a semigroup with a given generator set) is often given in arithmetical terms. In fact, the group G(S) being finitely generated, it is nothing but, up to isomorphism, one of type

Zd x Z/giZ x ... x Z/giZ for convenient integers d,l,gi,.. • ,giThus, if such an isomorphism is, a priori, considered, then TTQ becomes equivalent to the specification of the coordinate (d + /)-tuples (in the above product group) of the generators HI ,..., n^ of S. A semigroup given by such a specification is called a multinumerical semigroup. For the simplest case d = 1 and / = 0, they are usually referred to as numerical semigroups in the literature. What one would need, therefore, is to study toric varieties within arithmetics from multinumerical semigroups. This means to deduce the behaviour and geometrical properties of those varieties from arithmetic properties of the (d + Z)-tuples of integers or modular integers given by the semigroup generators. Such an arithmetical study becomes, nevertheless, difficult and it is an open problem except in rather few cases. The difficulties arising can be explained if one looks at the discussion in the above section on how combinatorics are involved in the development of toric geometry. In fact, using objects such as polytopes or simplicial or cellular complexes avoids having to deal with delicate relations among numbers. However, mathematically speaking, once that combinatorial methods have grown up and produced nice results, one can hope and try to interpret them in the framework of arithmetics. This strategy is used in [8] for affine and projective toric curves and in [5] for affine and simplicial projective toric surfaces. For the general case, good computational results dealing with equations are also derived by Vigneron in [18]. To show the possibilities of the above strategy, we will discuss, here, such results for curves. An affine toric curve is given by the numerical semigroup S given by a set A of h nonnegative integers. One has r = d = 1 and, since the cone C(S) has only one edge, also e = 1. Thus, this case is a simplicial Cohen-Macaulay one. Then, pick a partition of A in a set £ consisting of any single element s € A and as complementary set C the set of the h — 1 remaining elements. Now, consider the Apery set Q consisting of those integers q & S such that q — s £ 5, and from it construct the coloured graph QQ. It is not difficult to translate the graph structure into arithmetical relations, so that the homologies //j(A m ) = Hi(Qm) for

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the vertices m of QQ can be derived from such relations. One concludes that the minimal resolution for affine toric varieties can be obtained in complete arithmetical terms from the set of generators of the given numerical semigroup. See [8] for details. A projective toric curve of degree s is given by a subsemigroup of N2 generated by a set A = £ U C, where £ is the set consisting of the two elements (s, 0) and (0, s) and C consists of elements (GI, s — c\),..., (c^_2, s — 0^-2) for different values Cj with 0 < Cj < s. The semigroup S can be polarized by the function A given by \(c, c') — (c + c')/s, so that 5 defines an embedding of the projective toric curve in p/i-i_ Notice that one has d = e = 2 and that either r = 2 or r = 1, depending on the projective curve to be or not to be arithmetically Cohen-Macaulay. Let Si be the numerical semigroup generated by GI, . . . ,Q l _2,s, and for each c 6 Si denote by /z(c) the least number of the above generators of Si needed to achieve the sum c. Notice that the function fj, satisfies the property /j,(c) < /z(c—s) + l for every c e S whenever c — s S S. By translating into arithmetics the methods in [8], in [9] it is shown that the projective toric curve is arithmetically CohenMacaulay if and only if one has /i(c) = n(c — s) + 1 for every s e Si such that c - s e S. In general, from the knowledge of the function fj, it is easy to find the Apery set Q relative to the above partition A = £ U C as well as the set D consisting of those elements m in S such that m — (s, 0) € S, m — (0, s) e S, m — (s, s) ^ S. One can consider a coloured graph Q-p in an identical way as QQ but replacing Q by D. In [8] it is shown that the vector space Ki in the long exact sequence of the previous section can be identified with the homology Hi(Dm], where this last homology has also an identical construction than that for the case of the set Q. Thus, one deduces the long exact sequence

. . . -> Hi+1(Qm) -> Hi(Dm) -> tf z (A m ) -> Hi(Qm} - > . . . . The involved homologies as well as the image maps in this exact sequence can be given in aritmetical terms from the given data s, GI, ..., c^-i- From here, this is so for the reduced homologies H. (A m ), therefore, the minimal resolution of the projective toric curve is obtained from arithmetics.

9

APPLICATIONS

The development of toric geometry has provided applications to many problems in geometry. This is related to the fact that, quite often, toric varieties are objects on which one can determine and describe the main ingredients involved in the considered problems. Applications also occur to some problems external to geometry and algebra, in such a way that, toric geometry is becoming also an interesting topic of applied mathematics. Those external applications are mainly related to applied combinatorics or to applied optimization. We will end this paper by illustrating this situation with two examples of current research. The first one is the coin exchange problem, a classical problem of applied combinatorics. The approach and results are recently obtained by Campillo and Revilla in the paper [9]. Assume one has a coin system with coins of values c\ < c-2 < ... < Ch-i- Then, setting s = c^-i one has a projective toric curve Z, namely that of degree s given by the (polarized) subsemigroup of N 2 generated by the elements (0, s), (d, s - c i ) , . . . , (ch-i, s - ch-i) = (s,0).

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The exchange problem aims at achieving a value c in the semigroup S\ generated by the coin values in an appropriate way. One wants, in particular, to achieve the value c with the minimum number of coins, namely the integer p,(c) introduced in the above section. The problem can, therefore, be formulated as to give good ways or algorithms to achieve the value c with /z(c) coins in practice. Comments in the above section show how this problem is mathematically close to that of the determination of equations and syzygies for projective toric curves. Usually considered coin systems have a strong property, namely that for them the greedy algorithm to achieve the values c with //(c) coins works. The greedy algorithm achieves a value c e Si by first taking the largest coin Cj such that GJ < c and, then, restart with the value c — Cj and continue in the same way. If, for every c £ Si, the greedy algorithm uses fj,(c) coins then one says that it works for the system. From the discussion at the end of the last section, one deduces that if the greedy algorithm works then Z should be arithmetically Cohen-Macaulay. From this, one shows how toric geometry yields an interesting new class of coin systems with nice properties, namely the Cohen-Macaulay ones, i.e. those such that the associated projective toric curve Z is arithmetically Cohen-Macaulay. For them, in general, the greedy algorithm to achieve values with a minimal number of coins is not available, but one has an alternative new and good algorithm to do so (see [9] for details). The second application is to integer linear programming, also a classical problem, this time of applied optimization. Integer linear programming is related to multinumerical subsemigroups of Zd, which, for the sake of simplicity, will be assumed to be positive. Let S be such a subsemigroup and assume that it is generated by the elements HI, . . . , n^ 6 Zd. The integer linear programming problem consists in finding the optimal solution with non negative integral coordinates to one of type

which minimizes a linear map (the cost map)

Here, the coefficients p^ are real numbers in general. An integer linear program can be seen, therefore, as the specification of type (TTQ, p) where TTQ is the data of a semigroup and generators as above and p the cost function. For each m & S the solutions of the integer linear programming problem for ?n are among the elements in the fiber TTQl(m) and, moreover, among the vertices of the polytope O m . Now, notice that, once one fixes any monomial ordering on the variables xi, . . ,, Xh (for instance the reverse lexicographic ordering), the cost function gives rise to another monomial ordering by comparing two monomials, first, by the value of p on the exponents and, second, in case of equal values of p by the previous fixed ordering (i.e. the weighted ordering corresponding to the above one). Then, one can prove that the reduced Grobner basis of the ideal / given by TTQ relative to this new ordering provides a minimal test set for the integer programming as described in the sequel.

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In fact, the reduced Grobner basis is generated by binomials, therefore, it can be viewed as a subset Up of the lattice L. On the other hand, one has the property that if x = (xi,..., Xh) € Nh is in a fiber and 1 <E L (i.e. a feasible solution) then x — 1 is again a feasible solution whenever x — 1 g N fe . Then, the set U p is a test set for the program as it satisfies the following two conditions. First, if x is a feasible solution which is not optimal, then there exists 1 € U p such that x — 1 is also a feasible solution. Second, if x is an optimal solution to a program, then x — 1 is not a feasible solution for every 1 G U p . The condition on the Grobner basis to be reduced implies that U p is minimal among the subsets satisfying the above two conditions. Test sets provide nice algorithms, in the obvious way suggested by both conditions, to solve the integer linear programming problem. Non reduced Grobner bases provide non minimal test sets. In particular, the set U giving the universal Grobner basis in section 6, which is finite and the union of all Up for all cost functions, is a test set for all programs when p varies, i.e. it is a data which only depends on TTQ. See [16] and [17] for details. For algorithms involving cases of non positive semigroups see [3], or consider the Lawrence lifting (see for example [22]).

REFERENCES [1] A. ARAMOVA, J. HERZOG, Free resolution and Koszul homology. J. of Pure and Applied Algebra, 105 (1995), 1-16. [2] D. BAYER, B. STURMFELS, Cellular resolutions of monomial modules. J.reine angew. Math. (1998) 502, 123-140. [3] F. DI BIASE, R. URB ANKE, An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms, Experimental Mathematics, Vol.4, No. 3 (1995), 227-234.

[4] E. BRIALES, A. CAMPILLO, C. MARIJUAN, P. PISON, Minimal Systems of Generators for Ideals of Semigroups, J. of Pure and Applied Algebra, 124 (1998), 7-30. [5] E. BRIALES, A. CAMPILLO, C. MARIJUAN, P. PISON, Combinatorics of syzygies for semigroup algebras. Collet. Math. 49 (1998), 239-256. [6] E. BRIALES, P. PISON, A.VIGNERON, The regularity of a Toric Variety Preprint, University of Seville (1999). [7] A. CAMPILLO, C. MARIJUAN, Higher relations for a numerical semigroup. Sem. Theor. Nombres Bordeaux 3 (1991), 249-260. [8] A. CAMPILLO, P. GIMENEZ, Syzygies of affine toric varieties Journal of Algebra to appear. [9] A. CAMPILLO, M. REVILLA Coin exchange algorithms and toric projective curves. Preprint (1999). [10] M.P. CAVALIERE and G. NIESI, On monomial curves and Cohen-Macaulay type Manuscripta Math. 42, (1983), 147-159.

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[11] W. FULTON, Introduction to Toric Varieties, Princeton University Press (1993). [12] S. GOTO, N. SUZUKI, K. WATANABE, On affine semigroup rings, Japan.

J. Math. 2(1),(1976), 1-12. [13] J. HERZOG, Generators and relations of semigroups and semigroup rings, Manuscripta Math. 3, (1970), 175-193. [14] M. HOCHSTER, Rings of invariant of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Annals of Math. 96(2), (1972), 318-337.

[15] M. HOCHSTER, Cohen-Macaulay rings, combinatorics and simplicial complexes Lect. Notes in Pure and Appl. Math. Dekker 26(1977), 171-223. [16] S. HOSTEN, B. STURMFELS, GRIN, An Implementation of Grobner Bases for Integer Programming, In E. Balas and J. Clausen editors, Integer Programming and Combinatorial Optimization, LNCS 920, Springer-Verlag, (1995), 267-276. [17] S. HOSTEN, R. THOMAS Grobner basis and integer programming B. Buchberger and F. Winkler (editors) Grobner Bases and Applications, Lect. Notes Series 251, London Math. Soc., (1998), 144-158.

[18] P. PISON-CASARES, A. VIGNERON-TENORIO, N-solutions to linear systems over Z. Preprint of University of Sevilla (1998).

[19] P. PISON-CASARES, A. VIGNERON-TENORIO First Syzygies of Toric Varieties and Diofantine Equations in Congruence To appear in Comm. in Algebra. [20] U. SCHAFER, P. SCHENZEL Dualizing complexes of affine semigroup rings, Trans. A.M.S. 322(2),(1990), 561-582.

[21] B. STURMFELS, R. WEISMANTEL, G. ZIEGLER, Grobner basis of lattices, corner polyhedra and integer programming. Beitrage zur Algebra und Geometrie36(1995), 281-298. [22] B. STURMFELS, Grobner Bases and Convex Polytopes, Lectures Series, Vol. 8 (1995).

AMS University

[23] N.V. TRUNG, L.T. HOA, Affine semigroups and Cohen-Macaulay rings generated by monomials. Trans. A.M.S. 298(1), (1986), 145-167.

[24] A. VIGNERON-TENORIO, Semigroup Ideals and Linear Diophantine Equations. Linear Algebra and its Applications, 295 (1999), 133-144 •

Canonical Forms for Linear Dynamical Systems over Commutative Rings: The Local Case M. CARRIEGOS, Departamento de Matematicas, Universidad de Leon. 24071Leon. Spain. E-mail: demmcv@unileon. es

T. SANCHEZ-GIRALDA, Departamento de Algebra, Geometrfa y Topologia, Universidad de Valladolid. 47005-Valladolid. Spain. 1 E-mail:sanchezgiral@cpd. uva. es

Abstract In this paper we survey some results on the Brunovsky canonical form for the feedback equivalence of reachable linear dynamical systems over commutative rings. Besides the general definitions and well known facts we include some very recent results: A normal form for reachable linear dynamical systems over a local ring R is given. This normal form is a canonical form in the case of 2-dimensional systems. Finally, the case of discrete valuation domain R is specially treated: We obtain a complete set of invariants and a canonical form for a reachable n -dimensional linear dynamical system E with the property that the invariant Rmodules M-f, Mj% ..., M^ are free except at most one of them.

1

INTRODUCTION

The importance of the feedback action in Control Theory is well known: The modification of a dynamical system to achieve some desired behavior, since for example its stabilization, is very important by its applications. On the other hand, several authors have worked in the last years to extend linear control results to systems defined over a commutative ring R. Morse's paper [17] is one of the main initial works on the control of systems over rings and contains a constructive proof that reachability implies pole assignability for multi-input systems defined over a polynomial ring in a single indeterminate with coefficients in 113

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a field. Actually, the theory of systems over rings is well developed and of interest as an active and rich area of research. The influence in this area of R.E. Kalman from 1970 has been essential. We would like to call the reader's attention to [20] as a general reference in control theory and [3] and [5] as references in linear control theory from an algebraic point of view. When R is a field, the classification of reachable linear dynamical systems for the feedback action is due to P. A. Brunovsky [4], R.E. Kalman [14], and W.A. Wonham and A.S. Morse [23]. In this case, the feedback equivalence class of a reachable linear dynamical system E = (A, B) is characterized by a finite set of positive integers k\, ...,ks which are called "the Kronecker indices of E = (A,B)", because they are identical to the classical indices associated with the pencil of matrices (zldn — A B) In the general case the feedback classification problem is still open. Moreover the feedback classification problem is called "wild problem". Nevertheless, there are several results which determine canonical or pseudo-canonical forms for the feedback action of linear dynamical systems over special rings. This paper is organized as follows: In section 2 we review the feedback group, the feedback equivalence of linear dynamical systems over a commutative ring R, and the feedback invariant R- modules Np and Mf associated to E, where i is a positive integer lower than the dimension of the system E. Section 3 is devoted to study well known facts about feedback equivalence for systems over a field, such as the Brunovsky canonical form. For a commutative ring R, we show that every reachable linear dynamical system over R is equivalent to a Brunovsky canonical form if and only if .R is a field. Consequently, Brunovsky's classification is a complete classification of reachable linear dynamical systems over R if and only if .R is a field. In section 4, we review the notion of change of scalars in a linear dynamical system and we introduce some new results: A normal form is given for reachable linear dynamical systems over a local ring. This normal form becomes a canonical form in the case of m-input 2-dimensional reachable linear dynamical systems over a local domain, hence a feedback classification for those systems is given. The case of m-input, rx-dimensional reachable linear dynamical systems over a discrete valuation domain is also studied, and we obtain a canonical form for reachable systems E with the property that all invariant .R-rnodules Mp are free except at most one of them.

2

LINEAR DYNAMICAL SYSTEMS OVER COMMUTATIVE RINGS: THE FEEDBACK GROUP

Let R be a commutative ring with identity element. An m-input n-dimensional linear dynamical system E over R is a pair of matrices (A, B), where A is an n x n matrix and B is an n x m matrix with elements on R. The system E = (A, B) is said to be reachable if the columns of the n x mn matrix

A*B= ( which is called the reachability matrix of the system E, generate Rn.

Let X be a p x q matrix with entries in R. We denote by Uj(X)

the j-tii

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determinantal ideal of X, that is, the ideal generated by all j x j-minors of the matrix X. Note that the rn-input, n-dimensional system E = (A, B) is reachable if and only \iUn (A*B) equals R (see [3, Theorem 2.3.] for details). The feedback group Yn,m is the group, acting on m-input, n -dimensional linear dynamical systems E = (A, B) generated by the following three types of transformations: 1. A i—> A' = PAP~l ; B i—> B' = PB for some invertible matrix P. This transformation is consequence of a change of base in Rn, which is called the state module. 2. A i—> A' = A ; B i—> B' = BQ for some invertible matrix Q. This transformation is consequence of a change of base in Rm, which is called the input module. 3. A i—> A1 = A + BF ; B i—> B' = B for some mxn matrix F, which is called a feedback matrix. The system E' is feedback equivalent to E if it is obtained from E by an element of F n>TO .

Let E = (A, B) be a linear dynamical system of size (m, n) (i.e. m -input n-dimensional) over R. For i — l,...n we denote by Np the submodule of Rn generated by the columns of the (n x i- m)-matrix (jE?|AB| • • • \Al~lB^. We denote also by (S|>1S| • • • \Al~lB] the homomorphism of R -modules

(B\AB\ • • • |Ai-1B) : R*'"1 -> Rn denned by the matrix (B\AB\ • • • lA^B). We denote by Mp, for each i = 1, ...n, the quotient .R-module Mp — Rn/Np. Then note that by the Cayley-Hamilton theorem, the jR-modules Mp and Mp are equal for each i = n + l,n + 2,... Recall that the linear dynamical system E is reachable if and only if Np = Rn. Then note that E is reachable if and only if Mp = 0 for each i = n, n + 1,... LEMMA 2.1 Let E = (A,B) and E' = (A',B') be two feedback equivalent linear dynamical systems of size (m, n) over a commutative ring R. Then we have the following properties. 1. Np is isomorphic to Np for each 1 < i < n. 2. Mp is isomorphic to Mp for each 1 < i < n.

PROOF.- (See [12, Lemma 2.1.]) D

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REMARK 2.2 Note that by the previous Lemma one has that the sets:

and are two sets of invariants associated to the linear dynamical system £ by the action of the feedback group. The following example shows that the sets of invariants {./V,p}. ,. , and {Mp\ . ,. , are not sufficient to state the class of £ by the action l J *• ' l
Let R = TL be the ring of integers and set A = Put A' =

0

0

0 0

and B =

1 0 0 5

and B' = B and consider the linear dynamical systems £ =

We have:

N? = (B\AB) = Im

0

5

3

0

1 0 0 0 0 5 4 0

N? = (B\A'B) = Im

By [13, Theorem 2.4], the system E is not feedback equivalent to E' because the following congruence 3 = 4-u/i2 (mod5)

has no solution lih^TL and u is a unit in TL.

3

CANONICAL FORM FOR SYSTEMS OVER FIELDS

First, we recall a classical result due to P.A. Brunovsky [4].

THEOREM 3.1 Let E be an m-input, n-dimensional reachable linear dynamical system over a field k. Then there exists a finite set of positive integers k\ > k% > ... > ks > 0 uniquely determined by E with X/i=i k^ = n such that E is feedback equivalent to Ec = (Ac, Bc) where Ac and Bc are described below: 0

/

0 \ 0

0

Ac =

Es where Ei is the k; x L- matrix

( ° 1

0

0 0

1

0

V°

0 0

••• •••

... o \ '•. o '•- 1 0

0/

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and / e, 0

62

• • • •

\ 0

0

• • • es

Br =

0

0 0

0

• °0 \

0

•

0

• •

• • •0 /

where &i is the ki x 1 matrix ei=

( 0

...

0

1 )*

The integers {fc;}*=1 are called the Kronecker indices o/E, and £c = (AC,BC) described above is called the Brunovsky canonical form associated to the Kronecker indices {ki}*=l.

PROOF .- See [4], [9], [12] or [20]. D

REMARK 3.2 The set of positive integers {ki}1
or equivalently the set of decreasing non-negative integers

are two complete sets of invariants associated to E for the feedback equivalence.

PROPOSITION 3.4 Let k be afield and E = (A,B) a reachable linear dynamical system of size (m,n) over R. We put of — dimk(Mp) for 1 < i < n. Then the set of decreasing non-negative integers {crf}l
/

0 \

0

0

V o

E*es

0

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Note that E^Ci = 0 for h > ki. Moreover, the intersection of the vector subspaces lm(AhB) and lm(Ah' B) of kn is {0} when h ^ h'. If the sequence of Kronecker indices associated to E is {ki, ...,ks} where KS2 = • • • < KSt = • • • =

then the sequence {dim//?} is

s 2s 3s kss kss + si kss + s2 KSS -j~ (fe S

kss + • • • + (kst_l - kst_2) s t _i + st kss + • • • + (kat_1 - fc s t _ 2 ) st-i + (kst - fc^J st (The reader can see [8, Proposition 2.5.] for details of the calculation). Now, we conclude that we can obtain the Kronecker indices {ki}1
{erf }l
ar

e equivalent data to the set of Kronecker indices of the system S,

which completes the proof. D

REMARK 3.5 Suppose that R is an arbitrary ring and S = (A, B) is an m-input

n-dimensional reachable linear dynamical system over R. In general, E is not feedback equivalent to a Brunovsky canonical form Ec = (AC,BC). For example, in [13] it is shown that if R is a principal ideal domain and E is an TO -input, 2-dimensional reachable linear dynamical system over R, then E is feedback equivalent to a system E = (A, §} of the form 7 A

/ 0 0 \ ss

=(f

/ 1 0 •••

0

o j ' = ( o d ... o

where
R). REMARK 3.6 The above classification of reachable m-input 2-dimensional linear dynamical systems over a principal ideal domain R is generalized in [10] to a

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clasification of m-input 2-dimensional reachable linear dynamical systems over the quotient ring R/ (pn) where R a principal ideal domain, p a nonzero prime of R and n a positive integer.

DEFINITION 3.7 Let S = (A,B) be an m-input n-dimensional reachable linear dynamical system over R. We say that S is a Brunovsky system if S is feedback equivalent to a Brunovsky canonical form in the sense of Theorem 3.1. The following result characterizes the Brunovsky systems over a ring R which satisfies that all finitely generated projective modules over R are free.

THEOREM 3.8 Let R be a commutative ring. Let S = (A,B) be a m-input ndimensional reachable linear dynamical system over R. IfEisa Brunovsky system, then the R-module Mp is free for all i — 1,2, ...n. Moreover, if every finitely generated projective R-modules is free, then the system S is a Brunovsky system if and only if Mp is a free R-module for all i = 1,2, ...n

PROOF.- (See [12, Theorem 3.1.]) D It is natural to define a Brunovsky ring as a commutative ring such that every reachable linear dynamical system is equivalent to a Brunovsky canonical form. Note that for a Brunovsky ring the above theorem characterizes the feedback orbit of any reachable linear dynamical system. The following result proves that the class of Brunovsky rings equals the class of fields.

THEOREM 3.9 Let R be a commutative ring. The following are equivalent:

1. R is a Brunovsky ring. 2. R is a field.

PROOF.- Note that (2) => (1) is Brunovsky's Theorem (Theorem 3.1). Conversely assume that R is a Brunovsky ring. First, we claim that R is an absolutely flat ring (see [1] or [2]). Let a be a finitely generated ideal of R. a = (ai,...,a m ). Consider the (m + l)-input 2-dimensional linear dynamical system £( a ) given by 0

0 \

/ 1 0

0

...

0

Note that the reachability matrix of

A*B =

1 0 0

ai

0

...

0

0

0

0

...

0

a-2

...

ar,

1

0 0

...

0

verifies that the determinantal ideal U^ (A*B) is the whole ring R. Consequently S( a ) is reachable and therefore a Brunovsky system. Then, by the above Theorem 3.8, the .R-module Mf'"' = R2/lm(B) is free.

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Consider the finite free resolution of the ^-module M-p Rm+l

^R2^

M E<->

_

0

Since Mjp ° is free, it follows by [6] that the determinantal ideals associated to the matrix B are principal ideals generated by an idempotent element of R. In particular, the ideal Ui (B) = (a 1 ,a 2 ,...,a m ) = a, is a principal ideal generated by an idempotent and we have proven that every finitely generated ideal of R is principal generated by an idempotent. This is equivalent to R being an absolutely flat ring (see [1, ex. 27, p. 39]) Now let us prove by contradiction that R is a local ring. In fact, suppose that there exist two maximal m and m' ideals of R such that m ^ m'. Let x € m and x £ m' and consider the linear dynamical system over R given by 0 J'\ 0 x

The system £^ x ) is reachable, so £( x ) is feedback equivalent to a 2-input 2-dimensional Brunovsky canonical form. But note that the following two systems are the only 2-input 2-dimensional Brunovsky canonical forms: The form £{1,1} associated to the sequence of Kronecker indices "-1 = "'I ~ 1) "'S = ^4

=

' ' ' — 0,

and given by 0 0

0\ / 1 0) ' \ 0

And the form £{2} associated to the sequence of Kronecker indices

k± = 2, A;2 = &3 = • • • = 0, and given by 0 0

1\ / 0 0 0 J '\ 1 0

An easy calculation shows that the matrix

\

1 0 0 x

x

1 0 0 1

is not equivalent to the matrix

neither to the matrix 0 0 1 0 J '

Then the system £^ x ^ is neither feedback equivalent to the system £{1,1} nor to the system £{2}- This is a contradiction and the ring R is local. Collect the properties of R; that is, R is local and absolutely flat. This is equivalent to R being a field (see [1, ex. 28, p. 39]) D

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4

121

DEALING WITH THE LOCAL CASE

In this section we study some invariants for linear dynamical systems over a local ring. First note that if f : R —* R' is a ring homomorphism and £ = (A, B) is an minput n-dimensional linear dynamical system over R, where A — (<%) and B = (6^); then one can construct the linear dynamical system /* (E) = ((/(«ij)) , ( / ( & f c z ) ) ) over R' . The system /* (E) is called the extension of £ by change of scalars from R to R' via /. Next we review some properties of extension of scalars in linear dynamical systems:

THEOREM 4.1 Let f : R —> R' be a ring homomorphism and let £ and £' be two linear dynamical systems over R. Then, we have the following properties. 1. For each i = 1,2, ...n, there is a natural isomorphism

2. If £ and £' are feedback equivalents, then the systems /* (£) and /* (£') are feedback equivalent.

PROOF.- See [8, Lemma 2.1] D Now, let p be a prime ideal of R and consider the natural extension of scalars from R to Rf given by ip : R -> #p r i—> r/1 '

and the natural extension of scalars from R to R$/pRv given by TTP : R

->

In the sequel, we use the notation E p = i p (E) and £(p) = TT* (E). The next definition follows naturally from the extension of scalars in a linear dynamical system (see [8]).

DEFINITION 4.2 Let R be a commutative ring, E and E' two m-input n-dimensional reachable linear dynamical systems over R. We say that E and E' are locally (resp. pointwise) feedback equivalent if and only if £ p and E p (resp. E(p) and E'(p),) are feedback equivalent for each prime ideal p of R.

The following is straightforward from Theorem 4.1: E and E' are feedback equivalent ^ E and E' are locally feedback equivalent U E and E' are pointwise feedback equivalent At this point, two natural questions arise:

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• When do the reverse implications hold? • Study of feedback equivalence, local feedback equivalence and pointwise feedback equivalence. The pointwise feedback relation for reachable linear dynamical systems is treated in [8]. Now we introduce some original results in order to characterize the feedback equivalence of reachable linear dynamical systems over a local ring and hence to study the local feedback equivalence. The following proposition provides a normal form for reachable linear dynamical systems over a local ring.

PROPOSITION 4.3 Let (R,m) be a local ring and let S = (A,B) be a reachable m-input, n-dimensional linear dynamical system over R. Then, S is feedback equivalent to the system E^ = (Ah,Bh) where

Ah =

0 R (2) B

h

0 (2)

A A

h

0

,Bh =

xl

and in general, for each j = 2,..., s — 1

A? =

0

0

and finally 0

where all the entries of the matrices X\,..., -X"s_i are in the maximal ideal m of R. Moreover one has the following identities relating the invariant R-modules

[Mf]

Ki
to the positive integers {Ci}i • Ifi<s, then one has:

d:mR/m (Mp ®R R/m) =n-

• On the other hand, if s < i < n, then Mp = (0) . PROOF.- Let us prove the result by induction on n, the dimension of the system S. The case n = 1 is straightforward because every reachable m-input 1-dimensional linear dynamical system is feedback equivalent to the system

Suppose that the result holds for 1, ...,n — 1, and let us prove the case n. Consider the reachable m-input n-dimensional linear dynamical system £ = (A, B). Since £ is reachable, it follows that Un(A*B) = Un(B\AB\...\An~lB} = R,

Linear Dynamical Systems over Commutative Rings

123

consequently Ui(B) = R, and hence £1 = max{7 : U^(B) = R} > 1. Then by [18, Theorem 12] the matrix B can be transformed into B' 0

B' - PBQ - (' Mb 0

Y

by means of elementary row and column operations. Moreover, from the maximality of £1, we have that each entry of Y belongs to the maximal ideal m. Consequently the system E = (A, B) is feedback equivalent to the system

Idf

T2

0

Y Therefore S is equivalent to

£' = (A', B') = PAP'1 + PBQ 0

,PBQ}=

Idf

0

0

5(2)

Y

Note that the £1 -input, (n — £1 )-dimensional linear dynamical system

is reachable. Consequently, by the induction hypothesis, there exist invertible matrices P( 2 ),Q( 2 ) and a feedback matrix F^ such that p(2 ) j B (2) F (2)

=

and

Finally consider the invertible block matrices given by p/= ' p(2)

0

p(2)p(2)y

Id and a

F'= where

and

-i

F (2)p(2) B (2)g(2)

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Then, we have the equalities

P'B'F' = Ah and

P'B'Q' = Bh =

Idf

0

0

for some matrix X\ with all its entries in the maximal ideal of R. Consequently the system E is feedback equivalent to E/j. This completes the proof D

REMARK 4.4 Let k be a field and let E = (A, B) be an m-input, n-dimensional linear dynamical system over k. The above Proposition provides in this case a construction of a canonical form for the system E; this canonical form is not the Brunovsky canonical form (See [7, Chapter 1]). REMARK 4.5 Let E = (A, B) be a reachable linear dynamical system over a local ring R. When E = E^, we say that E is in normal form. Let m be the maximal ideal of R and consider the linear dynamical system E(m) obtained by natural change of scalars from R to R/m. Then the matrices B,B^2\ ... are in Hermite normal form and the reachability indices are (m)

m)

= dimR/m

In fact, the indices ai

= dim fi/m (Mp ® fl R/m) .

can easily be calculated from the equalities: E(m) _

_ ,f

f

,. s

COROLLARY 4.6 Let (R,m) be a local domain, let E = (A, B) be a m-input, ^-dimensional reachable linear dynamical system over R. Then two reachable linear dynamical systems E and E' are feedback equivalent if and only if the R-modules Mp and Mp are isomorphic.

PROOF.- By Proposition 4.3, we may suppose that the systems E and E' are in normal form, so consider the linear dynamical systems

B=

= \A =

1

0

and

E' = I A' =

B' —

0

where x,x' & m. First, suppose that the linear dynamical systems E and E' are feedback equivalent, then the matrices

B= and

B' =

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125

are equivalent. Hence the -R-modules Mf = R2/lm(B) and Mf' = R/Im(B') are isomorphic. Conversely suppose that the /Z-modules M-p and Mf are isomorphic. There are two cases: • Case 1: Suppose that x = x' = 0. It follows that E = £' and we are done. • Case 2: Suppose that x ^ 0 and x' ^ 0. Since R is a domain, it follows that x and x' are not zero-divisors. Consequently the linear maps B : R2 —> R2 and B' : R2 —» R2 determined by the matrices B and B' respectively, are injective. Therefore, by [16, Theorem II.13.(b)], there exists an invertible 2 x 2 matrix P such that PB = B1. Consider the matrix

p=(, p=

Pn

Pi2

P21

Pi'2

then we have the following equalities

Pn Pi2x

= 1 = 0

P21

=

0

p22X

=

x'.

Hence E and E' are feedback equivalent via the feedback action (P,Q,F) given by

P=(l

V 0

°

p22

Q = M>x2 ^=(0)2X2.

This completes the proof. D

The next result characterizes the feedback equivalence of some reachable linear dynamical systems over a discrete valuation ring. We use the following characterization of discrete valuation rings [2, VI.3.6.]: The local domain R is a discrete valuation ring if and only if R is a principal ideal ring (every ideal of R is a principal ideal). First we need a Lemma. LEMMA 4.7 Let (R,m) be a local ring and let S = S/j = (A,B} be an m-input, n-dimensional linear dynamical system in normal form as in Proposition 4-3. Consider a positive integer i with 1 < i < s, then if the invariant R-module Mp is free,

the matrix Xi is (0). PROOF.- Consider the following finite free resolution of the /J-module

Rl'm

i 1 V(B\AB\-\A - B) ;

-1

Rn

->

Mp

-* 0 •

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Since Mp is a free finitely generated .R-module, it follows that the determinantal ideals associated to the matrix [B\AB\---\Al~lB} are trivial ideals, that is Uj (B\AB\ • • • l^-1^) is either (0) or the whole ring R. (See [18]). On the other hand, a calculation on the determinantal ideals of

(B\AB\---\Ai~'LB') shows that:

• If j < £1 + •

then Uj (B\AB\ • • • \Al-lB] = R.

• If j > £1 + •

then Uj (B\AB\ • • • lA^B) C m.

Moreover one has that

Hi (X^ C %+...+?i+1 (B\AB\ • • • \Al~lB] , therefore U\ (Xi) — (0) and consequently Xi = (0). This completes the proof D

THEOREM 4.8 Let (R,m) be a discrete valuation ring, let E = (A,B) be an minput, n-dimensional reachable linear dynamical system over R. Suppose that every feedback invariant R-module Mp is torsion free except, perhaps, Mp.

Then, the

ordered set of R-modules {Mp,..., Mp} forms a complete set of invariants (up to isomorphism) associated to E for the feedback equivalence.

PROOF.- By the Proposition 4.3, we can suppose that the system E is in normal form, that is, where

A=

5(2)

•y

in general, for each j = 2, ..., s — 1 0

,B<» =

and finally 0

where the matrices Xi,...,Xs-i have all their entries in the maximal ideal m of R. First note that since Mp is torsion-free for all i > 2, it follows by Lemma 4.7 that Xi = (0) for all i > 2. Let T~im and H.n be the basis j" ;

and

127

Linear Dynamical Systems over Commutative Rings

of Rm and Rn respectively for the normal form E = E/i . Now, we construct new bases Jim and "Hn of Rm and Rn respectively such that the system S = E^ related to these new bases is 0

0

0

A?

.A. i

with X, = (0) for alii > 2 and

o \

( o

\ where d\\d^ • • • \dj are the elementary divisors associated to the quotient R-module

Mp. Consider the finite free resolution of the .R-module Mp (B\AB\--Ai~lB)

Rn

Mp

0

Since Mp = Rn/lm (B\AB\ • • • Ai~1B) is torsion-free for i > 2, it follows by Lemma 4.7 that Xi = (0) for all i > 2, and that Mp is generated by

Moreover, we claim that the /?-module tor(Mf') is in fact generated by

Consider the homomorphism of .R-modules

Mp

x + Np

-» H-» A(x) + Np '

if x + Np e tor(M E ), then A(x) + Np 6 tor(M2E) = 0. Then we have that

ABei + Np i tor(Mf) for each i = !,...,&. (Note that A(ABei) + Np = A2Bet + Np is an element of the basis Un and hence A(ABei) + Np ^ 0). Therefore the .R-module tor(Mf) is generated by {ABei:3+1+N?,...,ABei:2 + Np} . From the above facts, we have that 0

B (2)

Idf

0

xl

128

Carriegos and Sanchez-Giralda

where

and Xi — (0) for each i > 2. Define the free .R-modules

and note that FI is a free direct summand of Rn and F? is a free direct summand of Rm (because both are generated by a subset of a basis of Rn and Rm respectively) . Consider the .R-module homomorphism given by the matrix X* :

F2

X

-l

F1 •

Since R is a principal ideal domain, it follows that there exists a change of basis Q* and P* and a new basis {j^+i, ...,r/ m } of F2 and {ABe£3+1, ..., ABe^} of Fj which diagonalize Jf*, that is, the following diagram (with exact rows) conmutes

F2 Q* I

X

-l

_

F1 P' I ,

where di\d^ • • • \d7 are the elementary divisors of X^ and

d,

\

Now consider the basis

of Rm and the basis

Hn

= A2Bely...,

,, AsBei, ...,

Then, we claim that the system £ = E/j in these new bases has the desired form

E = (A,B\. Note that the matrix B related to the new basis l~im of Rm and 7in is on the form

B=

+1,..., Brjm)

=

Linear Dynamical Systems over Commutative Rings

V

\

129

(o)

/

m

And note that the matrix A related to the new basis 7im of R form

/

and Jin is on the

Now, since X% = (0) it follows that ABe^+i = 0, yl-Be^3+2, ..., ABe^ = 0, and consequently ABe^+i = 0, ^4_B^3+2, ••-, ABe^2 = 0. On the other hand, for each i > 3, the matrix Xi is the zero matrix, so AlBe^i+i = 0, ..., AzBe^i+1 = 0. Hence, the matrix A in the new basis "Hm of /?m and "Hn is of the form 0

0

=A

with Xj = (0) for alH > 2. This completes the proof D

COROLLARY 4.9 Let (R,m) be a discrete valuation domain, let S = (A,B) be an m-input, n-dimensional reachable linear dynamical system over R. Suppose that every feedback invariant R-module Mp is torsion free except, perhaps M?. Then, the index io, the R-module Mp and the ordered set of non negative integers

form a complete set of invariants associated to S for the feedback equivalence. PROOF.- The feedback invariant /^-modules Mf , ..., M^_1 are free, so the feedback equivalence class of E is characterized by the positive integers < ai I

>

and

) l<j<»o-l

by the feedback equivalence class of a (aig_2 ~ ^io-i H n P u t, (o"j 0 _i )-dimensional linear dynamical system 6l°~l (E) (see [7, Theorem 1.11]). The feedback invariant /^-modules Mof the system 8l°~l (E) are related to the feedback invariant /^-modules of E by the isomorphisms:

for all j — 1,2,... (see [7, Theorem 1.11]). Consequently all invariant .R-modules MJ

are free except Ml z

l

. Hence, by Theorem 4.6, the feedback equiv-

alence class of § °~~ (E) is characterized by the invariant .R-modules M-

up

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Carriegos and Sanchez-Giralda

to isomorphism. Therefore the feedback equivalence class of E is characterized by the J?-modules Mp up to isomorphism and the result follows in a straightforward way. D

REFERENCES [I] M.F. ATIYAH, I.G. MACDONALD: Introduction al Algebra Conmutativa. Reverte (1989). [2] N. BOURBAKI: Algebre Commutative, Chapitres 5 a 7. Masson (1985). [3] J.W. BREWER, J.W. BUNCE, F.S. VAN VLECK: Linear Systems over Commutative Rings. Marcel Dekker (1986). [4] P.A. BRUNOVSKY: A classification of linear controllable systems. Kybernetika, 3, 173-187 (1970).

[5] R. BUMBY, E.D. SONTAG, H.J. SUSSMANN, W. VASCONCELOS: Remarks on the Pole-Shifting problem over rings. Journal of Pure and Applied Algebra, 20, 113-127 (1981).

[6] A. CAMPILLO-LOPEZ, T. SANCHEZ-GIRALDA: Finitelly generated projective modules and Fitting ideals. Collectanea Mathematica, XXX , 2°, 3-8 (1979). [7] M. CARRIEGOS: Equivalencia feedback en sistemas dinamicos lineales, (Ph.D. dissertation). Universidad de Valladolid (1999).

[8] M. CARRIEGOS, J.A. HERMIDA-ALONSO, T. SANCHEZ-GIRALDA: The pointwise feedback relation for linear dynamical systems. Linear Algebra and its Applications, 279, 119-134 (1998)

[9] J.L. CASTI: Linear Dynamical Systems. Academic Press (1987). [10] J. FERNANDEZ-SUCASAS: Sistemas dinamicos lineales sobre anillos conmutativos y sobre modules, (Ph.D. dissertation). Universidad de Valladolid (1997).

[11] J. FERRER, Ma.I. GARCIA, F. PUERTA, Brunovsky local form of an holomorphic family of pairs of matrices. Linear Algebra and its Applications, 253, 175-198 (1997).

[12] J.A. HERMIDA-ALONSO, M.P. PEREZ, T. SANCHEZ-GIRALDA: Brunovsky's canonical form for linear dynamical systems over commutative rings. Linear Algebra and its Applications, 233, 131-147 (1996).

[13] J.A. HERMIDA-ALONSO, M.P. PEREZ, T. SANCHEZ-GIRALDA: Feedback invariants for linear dynamical systems over a principal ideal domain. Linear Algebra and its Applications, 218, 29-45 (1995).

[14] R.E. KALMAN: Kronecker invariants and Feedback, in Ordinary Differential Equations. Academic, 459-471 (1972).

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131

[15] E.W. KAMEN: Block-form control of linear time-invariant discrete-time systems defined over commutative rings. Linear Algebra and its Applications, 205, 805-829 (1994).

[16] B.R. MCDONALD: Linear Algebra over Commutative Rings. Marcel Dekker (1984). [17] A.S. MORSE: Ring models for delay differential systems. Automatica, 12, 529531 (1976).

[18] D.G. NORTHCOTT: Finite Free Resolutions. Cambridge University Press (1976). [19] M.P. PEREZ: Formas canonicas y clasificacion por feedback de sistemas lineales sobre anillos conmutativos, (Ph.D. dissertation). Universidad de Valladolid (1996).

[20]

E.D. SONTAG: Mathematical Control Theory. Springer-Verlag (1990).

[21] K. TCHON: On structural instability of normal forms of affine control systems subject to static state feedback. Linear Algebra and its Applications, 121, 95104 (1989).

[22]

W.V. VASCONCELOS, C.A. WEIBEL: BCS Rings. Journal of Pure and Applied Algebra, 52, 173-185 (1988).

[23] W.A. WONHAM, A.S. MORSE: Feedback invariants of linear multivariable systems. Automatica, 8, 33-100 (1972).

An introduction to Janet bases and Grobner bases F. J. CASTRO-JIMENEZ, Dpto. de Algebra, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla. Spain. E-mail: castro@cica. es

M.A. MORENO-FRIAS, Dpto. de Matematicas, Universidad de Cadiz, Apdo. 40, E-11510 Puerto Real. Spain E-mail: [email protected]

I

INTRODUCTION

The results of Buchberger ([7]) on Grobner bases in commutative polynomial rings have been generalized by several authors (see for example [6], [11] for early treatments) to the case of some rings of linear differential operators. Independently, the work of Riquier [43] and Janet [21, 22] on the algebraic approach to the systems of partial differential equations was discovered by Pommaret [40, 37] (see also [36],[42]). Since then, these works and the ideas behind them have been thoroughly explored, generalized and firmly established within the framework of effective methods for the resolution of systems of partial differential equations (see for example [44], [24], [15, 16], [46, 47], [48], [27], [41]). Despite this, as far as we know, there is still no systematic comparison between the Janet bases (called by him completely integrable systems) and Grobner bases approach. Most references in the literature accept that both of them are "essentially equivalent". This is the task we undertake in the present work. From this point of view, one can regard this paper as a "survey" on the subject. Concretely, we show that, under certain hypotheses, when the linear differential equations have their coefficients in a field, every completely integrable system is a Grobner basis and conversely (see 4.1.2, 4.1.3, 4.2.5, 4.2.6). This is particularly useful in the case of rings of differential operators with constant coefficients. This being a commutative polynomial ring, we think we can regard Janet bases as a precedent of Grobner basis (in the commutative 133

134

Castro-Jimenez and Moreno-Frias

case). Of course, we cannot directly apply the results of Janet to the case of differential equations with coefficients in a ring. This generalization does exists for Grobner bases theory, in the case of "general" rings of differential operators (see for example [6], [11, 12, 13], [23], [26], [20], [45]). Just a few words about references. Although the list is not exhaustive (on Grobner bases theory in the differential case), we have included firsthand, the references used in writing this paper and then some other references that we consider useful to the reader as a complement of the point of view we have developed here. The authors have greatly benefited from the work of J.-F. Pommaret, F. Schwarz and V.P. Gerdt while writing this article. We also wish to thank J.M. Tornero for a careful reading of the manuscript.

2

MONOMIALS

Let k be a field. Let us denote by M.(X) the set of monic monomials of the commutative polynomial ring k[Jf] = k [ x i , . . . , xn}. If a = (ai,..., an) € N" we will write Xa instead of a;"1 • • •<". In ([21]; (1920)) Janet gives a proof of the so-called "Dickson's lemma" and, as a consequence, he proves ([21], pp. 69-70) the following two lemmas:

LEMMA 2.0.1 Let I be an infinite subset of M(X).

Then there exists a finite

subset F C I such that for all Xa € I there exists X@ & F such that Xa is divisible byX?.

LEMMA 2.0.2 Let Si C 82 C • • • C S^ C • • • be an increasing sequence of subsets in A4(X) such that for all i, each monomial in <Si+i \ Si is not divisible by a monomial in Si. Then this sequence is finite. 2.1

Janet modules

DEFINITION 2.1.1 We say that a subset J of M(X) is a Janet module if either J — 0 or each multiple of a monomial in J lies in J:

\/Xa <= J,V/? e N" we have Xa+fi e J. REMARK 2.1.2 Let (f> : M(X)-^~Nn be the canonical map <j)(Xa) = ( o < i , . . . ,«„)-. J C M(X) is a Janet module if and only if (J}.

DEFINITION 2.1.3 Let J ^ 0 be a Janet module. A finite subset B of J is said to be a basis of J if each monomial in J is divisible by some monomial in B. PROPOSITION 2.1.4 Each Janet module has a basis.

PROF.- Apply 2.0.1.

Janet Bases and Grobner Bases

2.2

135

Multiplicative variables. Classes

Here we will give the definition (Janet, [21], p. 75-76) of multiplicative (and nonmultiplicative) variables and the definition of the class of a monomial.

DEFINITION 2.2.1 Let T be a finite subset of M(X) and Xa € f', 1. xn is said to be a multiplicative variable with respect to (or simply, for) Xa in J- if for all X@ G J- we have /3n < an. 2. Xj, 1 < j' < n — I , is said to be a multiplicative variable with respect to (or simply, for) Xa in J- if the following condition holds: for all X@ & J- with /3n = an, • • • ,/3j+i — otj+i, we have /3j < otj. We denote by mult(Xa,f) the set of multiplicative variables with respect to Xa in T. The variables x, £ mult(Xa,F) are called non-multiplicative variables for Xa in T.

DEFINITION 2.2.2 Let Xa be a monomial of the finite set T C M(X). class of Xa in F, noted by Ca,F, the set

Ca,r = {Xa+f>

We call

Each variable in X13 belongs to muh(Xa,F)}

Classes corresponding to different monomials are disjoint.

DEFINITION 2.2.3 ([21], p. 79) Let F be a finite subset of M(X) and denote by J the Janet module generated by F. The set F is said to be complete if the following holds: For all Xa € J- and for all Xi £ mult(Xa, F) there exists X@ € J- such that a eC0,f. XiX Let F be a complete subset of M.(X). Then for each Xa £ J- and for each Xi £ mult(Xa, F) the only X13 6 T such that XiXa € Cp^ verifies that (an,..., ai) is less than (/3n,... ,/?i) w.r.t the lexicographical order (see [21], p. 85).

3

COMPLETELY INTEGRABLE SYSTEMS. JANET BASES

Janet considers in [21] the degree lexicographical order (denoted by <^eK or -<): ( a < \p\ a <deg p ^ I or ( a| = \/3\

and

( « „ , - • • ,0^
where
k(X)[di,. ..,dn] and Q~n(k) = k((X))[<9i, ...,dn}. We denote by ft any of these three rings and by A any of the corresponding fields k,k(J 5 C), k((X)). Let A/" be a left 72.-module. Consider a system of (not necessarily homogeneous) linear differential equations: ( 5 ) : Pl ( « ) = / ! , . . . , Pr(u)=fr

where Pi g Tl, fi 6 J\f and the unknown u belonging to A/". Rewrite the equation PJ(U) — fj as

136

Castro- Jimenez and Moreno-Frias

with aa(j),a/3 <E A. The element aa^da^\u] (resp. ^2/3^a(j)a^(u} + /j) is called the first (resp. second) member of this equation. We will identify da(u) with da and with a. When fj = 0 and if no confusion is possible we will identify the equation Pj(u) = 0 with the linear differential operator Pj. DEFINITION 3.1.4 ([21], p.105) Let S be a system as above. We say that S is in canonical form (with respect to -<) if the following conditions hold: 1 a ) a(j) = 1 f°r allJ2) The first members of any two equations are distinct.

DEFINITION 3.1.5 ([21], p. 106) Given a system S in canonical form and f the set of its first members, we call principal derivative (with respect to S) each monomial da in the Janet module generated by J- . We call parametric derivatives the remaining ones.

DEFINITION 3.1.6 Let E = aada(u) = £^a a^(u)+f be a linear differential equation with a7 G A, f G A/" and aa ^ 0. We call support of E the set supp(E) = {7 6 N" | a7 ^ 0}. We call a the privileged exponent of E (with respect to -<) and we denote it by exp x (S) (or exp(E) for short, if no confusion is possible). DEFINITION 3.1.7 Let S be a system as above and denote by f the set of the first members of S . Let E be an element of S . We call multiplicative (resp. nonmultiplicative) variable of E (in S) any of the multiplicative (resp. non-multiplicative) variables of the first member of E (in J-). The class of E will be the class of its first member in f

(see 2.2).

DEFINITION 3.1.8 The system S is complete if F is complete (see 2.2.3). Let 5 = {Ei, . . . ,Er} be a complete system of linear homogeneous partial differential equations and suppose the Ei are in canonical form. Let us reproduce the definition of Janet ([21], p. 107): Si, par derivations et combinaisons, on ne pent tirer de (S) aucune relation entre les seules derivees parametriques (et les variables independantes) , on dira que le systeme est completement integrable. Denote by I the left ideal (in TV) generated by S and write

DEFINITION 3.1.9 ([21], p. 107) The complete system S is said to be completely integrable if the only element in I with support in Nn \ A(S) is the zero element.

DEFINITION 3.1.10 Let I be a left ideal ofR, generated by a finite homogeneous system S. The system S is called a Janet basis (of I) if S is completely integrable.

4 4.1

JANET BASES AND GROBNER BASES Homogeneous systems

The theory of Grobner bases developed by Buchberger [7] for commutative polynomial rings has been generalized to ideals in rings of differential operators and

Janet Bases and Grobner Bases

137

in particular to ideals in ft (see [12, 13], [23], [26], [20], [45]). If / is a left ideal of ft we denote by Exp(/) the set of privileged exponents exp(P) for P in / (see 3.1.6). A finite subset {Pi,...,Pr} C I is said to be a Grobner basis of / if Given a - ( a 1 , . . . , ar) E (N")r we define the partition { A i , . . . , A r , A} of Nn associated to a as follows: i-l

r

Fori = l , . . . , r ; A, = (a* + Nn) \ ({J A,-); A = N" \ ((J A<). j=l

i=l

If E_ = (Ei, • • • ,Er] E TV we call partition associated to E_ the partition associated to (exp(£ 1 ),...,exp(.E r )).

THEOREM 4.1.1 (Division theoremj/n 11). Consider (El,---,Er) E TV with Pi ji 0, i = 1, • • • ,r. Let {Ai, • • • , A r , A} be the associated partition of N". Then, for all E & ft, there exists a unique (Qi, • • • , Qr, R) € TV+l such that:

1-

E

= E[=i QiEi + R.

2. If R ^ 0, each monomial of R (in the variables di, • • • ,dn) lies in A.

3- If Qi 7^ 0, each monomial cda of Qi (with c E A) satisfies a + exp(Ei) C A,. In fact, this division theorem (and its proof) is explicit in Janet's work ([21], pp. 100 and 106) when the set {Ei,..., Er} is complete and in canonical form. PROOF.- The proof is analogous to the commutative polynomial ring case (see for example ([1], p. 28) because the coefficients of the differential operators belong to the field A and because Leibnitz's rule implies that for all a E A and a E N™, daaada is a differential operator of degree less or equal than |a — 1 = ai + • • • + an — 1.

THEOREM 4.1.2 Let I be a left ideal of Tl and B = {E±,... ,Er} C /. If B is a Janet basis of I then B is a Grobner basis of I, with respect to the monomial ordering -< on N™.

PROOF.- Write A = A(S) = Uj=i( ex P(£j) + N")-

Let

-P be in / and suppose

exp(P) ^ A. By the division theorem in Ti (see above) there exists R E Ti with

supp(.R) C N" \ A, such that P - R G / and exp(P) = exp(E). So R ^ 0. But this is impossible by the hypothesis (see definitions 3.1.9, 3.1.10). We say that a differential operator P 6 ft is monic if the coefficient of its privileged monomial is 1.

PROPOSITION 4.1.3 Let B = {El}..., Er} be a Grobner basis of a left ideal I of ft. Suppose exp(-Ej) ^ exp(Ej), for i ^ j, Ei is monic for all i and B is complete. Then B is a Janet basis of I. PROOF.- Let R be a non zero element of / with support contained in N™ \ A(S). Then exp(R) E Exp(7) = A(£?) which is a contradiction.

138

Castro- Jimenez and Moreno-Frias

THEOREM 4.1.4 (Criterion for complete integrability). Let

S = {E1,E2,---,Er} be a subset of monic elements in Ti. Suppose that for all i and for all nonmultiplicative variables d^ for Ei (in S) we have d^Ei = ^j=i ^^ -Ej such that the only variables in each monomial (in the variables di , • • • , dn) of A^ are multiplicative variables for Ej, Vj = 1, 2, • • • , r. Then we have: 1. For all HeU,

where the only variables of each monomial in Qi are multiplicative variables for Ei in S. Here (Ei, E2, • • • , Er) is the left ideal (ofR.) generated by the Ei.

2. S is completely integrable. PROOF.- We can suppose exp(Er) -< exp(£V-i) - < ; • • • - < exp(J5i). The hypothesis implies that S is complete, exp(dkEi) = exp(A^ '*' Ec^k^) for a unique integer

c(fc,z) < i (see 2.2) and exp(A Ej) -< exp(dkEi) for j ^ c(k,i). If H € (Ei,E2,---,Er) then we have H = £[=1 Gi^i- Each Gi ^ = l > - - - > can be written as Gi = G\ + Hi where G\ ' is the sum of the monomials of G with only multiplicative variables for Ei in S. In particular HI = 0. We have i-l

i=l

i=2

Let us denote 6 = (61, • • • , 5n) = max {exp(HiEi), i = 1, • • • , r} and ZG = max{z exp(HiEi) = S}. We call (6,io) the characteristic exponent of X^ r =i H j E j We will consider on N™ x {1, • • • , r} the well ordering defined as follows:

(6,i0) < (S',i'0) «=>

or {6 = 6'

and

ZQ < i'0.

Then we can write

where a € A, exp(a91ai •••d^Eio)=5 and e x p ( £ : i o ) -< 8. Suppose dk is a non- multiplicative variable for Et0, then by hypothesis we can write

HiQEio = oSf1 • • • d^~l • • • d^(dkElo) + HioEio =

Janet Bases and Grobner Bases

139

where the only variables in each monomial of A^ are multiplicative variables for EJ. Then rewrite

= HioEio i=2

where

for

j ^ io

Now we will compute the characteristic exponent of this new expression:

1. For i0 + 1 < j < r we have exp(#j£j) = exp(a5"1 • • • d%k~l • • • d^A^Ej + HjEj)

^ max{exp(a<9?1 • • • d^k~l ••• d^A^Ej), exp^Ej)}. We have first j ) -< 6, because of the definition of io, and then • ••

i = al} • • • , ak -

•••

, ••• ,a

( a i , - - - ,ak - 1, • • • , « „ ) +exp(dkEio) = S.

So, exp^'jEj)

-< S for iQ + 1 < j < r.

2.

~

and then exp(H'iQEi0) -< S. 3. For 1 < j < io — 1 we have

jEj) = exp(ad^ • • • d^1 • • • d^A^Ej + H^) -< •••

•••

The choice of j implies that exp(HjEj)

,-

X S and, on the other hand, we have

So, the characteristic exponent (6', i'0) of J^ . H'^Ej is less than (6, io) w.r.t the well ordering <], which implies the assertions of the theorem.

140

Castro- Jimenez and Moreno-Frias

REMARK 4.1.5 As a consequence of this theorem, Janet develops a finite procedure constructing a completely integrable system (i.e. a Janet basis) of a left ideal I 0/72., starting from an arbitrary system of generators of I. This algorithm should be compared to Buchberger's algorithm computing Grobner bases. Janet's procedure is as follows: a) We can suppose the starting system Si = {Ei, . , . , Er} to be complete and in canonical form (see 2.2.3,3.1.4). b) For each i = 1, . . . ,r and each k such that dk is non-multiplicative for Ei, write (see 4.1.1) d^Ei — ^ r =i ^ki ^j ~^~ ^ik where 1. Each monomial in A^ (in d\, . . . , dn) is formed only by multiplicative variables for EJ in Si .

2. e x p ( ^ J ) < exp(afcEi) for j = 1, . . . ,r. 3. The support of R^i is contained in N" \ A(5i). c) If all the Rki are zero, then S is completely integrable (see 4-1 -4)- d) V there exists Rki 7^ 0 then we consider the new system S% = SiU {Rki} and we restart. This procedure is finite. Indeed, let Si, i = 1, 2, . . . be the sequence of systems obtained applying Janet's procedure. Write Fi = (exp(J5) | E £ Si} C N n . By 2.0.2 this sequence is stationary and the procedure is finite. 4.2

Non-homogeneous systems

In this section we will explain how to extend the results of 3 and 4 to systems of linear non-homogeneous differential equations. Let 5 be a system of linear, non necessarily homogeneous, differential equations

where Pi 6 72, /j € A/" and the unknown u belonging to A/". We denote by Sh the homogeneous system PI(U) = • • • = Pr(u) = 0 associated to S. We will denote by Ei the equation Pi(u) = fi (or Pi(u) — fi = 0 ) . We identify the equation Pi(u) = fi (i.e. the equation Ei) with the couple (Pi, /i) S 72 © TV and we consider the 72.-sub-module M of 72. © M generated by DEFINITION 4.2.1 Let S = {Ei,---,Er} = {(Pi,fi),---,(PrJr)} be a complete system in canonical form. Let M be the 72.- sub-module ofR.® M generated by S. The system S is said to be completely integrable if the following holds:

1) If(OJ)

e M then f = 0.

2) If (P, f) € M and P ^ 0 then the support of P is not contained in N™ \ A(S^). DEFINITION 4.2.2 Let M be the U-sub-module ofU®N generated by S = {(-Pi, /i), • • • , (Pr> f r ) } • We call S a Janet basis of M if S is completely integrable.

Denote by £ the (left) 7?.-module 72. © J\f and by -n\ : £ —> 72. the canonical projection. As in 72 we have in £ the notions of privileged exponent and Grobner basis and we have a division theorem in £. We still denote by exp : £ \ ({0} ©A/") —> N" the map expx(P, /) = exp(P).

Janet Bases and Grobner Bases

141

THEOREM 4.2.3 (Division theorem in E). Consider (El,---,ET) € £r with Et = (Pi, f i ) and Pi ^ 0, i = 1, • • • ,r. Let {Ai, • • • , A r , A} be the associated partition of N™. Then, for all E = (P, /) € £, there exists a unique (Qi, • • • ,Qr, ( R , d ) ) 6 Tlrx£ such that:

2. If R ^ 0, each monomial of Ti (in the variables di, • • • , dn) lies in A. 3. If Qi ^ 0, each monomial cda of Qi (with c € A), satisfies a + exp(Ei) C PROOF.- Analogous to the proof of 4.1.1. We first write P = Y?i=\ QiPi + R then g = f- EL: Qi(fi). DEFINITION 4.2.4

Let M be a H- sub-module of£. A finite subset

of M is said to be a Grobner basis of M, with respect to -<, if the following two conditions hold: 1. {Pi, • • • ,Pm} is

a

Grobner basis o/7Ti(M) with respect to -<.

2. For all g £ M, if (0, g) € M then g = 0. THEOREM 4.2.5 Let M be an U-sub-module of £ = U ® M and suppose B = {(Pi, /i), • • • , (Pr, / r )} C M is a Janet basis of M . Then B is a Grobner basis of M, with respect to -<. PROOF.- Analogous to the proof of 4.1.2, using the division theorem 4.2.3. PROPOSITION 4.2.6 Let B = {El = ( P ^ f r ) , . .. ,Er = (Pr,fr)} be a Grobner basis of a left TL- sub-module M ofR.@J\f. Suppose exp(Pi) ^ exp(P7-) for i ^ j, Ei is monic for all i and B is complete. Then B is a Janet basis of M.

PROOF.- Suppose (P, /) e M and supp(P) c N" \ A(5 fe ). The family {Pi, . . . , Pr} is a Grobner basis of the ideal 7Ti(M) and then A(5 /l ) = Exp(?ri(M)). So, exp(P) 6 A(Sh) and then P = 0. THEOREM 4.2.7

(Criterion for complete integrability). Let

S = {E1,E2,---,Er} be a subset of monic elements in £. Suppose that for all i and for all non-multiplicative variables dk for Ei (in S) we have d^Ei = X^=i Ak^Ej where the only variables in each monomial (in the variables d\, 82, • • • , dn) of A^ are multiplicative variables for Ej, Vj = 1, 2, • • • , r. Then we have:

1. For all H e £,

1=1 where the only variables of each monomial in Qi are multiplicative variables for Ei in S. Here (Ei, E2, • • • , Er) is the left Tl-module generated by the E^.

2. S is completely integrable.

PROOF.- Analogous to the proof of 4.1.4.

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REFERENCES [1] W.W. ADAMS AND P.H. LOUSTAUNAU. An introduction to Grobner Bases. Number 3 in Graduate Studies in Mathematics. American Mathematical Society. (1994).

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[4] A. Assi, F. CASTRO-JIMENEZ AND J.-M. GRANGER. The standard fan of an analytic V-module. To appear, Journal of Pure and Applied Algebra. [5] T. DECKER AND V. WEISPFENNING. Grobner bases. A computational approach to commutative algebra. Springer-Verlag, 1993. [6] J. BRlANgON ET PH. MAISONOBE. Ideaux de germes d'operateurs differentiels a une variable, L'Enseignement Math., 30:7-38, (1984). [7] B. BuCHBERGER. Ein algorithmisches Kriterium fur die Losbarkeit eines algebraischen Gleichungssystems (ph. d. thesis). Aequationes Math., 4:374-383, (1970). [8] B. BUCHBERGER. Grobner bases: an algorithmic method in polynomial ideal theory. Multidimensional systems theory, 184-232. (1985).

[9] J.L. BUESO, F.J. CASTRO, J. GOMEZ-TORRECILLAS AND F.J. LOBILLO. An introduction to effective calculus in quantum groups, in: Rings, Hopf algebras and Brauer groups. S. Caenepeel, A. Verschoren (Eds,), Marcel Dekker, 1998. 55-83.

[10] J.L. BUESO, F.J. CASTRO AND P. JARA.

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computation of the

Gelfand-Kirillow dimension. Proc. Edinburgh Math. Soc. 40, 1 (1997), 111117. [11] F.J. CASTRO. Theoreme de division pour les operateurs differentiels des multiplicites. PhD thesis, Univ. Paris VII, Oct-1984.

et calcul

[12] F.J. CASTRO. Calculs effectifs pour les ideaux d'operateurs differentiels. In Travaux en Cours. Geometric Algebrique et Applications, Tome III. pages 1-19. Hermann (Paris). (1987). [13] F.J. CASTRO-JlMENEZ AND J.M. GRANGER. Explicit calculations in rings of differential operators. Prepublicaciones de la Facultad de Matematicas de la Universidad de Sevilla. n^ 35. (Junio-1997).

[14] F.J. CASTRO JIMENEZ, L. NARVAEZ MACARRO. Homogenising differential operators. Prepublicaciones de la Facultad de Matematicas de la Universidad de Sevilla. n^ 36, 1997. [15] V.P. GERDT. Grobner bases and involutive methods for algebraic and differential equations. Math. Comput. Modelling, n. 8-9, 75-90. (1997).

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[16] V.P. GERDT AND Y.A. BLINKOV Involutive bases of polynomial ideals. Math, and Comput. in Simulation, vol. 45, no. 5-6, 519-41 (1998).

[17] V. P. GERDT, N. A. KOSTOV, AND A. Y. ZHARKOV. Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra. In Solitons and applications (Dubna, 1989), pp. 120-128. World Sci. Publishing, River Edge, NJ, 1990.

[18] V. P. GERDT, A. B. SHVACHKA, AND A. Y. ZHARKOV. Computer algebra application for classification of integrable nonlinear evolution equations. J. Symbolic Comput, 1(1):101-107, 1985.

[19] V. P. GERDT AND A. Y. ZHARKOV. Computer classification of integrable coupled KdV-like systems. J. Symbolic Comput, 10(2):203-207, 1990. [20] M. INSA AND F. PAUER. Grobner bases in rings of differential operators. In Grobner Bases and Applications. London Math. Soc. L.N.S. 251. Cambridge University Press, pp. 367-380. (1998).

[21] M. JANET. Sur les systemes d'equations aux derivees partielles. J. de Math., 8e serie, III: 65-151. (1920). [22] M. JANET. Legons sur les systemes d'equations aux derivees partielles. Gauthiers-Villars. Paris. (1929).

[23] A. KANDRI-RODY AND V. WEISPFENNING. Noncommutative Grobner bases in algebras of solvable type. J. Symbolic Computation 9,1, 1-26. (1990). [24] E.R. KOLCHIN. Differential (1984).

algebraic groups. Academic Press. New York.

[25] M. LEJEUNE-JALABERT. Effectivite Univ. Grenoble. (1984-85).

de calculs polynomiaux. Cours de DEA.,

[26] F.J. LOBILLO. Metodos Algebraicos y Efectivos en Grupos Cudnticos. Tesis Doctoral de la Universidad de Granada. (1998). [27] E.L. MANSFIELD. Differential Sydney. (1992).

Grobner Bases. Ph. D. Thesis. University of

[28] T. MORA. An Introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science. 134 (1994), 131-173. [29] T. OAKU. Algorithmic methods for Fuchsian systems of linear partial differential equations. J. Math. Soc. Japan, 47(2):297-328, 1995. [30] T. OAKU. Grobner bases for D-modules on a non-singular affine algebraic variety. Tohoku Math. J. (2), 48(4):575-600, 1996.

[31] T. OAKU. An algorithm of computing 6-functions. Duke Math. J., 87(1):115132, 1997. [32] T. OAKU. Some algorithmic aspects of the D-module theory. In New trends in microlocal analysis (Tokyo, 1995), pp. 205-223. Springer, Tokyo, 1997.

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[33] T. OAKU AND T. SHIMOYAMA. A Grobner basis method for modules over rings of differential operators. J. Symbolic Cornput., 18(3):223-248, 1994.

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new perspectives in two-dimensional conformal geometry. Physics 10, 47-91. (1993).

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Editorial Kapelusz. Buenos Aires.

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algebra. Dover Publications, Inc. New York. (1966).

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Invariants of Coalgebras J. CUADRA, Dpto. Algebra y Analisis Matematico, Universidad de Almerfa. 04120-Almen'a. Spain.

F. VAN OYSTAEYEN, Department of Mathematics, University of Antwerp (UIA). B-2610 Wilrijk (Antwerp). Belgium.

1

INTRODUCTION

In this paper we give a semi-survey of the known results for certain invariants of coalgebras, in particular the Picard group and the Brauer group. We also include some proofs and enriched the theory with some new results and observations, in particular Theorem 3.2.9 and consequents, Corollary 4.1.13 and Examples 4.1.14, Theorem 4.2.5 and Corollary 4.2.6, Proposition 4.3.10 and Corollaries 4.3.11, 4.3.12. After some preliminaries in Section 2 we survey the results on the Picard group in relation with Morita-Takeuchi theory in Section 3. Not so many Picard groups have been explicitly calculated. This is possible sometimes when the calculation may be reduced to dealing with automorphisms, more correctly to outer automorphisms; this is the topic of Section 3.2. Good cases are obtained for basic coalgebras (Theorem 3.2.5) and pointed coalgebras (Corollary 3.2.7). The coalgebra case sometimes reduces to a calculation for the dual algebra in case the coalgebra has finite dimensional coradical (Theorem 3.2.9). We review the general theory for the Brauer group in Section 4.1, again involving Morita-Takeuchi theory, and we add new examples to the theory. The Brauer group of a cocommutative coalgebra may be calculated from the Brauer group of the irreducible components. Several questions about the torsioness of the Brauer group of irreducible coalgebras were pointed out in [25]. In particular, it was conjectured that the Brauer group of an irreducible coalgebra is torsion. We solve this question in Corollary 4.2.6. In fact, we proved that the Brauer group of an irreducible coalgebra embeds in the Brauer group of the dual algebra. 147

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Finally we describe some subgroups of the Brauer group which are versions of the Schur subgroup and the projective Schur subgroup of the Brauer group of a ring. These have good properties mainly because they define torsion groups whereas the Brauer group itself is not necessarily torsion. We prove that the Schur group of a cocommutative coalgebra over field of characteristic positive is trivial. Also for irreducible coreflexive coalgebras with separable coradical, these subgroups may be computed from the ones corresponding to the coradical.

2

PRELIMINARIES

Throughout k is a fixed ground field and M-k denotes the category of k- vector spaces. All coalgebras, vector spaces and unadorned (g>, Horn, etc., are over k. Coalgebras and Comodules (See [1], [21]): For a coalgebra C, A and e denote the comultiplication and the counit, respectively. The category of right C-comodules is denoted by Mc; for X in M.G , px is the comodule structure map. For X, Y € M.c , Com^c(X, Y) is the space of right C-comodule maps from X to Y. Similarly, c M. denotes the category of left C-comodules. If D is another coalgebra, then X is a (D, <7)-bicomodule if X e Mc via px, X e DM via xP and (1 px)xP = Cotensor product : Let M be a right <7-comodule and N a left C-comodule with structure maps PM and p^ respectively. The cotensor product MOCN is the kernel of the map

The functors MO^— and —D^N are left exact and preserve direct sums. If cMrj and oNs are bicomodules, then MO^N is a (C, £?)-bicomodule with comodule structures induced by those of M and N. Let X 6 Mc , we say that X is finitely cogenerated if there is a finite dimensional vector space W such that X is isomorphic to a subcomodule of W ® C. W ® C is nothing else but the direct sum of C dim(W) times. Sometimes we will write C^ instead W ® C with n = dim(W). X is free if X is isomorphic to C^ for some set I. X is said to be a cogenerator if for any comodule M e M.c , M <—> X^\ for some set /, as comodules. X is injective if the functor XDC— is exact. Morita-Takeuchi Theory (See [23]): A comodule X € M.c is called quasi-finite if Com-c(Y, X) is finite dimensional for any finite dimensional comodule Y e M.c ' . Quasi-finite comodules can be characterized by looking at the socle, cf. [23, Prop. 4.5]. PROPOSITION 2.0.1 For a comodule X, the following assertions are equivalent: i) M is quasi-finite. ii) soc(M) is isomorphic to (Big/iS,where {5i}ig/ is a complete set of representatives of isomorphism classes of simple comodules and Hi are finite cardinal numbers.

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For a quasi-finite comodule X e Mc and any Y € Mc the co-hom functor is defined by

where {Y\} is a directed system of finite dimensional subcomodules of Y such that Y = lim— Y\. When Y = X then h^c(X, X) is denoted by e_c(X) and it becomes A a coalgebra, called the co-endomorphism coalgebra.

A Morita-Takeuchi context (M-T context) (C, D,cPo, oQc, /,5) consists of coalgebras C, D, bicomodules cPo, oQc, and bicolinear maps / : C —> POrjQ and g : D —> QO/^P satisfying the following commutative diagrams:

P ——^—> PODD CP

10g

Q —————> QUO

in/

\/ \/ Cacp —-> poDQacp /ni

The context is said to be strict if both / and g are injective (equivalently, isomorphism). The following result, due to Takeuchi, characterizes the equivalences between two categories of comodules, [23, Prop. 2.5, Th. 3.5]:

THEOREM 2.0.2 LetC,D be coalgebras. a) If F : Mc —> M is a left exact linear functor that preserves direct sums, then there exists a (C,D)-bicomodule M such that F(-) = — O^M. b) Let M be a (C,D)-bicomodule. The following assertions are equivalent: i) The functor — ^cM.:Mc-+MD is an equivalence, ii) M is a quasi-finite injective cogenerator as a right D-comodule and e__£>(M) = C as coalgebras. Hi) There is a strict M-T context (C, D,cPo, oQc, /, Mc. The coalgebras C and D are called Morita-Takeuchi equivalent coalgebras. Morita-Lin Theory (See [15]): There is another Morita theory for coalgebras, due to Lin. The category of right comodules is embedded in the category of left C"*-modules and using the classical Morita theory for rings one characterizes the equivalences in the category of comodules. A right (7-comodule M is said to be an ingenerator if it is a finitely cogenerated injective cogenerator. We say that C is strongly equivalent to D if Mc is equivalent to MD via / : Mc -> MD, g : MD -> Mc, f ( C ) is an ingenerator in MD and g(D) is an ingenerator in M.c'. If both coalgebras have finite dimensional coradical, then strongly equivalent is the same as equivalent. Here is the theorem characterizing strong equivalences:

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THEOREM 2.0.3 Let C and D be coalgebras. a) If Mc is strongly equivalent to M,D via f : M.c —> MD and g : M.D —> Jv[° , then there are ingenerators P € M° and Q € MD such that C'-M. is equivalent to D*M via

F(-) =D.P£.®C'-: c-M^ D-M, G(-) = c-Q*D.®D'-: D-M-* c-M. Moreover, f and g are naturally isomorphic to F and G respectively.

b) U c*M is equivalent to rj*M. via functors F : C'-M- —> D*-M, G : D'M^ C'M such thatF(Mc) C MD andG(MD] C Mc , then Mc is strongly equivalent to MD .

The cocenter (See [25]): Let C be a coalgebra. If we view C as a right Cecomodule (Ce = C°p 0 C), then C is quasi- finite. The coendomorphism coalgebra has the following universal property: i) e-C" (C) is a cocommutative coalgebra with a surjective coalgebra map ld : C —> e-ce(C) which is cocentral, i.e., for all c € C,

ii) For any cocentral coalgebra map / : C —> D there exists a unique coalgebra map g : e_c e (C < ) —* D such that / = gld. In particular, an injective coalgebra map induces a coalgebra map from e_c=(C f ) to e_£>e(D).

e_ce(C) is denoted by Z(C) and it is called the cocenter of C. Let C be a cocommutative coalgebra, a coalgebra D is said to be a C-coalgebra if D is a coalgebra together with a cocentral coalgebra map e : D —> C, called C-counit. A map of C-coalgebras is a coalgebra map which respects the C-counits. Let D,E be C-coalgebras with C-counits CD^E respectively, and M a (D,E)bicomodule. M is said to be a bicomodule over C if the following diagram is commutative.

M ——f^^M®E

<8>6g

. M0C

PD

CD <8> 1

T

where T is the twist map. Let F : .M0 —-> A^ B be an equivalence of categories. We have seen that F is of the form F(-) = -DDM for a (£>,.E)-bicomodule M. We say that F is an equivalence over C if M is a bicomodule over C. In this case, we say that D and E are Morita-Takeuchi equivalent coalgebras over C.

3

THE PICARD GROUP

From Morita-Takeuchi theory we retain that every autoequivalence of M.c is given by a (C, C')-bicomodule M, which is called an invertible bicomodule. The Picard group of C, denoted by Pic(C), was introduced in [24] by taking isomorphism classes of invertible bicomodules and cotensor product over C. In that paper, an

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exact sequence connecting the group of coalgebra automorphisms of C, Aut(C), and Pic(C) was given. This sequence yields that the group of outer automorphisms, Out(C), embeds into Pic(C}. In this section we find coalgebras such that Out(C) is the whole Pic(C). So, for these coalgebras the computation of Pic(C) reduces to the computation of automorphisms of C. This result allows us to compute new examples of Picard group of coalgebras. 3.1

Definitions and properties

In this subsection we recall from [24] the definition of the Picard group of a coalgebra and some of its properties. We refer to the reader to [24] for the proofs.

DEFINITION 3.1.1 A (C,C)-bicomodule M is called invertible if the functor — OCM : M.c —* M° defines a Morita-Takeuchi equivalence. This is equivalent to the existence of a (C, C)-bicomodule N and the bicomodule isomorphisms MDCN ^ C and NncM ^ C.

DEFINITION 3.1.2 The Picard group of C, denoted by Pic(C), is defined as the set of all bicomodule isomorphism classes [M] of invertible (C,C)-bicomodules. Pic(C) becomes a group with the multiplication induced by the cotensor product, that is, for [M], [N] £ Pic(C), [M}[N] - (MUCN\. The identity element is [C] and(M}-1 = [h_c(M,C)}. Let M be a (C, C')-bicomodule with right and left structure maps pM and MP respectively, ld : C —* Z(C) the universal cocentral map from C in its cocenter and T ; M ®C —> C* <S> M the twist map. The set

Picent(C) = {[M] e Pic(C) : r(ld ® l)pM = (1 <8> l d ) uP\ is a subgroup of Pic(C) called Picent of C. Remark: Let Co be the coradical of C. If Co is finite dimensional, then every autoequivalence in Mc induces an autoequivalence in c*M by Theorem 2.0.3. In this case Pic(C) C Pic(C*) via the map [M] i—» [M*]. When C is finite dimensional this map is surjective and both groups coincide.

PROPOSITION 3.1.3 Let C,D be Morita-Takeuchi equivalent coalgebras, then Pic(C) ^Pic(D). This proposition shows that Pic(~) is an invariant up to the M-T equivalence relation.

Let Aut(C) denote the group of automorphisms of the coalgebra C. An automorphism / 6 Aut(C) is said to be inner if there is a unit u & C* such that f ( c ) = S(c) U ( c i) c 2 u ~ 1 ( c s) f°r all c& C. We denote by Inn(C) the group of inner automorphisms of C. Inn(C) is a normal subgroup of Aut(C) and the factor group Out(C) = Aut(C)/Inn(C) is called the group of outer automorphisms of C. Let M be a (C, C*)-bicomodule and /,g 6 Aut(C). We denote by f M g the bicomodule constructed in the following way: as a vector space /Mg = M and

, fMgP = ( f ®

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THEOREM 3.1.4 There is an exact sequence

where w(/) = [ /Cj] for all / £ Aut(C). Hence, u> induces a monomorphism from Out(C) to Pic(C}. The following result is used in the proof of the further results. PROPOSITION 3.1.5 Let [M], [N] e Pic(C). Then, M ^ N as right comodules if and only if there exists f 6 Aut(C) such that N = /Mi as bicomodules. 3.2

The Aut-Pic property

Now we investigate when the map w : Aut(C) —> Pic(C) is surjective. In this case Pic(C) = Out(C), and thus we reduce the computation of Pic(C) to the computation of the automorphisms of the coalgebra C. DEFINITION 3.2.1 A coalgebra C has the Aut-Pic property ifuj of Theorem 3.1.4 is surjective. The first example of coalgebra with the Aut-Pic property was given in [24, Th. 2.10]:

PROPOSITION 3.2.2 Let C be a cocommutative coalgebra, then C has the AutPic property. Consequently, Pic(C) = Aut(C). Next we study more general coalgebras with the Aut-Pic property. PROPOSITION 3.2.3 Let C be a coalgebra such that every right injective comodule is free. Then C has Aut-Pic.

PROOF: Let M be an invertible (C, C)-bicomodule with inverse N, then M, N are quasi-finite injective cogenerators as right comodules. By the hypothesis, we know that M = C (n) and N = (7(m) for some n, m > 1 as right comodules. Now,

C ^ MucN ^ C(n]ocN ^ N(n*> ^ c(nm\ as right comodules. Since C has the IBN property, cf. [17, Prop. 4.1], then nm = 1 and so M = C as right comodules. From Proposition 3.1.5, it follows that there exists / e Aut(C) such that M = $Ci as bicomodules. Hence [M] 6 Im(u>). I More examples of coalgebras with Aut-Pic were given in [8, Prop. 2.5]; these derive from the consideration of matrix coalgebras. PROPOSITION 3.2.4 Let C be a coalgebra with Aut-Pic. Then the matrix coalgebra over C, Mc(n, C), has Aut-Pic for all n > 1. We recall from [5] that a coalgebra is basic if every simple subcoalgebra is the dual of a division algebra over k. The following theorem shows that basic coalgebras also have the Aut-Pic property.

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THEOREM 3.2.5 Let C be a basic coalgebra, then C has Aut-Pic. PROOF: We first prove that for basic coalgebras the isotypic component of a simple right coideal only contains that coideal. Let / be a non-zero simple right coideal of C and J its isotypic component. It is well-known that J is a simple subcoalgebra, cf [13]. By the hypothesis, there is a finite dimensional division algebra D over k such that J = D* as coalgebras. I is a right coideal of J, so that 1^ is a right ideal of D. Since D is a division algebra it must be /-1 = {0} and hence I = J. We know that C is a quasi-finite injective cogenerator. Using Proposition 2.0.1 we have that C = ® aer E(Sa)^na'i where {Sa} is a representative family of right simples comodules and na are finite cardinal numbers. Every 5a is isomorphic to a right simple coideal Ia of C . Taking into account that the isotypic component of a coideal only contains that coideal, it must be na = 1 for all a <E F, and so

Now, let [M] £ Pic(C), then M is a right quasi-finite injective cogenerator. Since M is quasi-finite, again by Proposition 2.0.1, soc(M) = ® agr Sa a , and since M is an injective cogenerator, then M = (J)aer E(Sa)^ma^ where ma > 1 for all a € T. Set P = 0Q6r E(Sa)(m°-^, then M ^ C ® P as right comodules. Let N be the inverse of M, then C = MacN ^ (C'ncN) 0 (POCN) as right comodules and so soc(C) = soc(N) 0 soc(POGN). Writting N ^ ® Q€r E(Sa)(ta\ from the above isomorphism it follows that ta < 1 for all a e F. But, since N is a cogenerator, ta > 1, therefore ta — 1 for all a £ F. Thus, N = C as right comodules and from C = NOCM, we have that M = C as right comodules. Now, by Proposition 3.1.5, there is / e Aut(C) such that M = jC\ as bicomodules. Hence [M] e Im(w). I Cocommutative coalgebras are examples of basic coalgebras, so we rediscover [24, Th. 2.10].

COROLLARY 3.2.6 Every cocommutative coalgebra has the Aut-Pic property. Noting that pointed coalgebras are basic coalgebras we have:

COROLLARY 3.2.7 Every pointed coalgebra has Aut-Pic. Thus we have found a big family of coalgebras having the Aut-Pic property. Using this result we are able to give new examples of Picard groups of coalgebras.

EXAMPLE 3.2.8 1.- Let C be the Sweedler coalgebra, i.e., the vector space generated by the set {dn,sn '• n € IN} with comultiplication and counit given by

= gn = gn <S>s n + sn ®gn+i

e(sn)=Q

C is a pointed coalgebra, and by the above corollary, Pic(C) = Out(C). We show that Pic(C) is trivial. Before computing Aut(C), it is easy to observe: For i,j € IN with j ^ i + 1 the (gi,
X & k. For all n 6 W, the (gn, g ra+ i)-primitive elements are angn + (3nsn — angn+i with an, /3n G k.

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Let / £ Aut(C), then / maps group-like elements to group-like elements, (
with on £ k and /% £ k* for all i £ IN. We next check that every automorphism is inner and hence Pic(C) is trivial. Let / be as above, we define u, v : C —» k by:

u(gi) = 1

v(gi) = I

It is not hard to prove by induction that u is a unit with inverse v and that 2.- Let r G IV such that r > 2 and C the vector space generated by the set j, ..., a£, 6^, ..., b£ : n £ IV} with comultiplication and counit defined by:

A(<) = oj, ® 4

e(4) - 1

for alH = 1, ..., r and n € IV. C is a pointed coalgebra and then Pic(C) = Out(C). A similar argument to the above one yields: Pic(C) = Sr, the symmetric group of r letters. 3.- We consider the same vector space as above but with comultiplication and counit given by:

= 0

for all i = 1, ...,r — 1 and n £ IV. C is a pointed coalgebra and hence Pic(C) = Out(C). Using a similar reasoning as in 1, it can be computed that Pic(C) = 2Zr, the cyclic group of order r. Suppose that the coradical of C is finite dimensional. If C has the Aut-Pic property, does it follow that C* the Aut-Pic property? The following result answers this question. THEOREM 3.2.9 Let C be a coalgebra with coradical of finite dimension.

i) If C has Aut-Pic and Pic(C) = Pic(C*) via [M] -> [M*], then C* has the Aut-Pic property. ii) If C* has the Aut-Pic property, then C has the Aut-Pic property. PROOF: i) Let [M] £ Pic(C*), since Pic(C) ^ Pic(C*), then there is a (C, C}bicomodule JV such that M = N* as (C* , C**)-bimodules. As C has the Aut-Pic property, we can find g £ Aut(C) such that N = 3Ci. Then N* ^ g,C{ and thus M* £* g.Ct as (C*,C'*)-bimodules.

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ii) Let [M] € Pic(C), then [M*] e Pic(C*} and as C* has the Aut-Pic property, there is an automorphism / in C* such that M* = fC^. Let $ : C* —> M* the isomorphism of right C"*-modules. We put n* = $(e) and we define * : M —> C, m i—» E( m )( n *> m (o)} m (i)- Then ^ is a map of right C-comodules which verifies that <]/* = $. Thus, \& is an isomorphism of right (7-comodules and hence there is g 6 Aut(C) such that M = 5Ci as bicomodules. So, C* has the Aut-Pic property.

COROLLARY 3.2.10 Suppose that C is a finite dimensional coalgebra. Then, C has Aut-Pic if and only if C* has Aut-Pic.

COROLLARY 3.2.11 Let C be a cocommutative coalgebra with finite dimensional coradical. Then, C* has the Aut-Pic property. Moreover, if C is coreflexive, then PROOF: We know that J = CQ is the Jacobson radical of C* and C* / J = CQ. Since CQ is finite dimensional, we have that C* is a semilocal ring and hence Picent(C*) is trivial. Now, [4, Prop. 1.5] claims that a commutative ring has the Aut-Pic property if and only if its Picent is trivial. Hence, C* has the Aut-Pic property and so Pic(C*) ^ Aut(C*). If C is in addition coreflexive, by [22, Prop. 7.1], the map from Aut(C) to

Aut(C*), f ^ f* is an isomorphism. Thus, Pic(C) £* Aut(C) ^ Aut(C*} ^ Pic(C*). I

EXAMPLE 3.2.12 Let C be the power divided coalgebra, that is, the fc-vector space generated by the set {co,ci,C2, ....} with comultiplication and counit given by:

A(c n ) = £)"=! c% 0 cn^i,

e(cn) = S0
for all n e W where S means the Kronecker symbol. It is well-known that C* is isomorphic to the power series ring fe[[x]] and that C is a coreflexive coalgebra. Using the above proposition Pic(C) £* Aut(C) ^ Aut(k[[x}}) ^ Pic(k[[x]]). When a coalgebra C has the Aut-Pic property, the Picent of C is also described in terms of automorphisms. Set Autz(c)(C~) and Innz(c}(C] for the groups of Z(C}automorphisms and Z(C*)-inner automorphisms, respectively and Outz(c)(C) =

Autz(c)(C}/InnZ(C](C}. PROPOSITION 3.2.13 If C has the Aut-Pic property, then

Picent(C) ^ Outz(C)(C). PROOF: Let [M] € Picent(C) C Pic(C). By hypothesis there exists / e Aut(C) such that M = SC\ as (C, C)-bicomodules. Since [M] e Picent(C), r(ld ® !)(/ igi 1)A = (1 ® l d )A, that is, for c G (7, we have:

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Applying e 1, l d f ( c ) = ld(c) and this just means that / is a map of Z(C)coalgebras. Hence Picent(C] = Outz(c)(C). I To finish this section, we observe that in the cocommutative or cosemisimple case the Picent is trivial. This shows lights differences with the ring case. In the cocommutative case, this difference arises from the decomposition of any cocommutative coalgebra as direct sum of irreducible subcoalgebras. We will see in the next section that this fact is also very important for the Brauer group of a cocommutative coalgebra.

PROPOSITION 3.2.14 Let C be a cosemisimple or cocommutative coalgebra, then Picent(C) is trivial. PROOF: From [24, Lem. 2.12] we get that Picent(C) ^ Hi€l Picent(Ci) when {Ci} is a family of subcoalgebras such that C = ® i 6 /Ci- Suppose that C is cosemisimple, using this remark, it is enough to prove that Picent(C) is trivial when C is a simple coalgebra. But if C is simple, C is finite dimensional and Picent(C) £* Picent(C*). Now, as C* is simple, by [9, p. 375], Picent(C*) = {!}. Hence, Picent(C) = {!}. As C is cocommutative, C has. Aut-Pic and by the above proposition

Picent(C) ^ Autc(C) = {!}.

4 4.1

THE BRAUER GROUP OF A COCOMMUTATIVE COALGEBRA Definitions and properties

We recall from [25] the construction of the Brauer group for a cocommutative coalgebra and its most important properties. Proofs of the results may be looked up in [25].

In this section, C is a cocommutative coalgebra. Let I? be a C-coalgebra with C-counit denoted by e. The universal property of the cocenter implies that there is a unique coalgebra map TJ : Z(D) —> C such that e = r j l d .

DEFINITION 4.1.1 D is said to be cocentral if the map r] is an isomorphism of coalgebras. Putting De — DOcDop, D may be viewed as a right or left De-bicomodule. DEFINITION 4.1.2 D is coseparable over C or C-coseparable if it verifies one of

the following equivalent conditions: i) There is a map TT : DOCD —> D such that ?rA = Irj. ii) D is injective as a right De-bicomodule. DEFINITION 4.1.3 A C-coalgebra D is called Azumaya over C or a C-Azumaya coalgebra if it is C-coseparable and cocentral. The following proposition lists some properties of Azumaya coalgebras.

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157

PROPOSITION 4.1.4 LetD,E be Azumaya coalgebras over C . Then, i) Dop and DOCE are Azumaya coalgebras. ii) If P is a quasi-finite injective cogenerator, then e_cr(P) is an Azumaya coalgebra. in) If C1 is a cocommutative coalgebra and f : C' —> C is a map of coalgebras,

then Da^C' is an Azumaya coalgebra over C". Let D be a coalgebra and Z(D) its cocenter. Since Z(D) = e_£>e(D), D may be viewed as an (Z(D), De)-bicomodule. Let /, g denote the canonical maps

/ : Z(D) -* DDD,h^D.(D,De),

g : De -+ h_D*(D,D*)Uz(D]D.

Then (Z(D), De, D, hDe(D,De), f , g ) forms a M-T context. Azumaya coalgebras can be characterized in terms of Morita-Takeuchi equivalences.

THEOREM 4.1.5 Let D be a C -coalgebra. The following are equivalent: i) D is an Azumaya coalgebra. ii) The above M-T context is strict. Hi) D is a quasi-finite injective cogenerator as left C -module and ec-(D) = De. Denote by B(C) the set of the isomorphism classes of Azumaya C-coalgebras. An equivalence relation is introduced in B(C) as follows: if E, F £ B(C), then E is equivalent to F, denoted by E ~ F , if there exist two quasi-finite injective cogenerators M, N in M° such that

^ FDce_c(N), as C-coalgebras.

THEOREM 4.1.6 The quotient set B(C)/ ~, denoted by Br(C), is an abelian group with the multiplication [E][F] = [EO^F], unit element [C] and for [E] the inverse is [E°p\. The group Br(C) is called the Brauer group of the cocommutative coalgebra C.

If T] : C' —> C is a map of cocommutative coalgebras, then 77 induces a group homomorphism 77, : Br(C) -> Br(C') given by rj*([E]) = [EUCC'} for all [E] € Br(C). Thus we have defined a contravariant functor from the category of cocommutative coalgebras to the category of abelian groups. The following proposition claims that the equivalence relation in B(C) is indeed the Morita-Takeuchi equivalence relation.

PROPOSITION 4.1.7 [D] = [E] € Br(C) if and only if D and E are MoritaTakeuchi equivalent coalgebras over C . If the coalgebra C is of finite dimension, then the study of the Brauer group of C is equivalent to the study of the Brauer group of the dual algebra C*.

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PROPOSITION 4.1.8 Let C be a coalgebra of finite dimension and D a C-coalgebra. Then, i) D is an Azumaya coalgebra if and only if D* is an Azumaya algebra over the dual algebra C* . ii) The map (— )* : Br(C] —> Br(C*), [D] t—> [D*} is a group isomorphism. The following results study the behaviour of the functor Br(-) with respect to taking direct sums.

LEMMA 4.1.9 Suppose that there is a family of co commutative coalgebras {Ci}j6/ such that C = ® ie / Ci. Then, i) Every C -coalgebra D is of the form ® i6 / A where Di = DOCC^ is a C;coalgebra. Conversely, if the Di's are Ci- coalgebras, then D = (Big/ A is a Ccoalgebra and Di = Docd. ii) Let D,E be C -coalgebras. Then D is Morita-Takeuchi equivalent over C to E if and only if Di is Morita-Takeuchi equivalent over Ci to Di for all i £ I. in) A C -coalgebra D is C- Azumaya if and only if all the Di are Ci- Azumaya.

THEOREM 4.1.10 Let C = ®ieICit then Br(C] ^Hi€l Br(Ci). PROOF: In light of the above lemma, we may define a map

13 : [D] i—> IIig/[A] with Di = DO^Ci. This map is well-defined in view of the above lemma and because the relation in B(C] is a M-T equivalence relation. Suppose that n ie /[A] = 1 € Hie/ Br(Ci). Then, from Proposition 4.1.7, it follows that A is MT equivalent to Ci for all i e /. Now Lemma 4.1.9 entails that D is M-T equivalent to C and then [D] = [1] in Br(C). Lemma 4.1.9 also yields that j3 is surjective. That J3 is a group homomorphism is deduced from the fact (DOcE}i = DiO^Ei for alH € / and Azumaya coalgebras D, E. I Remark: This decomposition has two important consequences:

1) In general Br(C) is not torsion. Let <$ be the rational number field and C the group like coalgebra C = @n^j^
In view of 2) we may assume that C is cocommutative irreducible and study the Brauer group in this case. For an irreducible C, the following appeared in [25, Prop. 4.10]:

LEMMA 4.1.11 Let C be an irreducible coalgebra, and D an Azumaya C -coalgebra with C-counit e. i) D is irreducible and D* is Azumaya over C* . ii) D is finitely cogenerated free as right C-comodule via e. iii) The map (—)* : Br(C) —> Br(C*), [D] H-> [D*] is a group homomorphism.

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159

If the coalgebra C is in addition coreflexive, it was proved in [25, Prop. 4.12] that the map (-)* : Br(C) -» Br(C*} is injective. Moreover, in this case, if C0 denotes the coradical of C, then induced map z* : Br(C) —> Br(Co) for the inclusion map i : CQ —> C is injective. In [26] a cohomological interpretation of the Brauer group allowed to prove that both these maps are isomorphisms.

PROPOSITION 4.1.12 Let C be a cocommutative coreflexive irreducible coalgebra. Then, the maps i* : Br(C) -> Br(C0) and (-)* : Br(C) -> Br(C*) are isomorphisms. COROLLARY 4.1.13 If C is a cocommutative coreflexive coalgebra, then

PROOF: We write C as direct sum of irreducible coalgebras C = © i€ / Ci. Every Ci is irreducible and coreflexive. From the foregoing proposition it follows that Br(Ci) = Br(Cio) where CJQ denotes the coradical of Ci. Now,

Br(C] =

Br(Ci) = iel

since C0 = @ieICiQ.

Br(Ci0) ~ 5 r ( C i 0 ) ^ Br(C0), iel

i€l

I

With these results we can provide new examples of Brauer groups of coalgebras.

EXAMPLE 4.1.14 1.- The Brauer group of a coreflexive coalgebra over the real number field 1R is isomorphic to a direct product of copies of Zj2- Let C be a cocommutative coreflexive coalgebra, it admits a decomposition as direct sum of irreducible coreflexive coalgebras C = ® ie/ C;. From Theorem 4.1.10, Br(C) ^ Y[iel Br(Ci). For every irreducible subcoalgebra its coradical is isomorphic to JR or (F, the complex number field. Using that Br(]R) ^ %2, Br((H) is trivial and Br(Ci) ^ Br(Ci0) by the above proposition, Br(Ci) is trivial or isomorphic to -2^2 for all i 6 /. 2.- If C is a coreflexive coalgebra over either an algebraically closed field or a finite field, then Br(C) is trivial. It is proved using a similar argument as above and noting that the Brauer group of either an algebraically closed field or a finite field is trivial. 3.- Let V be a finite dimensional vector space over a field of characteristic zero and C the symmetric algebra over V. From [21, Prop. 11.0.10], [14, Th. 3.4.3], C is connected and coreflexive. From Corollary 4.1.13, Br(C) == Br(k). 4.- Let i be a Lie algebra over a field of characteristic cero and C its universal enveloping algebra U(L). By [21, Prop. 11.0.9], [14, Th. 3.4.3], C is connected and

coreflexive. Then, Br(C) ^ Br(k). 5.- Let A be a commutative noetherian algebra. From [14, Th. 3.3.3], the finite dual coalgebra A° is a coreflexive coalgebra. Using 4.1.13, Br(A°) = Br(A^). If in addition A is pointed, then A° is pointed and so Br(A°] = FJj Br(k) for a suitable set /.

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Several questions were pointed out in [25]. 1.- Is the Brauer group of an irreducible coalgebra torsion? It is tempting to conjecture that for a cocommutative coalgebra, the non-torsion aspect comes from the product decomposition but that for each irreducible subcoalgebra the Brauer group is a torsion group.

2.- For an irreducible coalgebra C, is the map (-)* : Br(C) When is it an isomorphism?

—> Br(C*) injective?

In the coming subsection we prove that these questions have affirmative answers. We finish this subsection with a version of the Auslander's problem for coalgebras. The Auslander's Problem for Coalgebras

Let C be a cocommutative irreducible coalgebra. When is the map z* : Br(C) —» Br(Co) an isomorphism? We know that it is true if C is coreflexive. We conjecture that this map is always an isomorphism. 4.2

Torsioness in the Brauer group

In this section we proved that the map (—)* : Br(C) —> Br(C*) is injective. Since Br(C) embeds in a torsion group, Br(C) is torsion. Thus we solve positively the questions presented before. In order to do this we need some preliminaries. Let C be a coalgebra with dual algebra C*. Every right C-comodule may be viewed as a left C"*-module (see [21],[1]). The left C"*-modules arising from a right comodule are called rational C*-modules and the full subcategory of rational left C*-modules is isomorphic to M.c.

Let M be a left C*-module and m G M. We say that the element m is rational if there are families {mi}f=1 C M and {ci}™=1 C C such that • in =

The set of rational elements of M, Rat(M), is a submodule of M. The functor Rat(-) : c'M —»• c*M, M i-» Rat(M) is a left exact preradical (see [16, page 371], [12]). It is well known that there is a bijective correspondence between left exact preradicals, hereditary pretorsion classes and left linear topologies (see [20, VI, Prop. 4.2]). The left linear topology associated to Rat(-) is the class f of all cofinite closed left ideals of C*. We remember that a left coideal / of C* is called cofinite if C* /I is finite dimensional as vector space. / is called closed if there exists a subspace W of C such that I = W^^ '. Hence / is cofinite closed if and only if there is a finite dimensional subspace W of C such that I = W^c \ The following was observed in [16, page 371].

PROPOSITION 4.2.1 Let C be a coalgebra and C* its dual algebra. i) The class J- of all cofinite closed left coideals of C* is a linear topology. That is: a) If I e J- and J is any left ideal of C* such that I C J, then J € J-.

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161

b) If /, J € JF, then I n J £ JF. cj /// 6 JF ana7 c* £ C*, i/ien (/ : c*) e JF.

nj If M is a left C*-module, Rat(M) = {m € M : annc- (TO) e JF}.

mj Aic = {M 6 C..M : Rat(M) = M}. Observe that if C is an infinite dimensional coalgebra, {0} ^ JF. In the sequel we will assume this. J- is also symmetric, that is, any ideal in JF contains a two-sided ideal that belongs to J-. PROPOSITION 4.2.2 Let C be a coalgebra, C* its dual algebra and M & C,M. Then, Rat(M) = {m £ M : annc.(C* • m) £ JF}. PROOF: Let m £ M such that annc-(C* • m) £ T . Since annc*(C* • m) C annc'(m), it follows that annc- (TO) £ JF. By the above proposition, m € Rat(M). Conversely, let TO £ Rat(M), then annc»(m) 6 JF. Let {i>*}i€/ be a basis of annc'(m). We extend this to form a basis of (7*. We only have to add a finite number of elements {y*, ...,y£} because annc*(m) is cofinite. We claim that annc*(C* • m) = n™=i(annC" (m) : y*). Let d* £ annc'(C* • m), then 0 = d* • (y* • TO) = (d* * y*) • TO for all z = l,...,n, where * denotes the product in C* . Hence d* * y* £ annc* (TO), that is, d* £ (annc-(m) : j/*) for all z = l,...,n. Thus, d* £ P|"=1(annc.(m,) : T/*). Suppose now that e* £ Pl™_1(annC7. (TTI) : y*) and let c* be in C*. We can write c* = ELi M, + £?=! Ay* with A,, ft e fc. Then, r

n

e* • (c* • TO) = V^ Aje* • (v*. • TO) + Y^ A(e* * y*) • m = 0, i=l

i=l

since Vji , e* * y* £ annc* (TO.). Hence e* G annc* (C* • m). Since annc*(m) S JF; (annc* '• y*) 6 -^ f°r all * = 1) • • • > n (m) : y*) e JF. I

an

d so annc*(C* •

LEMMA 4.2.3 Le^ C be a cocommutative coalgebra, D a C-coalgebra with C-counit e, and suppose that D is finitely cogenerated free as right C-comodule via e. If I is a cofinite closed ideal in C* , then D*e*(I) is a cofinite closed two-sided ideal in D* . PROOF: Since D is finitely cogenerated free as right C-comodule via e, there is a finite dimensional vector space W such that D = W C as right C-comodules. Let h : D —> W ® C be this isomorphism. Let d £ D, and suppose that h(d) = Y?J=IWJ®CJ with Wj e W,Cj e C, then pw®c(h(d)) = (h®l)pD(d) = (/i<8>e)A(cZ). This yields the equality:

Let / be a cofinite closed ideal in C* , then there is a finite dimensional subspace V of C such that / = V^ . J is a two-sided ideal of C* because C* is commutative. By [21,

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Prop. 1.4.3], V is a subcoalgebra of C. We are going to prove that h*(W* 0 V^) = D*e*(V -1). Let (p e h*(W* 0 I/-1), then there is ^ e VF* 0 V1- such that p - h*(ip). We can write i/j = Y^i=i wl ® & with w* £ W7* and ^ € V1- for all z = 1, ..., n. Then,

Set (i* = h*(w* 0e c ) for i = l,...,n. Then,

= E"=i E(d)« ® £c-> h(d(i)))(ipi, e(d(2))) = E"=i EJ=i E(C.,.)

Hence, ^ = ^=1 dje*(^) e ^e^^1). Thus, /i*(W* ® I/-1) C D*e By linear spaces arguments, it is easy to check that h*(W* ® F-1) = /i*((W ® V)^) = h'l(W ® F)1. We check that D*e*(V- L ) C h~l(W ® V}^ . Let x* e D*e*(VjL], then there is d* e -D*,Vi € y x ,i = l,...,n, such that z* = E"=i d * e *(Vi)- Let d^ h~l(W ®V), then /i(d) e W I/. Set /i(d) = Yfj=\ wj ® ci with wi € ^' c j e ^- Since ^ is a subcoalgebra of C and Cj e V, then Cj(2) € V for all j = 1, ..., s.

= = E"=i EJ=i We conclude that /i-^W7 0 K)1 = ^(V^D*. The space /j-^W 0 F) is finite dimensional because W 0 V is so. Thus we have proved that D*e*(I) is a cofmite closed two-sided ideal. I LEMMA 4.2.4 Let C be a cocommutative coalgebra, D,E C-coalgebras and M a (E, D)-bicomodule. M is a (E,D)-bicomodule over C if and only if M* is a (D* ,E*)-bimodule centralized by C* . PROOF: Let erj : D —> C and e^ : E —> C the C-counits of D and £ respectively. D* and S* are C"*-algebras via e*D and ej; respectively. M* is a (D*, S*)-bimodule via, ,

m-e,m =

(m)

for all TO e M,tf* G Z?*,e* e S* and m* e M*.

(m)

Invariants of Coalgebras

e

163

If M is a bicomodule over C, then Y^(m)m(o) ® £^(^(-1)) = £( m ) m(o) ® z?( m (i)) f°r aH rn £ M. Given c* e C*,m* £ M* and TO e M, we have, {ej;(c*)-m*,m}

= £(m){e|;(c*), TO ( _I))(TO*, TO (O) } = E(m){ C Vs( m (-l)))( m *> m (0)} = E(m)(C*>eD(m(l)))(TO*>m(0)}

= £( m ){ e £>( c *)> m (l))( m *> T O (0)}

= (m* •e*D(c*),m).

Thus e^c*) • m* = m* • e^(c*) for all c* e C*, m* e M*. That is, M* is centralized by C*.

Suppose now that M* is centralized by C*, then f*E(c*) • m* = m* • e*D(c*) for all c* e C*,m" e M*. So,

(m)

(m)

for any m € M. We can view M ® C embedded in (M* C*)* via the map A : M C -+ (M* OC'*)*,(A(m(8)c) ! m* ® c*) = (TO*,TO}{C*,C) for all m* e M*,c* € C*. For m e M, i))),ra*®c*}

=E ( m ) (m*,TO ( 0 )}{c*,£ E (m ( _ 1 ) )} ) TO(0) ® £D(m ( 1 ) )), m* ig) c*},

for all 77i* € M* and c* £ C*. Since A is injective, we obtain

(m)

(m)

for all TO € M. This just means that M is a bicomodule over C.

I

THEOREM 4.2.5 Let C be cocommutative irreducible coalgebra. The group homomorphism ( — )* : Br(C] —> Br(C*), [A] H^ [A*] is injective. PROOF: Let A be an Azumaya C-coalgebra such that [^4*] = [C*] in Br(C*}. The equivalence relation in Br(C*) is the Morita equivalence relation over C*, see [19, Ex. 2.19]. So that, A* and C* are Morita equivalent algebras over C*. We have equivalences over C1*, F

c>M-+ ——- A'M. G Morita's theorem yields the existence of a (A*, C*)-bimodule P and a (C*, A*)bimodule Q such that:

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Since F and G are equivalences over C* , then P and Q are centralized by C* (see [3, page 57]). Let e : A —> C be the C-counit of A, then A* is a C"*-algebra via the injective map e* : C* —> A*. We see that F(Ai c ) C At A . Let M £ Mc arbitrary. By Proposition 4.2.1 ii),iii) we are going to check that ann^*(x} is cofinite and closed for all x 6 F(M). Let

x be in F ( M ) , then x = Y^i-\Pi ® mi f°r Pi € P, TOJ e M. From Proposition 4.2.1 ii), iii), annc> (mi) is cofinite and closed for all i = 1, ...,n. I = P)r=i annc*(ci) is a cofinite closed ideal of C*. We write / = e*(I)A*. Since ^4 is an Azumaya coalgebra over (7, by Proposition 4.1.11, A is finitely cogenerated free as right C-comodule. By Lemma 4.2.3, / is a closed two-sided ideal of A* . Let y € /, then there is aj e A* and c* e 7 such that y = ]CJli a j e *( c j)y x=

m n ^— ^ ^— ^

m

n

m

n

'

J — 1 Z— 1

where in the second equality we have used that P is centralized by C* , and in the last equality that c* vanishes all the m^s. Hence / is a cofinite closed ideal in A* such that I • x = {0}. Then / C anriA->(x), and we conclude that annA,(x] is cofinite and closed. So we have proved that F(M) & MA, and thus, F(M°) C MA. We now check that G(MA) C Mc '. Let N e MA and a; e G(N), then x = Y^i=i 1i ®ni with qtj e Q,rii € ^V for all i = 1, ...,n. Since .AT € Ai"4, ann_4.(A* • nj) are cofinite closed two-sided ideals by Proposition 4.2.2. / = HILi annA*^* -Hi) is a cofinite closed two-sided ideal of A*. Since A* is an Azumaya C*-algebra, there is a bijective correspondence between ideals of C* and two-sided ideals of A* given by (see [19, Cor. 2.11]):

C* -* A*, JH-> JA*,

A* ^C*, A - ^ i f n e * ( C * ) .

Let / = /ne*(C*). Let J be an ideal of C* such that e*(J) = /. Then, given y e J, e*(y) € / and, n

n

ft (g) nj = P gj£* (y) (8)^ =

n

^(81 (e* (y) • n;) = 0,

where we have used that Q is centralized by C* and that e*(j/) 6 /. Hence it vanishes all the o^s. So we have that J C annc*(x). We see that J is cofinite and closed. Since / is cofinite and closed, there is a finite dimensional subspace W of A such that / = W^A'\ We claim that J = e(W)^c"> . Let 9? e e(W)^c"> C C* and 10 e W, then (e*^),™} = () € /ne*(C7*) = 7 and then ),w) = 0. e(W) is finite dimensional because W is so. Hence J is cofinite and closed. It follows that annc* (x) is cofinite and closed. Thus G(N) e Mc ', and so G(MC) C Al-4. Finally, F(MC) C A^A and G(M A ) C yW c . By Theorem 2.0.3 b), C is strongly equivalent to A via F' = F\Mc and G' = G\MA. In view of Theorem 2.0.2 a),

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165

let M be a (C, A)-bicomodule such that F'(-) = -DCM. From Theorem 2.0.3 a), A*M*j. ®c* - = P ®c* -• Then, M* ^ P as (A*, C*)-bimodules. Since P is centralized by C* , M* is centralized by C* . Lemma 4.2.4 yields that M is a (C, j4)-bicomodule over C and so F' is a Morita-Takeuchi equivalence over C. From the Proposition 4.1.7, [A] = [C] in Br(C). Thus we have proved that the map (-)* : Br(C) -> Br(C*) is injective. I It is known that the Brauer group of a commutative ring is torsion, see [19, Th. 12.9]. Then Br(C*) is a torsion group. By the above theorem we have that: COROLLARY 4.2.6 The Brauer group of a cocommutative irreducible coalgebra is a torsion group. 4.3

Subgroups of the Brauer group

In the classical theory of the Brauer group of a commutative ring important subgroups related to group theory appear, the Schur and the projective Schur subgroups. It is natural to ask which are the corresponding subgroups to these in the Brauer group of a coalgebra. A Schur and a projective Schur subgroup can be defined for cocommutative coalgebras by using crossed coproducts. The interest of these subgroups is that Azumaya coalgebras represented by crossed coproducts have a nice structure which comes from the good structure of the crossed coproducts. For example, the Brauer group of a coalgebra may not be torsion, however the Schur subgroup is a torsion group. Let C be a cocommutative coalgebra and G a finite group with identity element e. We regard the Hopf algebra H = (kG)* with basis {pg : g S G} dual to the basis of kG; that is pg(h) = Sg h V
Let a : C —> (kG)* <8> (kG)* be a linear map expressed in the following way: Vc G C,

with ax>y e C* for all x,y £ G. Let C XQ (kG)* be the vector space C <S> (kG)* and we define a map: A a ( c Xph)

= E a a , a - i f t ( c 3 ) c i X Pa. ® C2 X p a - i / i .

C Xa (kG)* is said to be a crossed coproduct if A Q is coassocitative and £c®£(kG)* is a counit. LEMMA 4.3.1 C XQ (kG)* is a crossed coproduct if and only if the following conditions hold: (CU) Normal cocycle condition ag<e(c) = ae,g(c) = e(c) \/g^G,c&C. (C) Cocycle condition

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Vs, r, q e G, c € G. This construction is a particular case of a more general construction where (kG)* is replaced by an arbitrary Hopf algebra H and H coacts weakly on C. All the details of this and an interesting connection of crossed coproducts with Galois coextension may be found in [10]. A map a : C —> (kG)* ® (kG)* verifying (C) and (CU) is said to be a cocycle. If the cocycle a is convolution invertible we say that C Xa (kG)* is a twisted group coalgebra. The map ea = (1 <8> e) : C Xia (kG)* —> C makes C XQ (kG)* into a G-coalgebra. The following proposition lists some properties of twisted group coalgebras, see [6].

PROPOSITION 4.3.2 Let C,D be cocommutative coalgebras, G,H finite groups and a : C —> (kG)*®(kG)*, /3 : C —> (kH)* ® (kH)* convolution invertible cocycles. Then,

i) aop : C -> (kG)*°P ® (kG)*0? £* (kG°P)* (kG0?)* defined by aop(c) = Sz yeG ay,x(c)Px ®py is a convolution invertible cocycle. ii) (C XQ (kG)*)op ^ C Xa°P (kGop)* as C-coalgebras. Hi) The map a x /? : C —» (kG x H)* (/cG x H" )* defined & 2 / a x / 3 = ( l < g > T < g > l)(a /3)A, where r is the twist map, is a convolution invertible cocycle. iv) (C Xa (kG)*)Oc(C Xp (kH)*) ^ C X Q X /3 (kG x H)* as C-coalgebras. v) If f : D —> C is a coalgebra map, then the map a : D —> (kG)* (kG)* given by a(d) = a f ( d ) is a convolution invertible cocycle.

vi) (C Xa (kG)*)OcD ^ D X5 (kG)* as D-coalgebras. vii) a* : kG ® kG —> C* is a normalized cocycle and a*(kG ® kG) C U(C*). viii) (C X\a (kG)*)* ^ C* *Q. kG as C* -algebras. ix) C XQ (kG)* is C- co separable if and only if [G]"1 6 k. From now on, unless otherwise stated, Ji is a class of groups closed under finite products and taking opposites.

DEFINITION 4.3.3 Let A be a C-coalgebra. We say that A is a projective Schur C -coalgebra relative to TL (H-PSC) if A is C-Azumaya and there exists a twisted group coalgebra C XQ (kG)* with G € H and an injective C-coalgebra map i : A —>

C XQ (kG)* . When a is trivial, i.e., a(c) = J^ yeG£(c)px ®py, A is called Schur C-coalgebra relative to H (H-SC).

PROPOSITION 4.3.4 The set PSH(C) = {[A] 6 Br(C) : A is H - PSC} is a subgroup of Br(C). This subgroup is called the H-proyective Schur subgroup of C. PROOF: Let [A], [5] e PSH(C), then there are twisted group coalgebras C Xa (kG)* , C x/3 (kH)* with G, H 6 H. and injective G-coalgebra maps i : A —> C XQ (kG)* and j : B -> C Xp (kH)* . The map J°P : Bop -> (G Xp (kH)*)op is an injective G-coalgebra map and (G Xp (kH)*)op ^ C X()°P (kHop)* by Proposition 4.3.2 ii). Using the left exactness of -DCB°P and C XQ (kG)*ac- it follows that zDl : AOCB°P -> C Xa (kG)*ncBop and lOjop : C Xa (kG)*ncBop ->

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167

C XQ (kG)*OcC Xpop (kHop)* are injective C-coalgebra maps. From Propo-

sition 4.3.2 iv), C XQ (kG)*acC X^P (kHopY op

= C Xaxf3oP

(kG x Hop)*

and the composition (lDj )(iOl) is an injective C-coalgebra map from AOCB°P to C XJ QX /3°p (kG x Hop)*. By hypothesis on H, G x Hop e U and hence

[A][B]°P = {ADCB°P} e PSH(C).

I

COROLLARY 4.3.5 The set SH(C) = {[A] e Br(C) : A subgroup of Br(C) called T-i-Schur subgroup of C .

is

U- SC} is a

If H is the class of finite groups then we denote PSH(C) and SH(C) simply by PS(C) and S(C). PS(C) is called the projective Schur subgroup of Br(C) and S(C) the Schur subgroup. Let [D] e PSU(C] with injective C-coalgebra map i : D -> C XQ (kG)* where C Xa (kG)* is C-coseparable. The set of this classes, denoted by PS^(C), is a subgroup of PS(C) since the cotensor product of two C-coseparable coalgebras is again C-coseparable. All the above definition hold for

PROPOSITION 4.3.6 Let rj : D —=> C be a map of cocommutative coalgebras, then t] induces group homomorphisms 77* : PSW(C) —> PSH(D) and 77*, : 5 W (C) —* S^(D). Hence, P5 W (— ) and 5 W (— ) are contravariant functors from the category of cocommutative coalgebras to abelian groups.

PROOF: We know that there is a group homomorphism 77* : Br(C) —+ Br(D), [A] H-> [AOCD]. We show that the restriction of 77* to PSH(C) has its image contained in PSH(D). Let [A] & P5W(C), then there is a twisted group coalgebra C X\a (kG)* and an injective C-coalgebra map i : A —> C Xa (kG)* for some G 6 7Y. By the left exactness of — OCD, we have an injective Z?-coalgebra map

iOl : AncD -> C XQ (fcG)*D c £). Since C xa (fcG)*D c D ^ D Xfe (fcG)* by Proposition 4.3.2 vi), it follows that [AUCD] € PSH(D). I As in the Brauer group, when the coalgebra is finite dimensional, the study of the 7i-projective Schur is equivalent to the study of the W-projective Schur subgroup

of the dual algebra C* .

PROPOSITION 4.3.7 Suppose thatC is a finite dimensional cocommutative coalgebra and let A be a C-coalgebra. Then, i) A is a projective Schur (resp. Schur) C-coalgebra relative to Ti if and only if A* is a projective Schur (resp. Schur) C* -algebra relative to TL. n) PSH(C) ^ PSH(C*) (resp. SH(C) ^ Sn(C*)) mapping [A] ^ [A*]. The behaviour of the Ti-projective Schur functors is not so good as the Brauer one as the following proposition shows. However, this result will be useful sometime.

PROPOSITION 4.3.8 Let {Ci}i€l be a family of subcoalgebras of C such that C = ®i€lCt. Then, PSH(C) -> Tli6/ PSH(d) and SH(C) <-+ Yli&1 ^(Q).

PROOF: By Theorem 4.1.10, the map 77 : Br(C) -> Hi€l Br(Ci), [D] >-> Y[i€l[Di] is a group isomorphism. Viewing 77 restricted to P5W(C) we obtain a group monomorphism, and its image is contained in Hig/ P5^(Cj). Let D be a Jiprojective Schur coalgebra, reasonig as in Proposition 4.3.6 for each C{ we obtain

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that DI = DOcd is a proyective Schur Gj-coalgebra relative to H for all i s /. Therefore IL e /[A] € lL6, ^«(G Z ).

•

The nice structure of the twisted cogroup coalgebra makes the following result possible though the Brauer group is not torsion.

PROPOSITION 4.3.9 Let C be a cocommutative coalgebra over a field k, then is a torsion group. If char (k) — 0 then S (C) is torsion. PROOF: Let [D] € S^(G), then there exists an injective G-coalgebra map i : D —> C® (kG)* with G G T~i and JGJ" 1 G k. Using the universal property of the cocenter it may be proved that the cocenter of C ® (kG}* is C Z(kG)*, where Z(kG) denotes the center of kG. kG is Azumaya over Z(kG) because \G\~l € k. Then we can find n 6 IN such that (kG)n ^ Mm(Z(kG)) = Z(kG) ® Mm(k). Denote by Dn the cotensor product of D n times. By Proposition 4.3.2 vi), there is an injective G-coalgebra map

i' : Dn -> C (kGn)* ^ G Mm(Z(kG))* ^ C Z(kG)* Mm(k}* . We can consider Dn as a subcoalgebra of C ® Z(kG)* Mm(k)* . Since G Z(kG)*®Mm(k}* is an Azumaya G®Z(A:G)*-coalgebra, by [25, Cor. 3.17] there exists a subcoalgebra C' of C®Z(kG)* such that Dn = C' ®Mm(k)* as C-coalgebras. Hence their cocenters are isomorphic, i.e, C = C' and thus Dn = C ® Mm(k)* as G-coalgebras. This means that [L>n] = [C] in Br(C). If char(k) = 0, then the group coalgebra C <8> (kG)* is always C-coseparable by Proposition 4.3.2 ix). Hence 5^(C) = SH(C). I

In the irreducible case, the injectivity of the map (— )* : Br(C) entails the following:

—> Br(C*)

PROPOSITION 4.3.10 Lei C be a cocommutative and irreducible coalgebra then there are injective group homomorphisrns PSU(C) -+ PSH(C*), and SH(C) -> SH(C*). PROOF: Theorem 4.2.5 yields the existence of an injective group homomorphism (-)* : Br(C) -» Br(C*), [A] ^ [A*]. The restriction of this map to P5 W (G) is an injective group homomorphism and its image is contained in P5'W(C'*). Let [A] e PSH(C), then there is a twisted group coalgebra C XQ (kG)* with G <E H and an injective C*-coalgebra map i : C X (kG)* —> A. Dualizing this map and using Proposition 4.3.2 viii) we obtain a surjective C*-algebra map from C* *Q* G into A*. This fact combined with the fact that A* is G*- Azumaya (by Proposition 4.1.11) leads to [A*] € PSH(C*). For SH(C) we use the same argument taking into account that a* is trivial when a is trivial. I

We may deduce some properties of S(C) from properties of S(C*). The following result is an example of this.

COROLLARY 4.3.11 If C is a cocommutative coalgebra over a field with characteristic different from zero, then S(C) is trivial.

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169

PROOF: We suppose that C is irreducible, then S(C) ^-> S(C*}. Since k has non zero characteristic, C* is a fc-algebra of positive characteristic and applying [11, Prop. 1], we have that S(C*) is trivial. Hence S(C) is trivial. If C is not irreducible then, since it is cocommutative, there exists a family of irreducible subcoalgebras {Ci}iel such that C = ® ie /C;. From Proposition 4.3.8, S(C) ^-> Y[i€lS(Ci), but S(d) = {0} for all i 6 /, thus S(C) = {0}. I

COROLLARY 4.3.12 Let C be a cocommutative coreflexive irreducible coalgebra. If C0 is separable, then PSH(C) ^ PSH(C0) and SH(C) ^ SH(C0). In particular, ifC is connected, then PSn(C) ^ PSH(k) and SH(C) ^ SH(k}. PROOF: Proposition 4.1.12 claims that the inclusion map i : CQ —-> C induces an injective group homomorphism i* : Br(C~) —> fir (Co). Then, the restriction to PSH(C), z*» : PSH(C) -> PSH(C0) is also injective. Since C0 is separable, the Malcev-Wedderburn decomposition for C yields the existence of a coalgebra map TT : C —> Co which splits the inclusion map i : CQ —> C, i.e., wi = Ic0- By the functorial properties of P5 W (—), i**?!-** = \PSn^c}- Then z** is surjective and hence an isomorphism. I

EXAMPLE 4.3.13 1.- Let V be a finite dimensional vector space over a field of characteristic zero and C the symmetric algebra over V. We know that C is connected and coreflexive. By the above corollary, PSH(C) ^ PSH(k), and SH(C) ^ SH(k).

2.- Let L be a Lie algebra over a field of characteristic cero and C its universal enveloping algebra U(L). Since C is connected and coreflexive, PS'W(C') = PS'!~i(k),

and SH(C) ^ SH(k).

Acknowledgments J. Cuadra is grateful to Professors J.R. Garcfa Rozas and B. Torrecillas for their comments and ideas in the proof of Theorem 4.2.5.

REFERENCES [1] E. Abe, Hopf Algebras, Cambridge University Press, 1977.

[2] M. Auslander and O. Goldman, The Brauer Group of a Commutative Ring, Trans. Amer. Math. Soc. 97 (1960), 367-409. [3] H. Bass, Algebraic K-Theory, Benjamin, New-York, 1968.

[4] M. Bolla, Isomorphism between Endomorphism Rings of Progenerators, J. Algebra 87 (1984), 261-284. [5] W. Chin and S. Montgomery, Basic Coalgebras, Collection Modular interfaces (Riverside, CA, 1995), 41-47.

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[6] J. Cuadra, J.R. Garci'a Rozas and B. Torrecillas, Subgroups of the Brauer Group of a Cocommutative Coalgebra. J. of Algebras and Representation Theory, 3 (2000), 1-18. [7] J. Cuadra, J.R. Garci'a Rozas, B. Torrecillas and F. Van Oystaeyen, On the Brauer Group of a Cocommutative Coalgebra, to appear in Comm. in Algebra, 2001.

[8] J. Cuadra, J.R. Garci'a Rozas and B. Torrecillas, Outer Automorphisms and Picard Groups of Coalgebras, to appear Revue Roumaine de Mathematiques Pures et Appliquees, 2000. [9] C.W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. II. Wiley, 1987. [10] S. Dascalescu, S. Raianu and Y.H. Zhang, Finite Hopf-Galois Coextensions, Crossed Coproduct and Duality, J. Algebra 178 (1995), 400-413.

[11] F. DeMeyer and R. Mollin, The Schur Subgroup of a Commutative Ring, J. Pure and Applied Algebra 35 (1985), 117-122. [12] J. Gomez Torrecillas, Coalgebras and Comodules over a Commutative Ring, Revue Roumaine de Mathematiques Pures et Apliquees 43 (1998), 591-603. [13] J.A. Green, Locally Finite Representations, J. Algebra 76 (1982), 111-137.

[14] R.G. Heyneman and D.E. Radford, Reflexivity and Coalgebras of Finite Type, J. Algebra 28 (1974), 215-246. [15] B. I-Peng Lin, Morita's Theorem for Coalgebras, Comm. in Algebra 1 No. 4 (1974), 311-344. [16] B. I-Peng Lin, Semiperfect Coalgebras, J. Algebra 49 (1977), 357-373. [17] C. Nastasescu, B. Torrecillas and F. Van Oystaeyen, IBN for graded Rings. To appear in Comm. Algebra.

[18] P. Nelis and F. Van Oystaeyen, The Projective Schur Subgroup of the Brauer Group and Root Groups of Finite Groups, J. Algebra 137 (1991), 501-518. [19] M. Orzech and C. Small, The Brauer Group of a Commutative Ring, Lecture Notes in Pure and Applied Mathematics 11, Marcel-Dekker, New-York, 1975.

[20] B. Stentrom, Rings of Quotients, Springer-Verlag, 1975. [21] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. [22] E.J. Taft, Reflexivity of Algebras and Coalgebras, Amer. J. Math. 94 (1972), 1111-1130.

[23] M. Takeuchi, Morita Theorems for Categories of Comodules, J. Fac. Sci. Univ. Tokyo 24 (1977), 629-644. [24] B. Torrecillas and Y.H. Zhang, The Picard Groups of Coalgebras, Comm. in Algebra 24 No. 7 (1996), 2235-2247.

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[25] B. Torrecillas, F. Van Oystaeyen and Y.H. Zhang, The Brauer Group of a Cocommutative Coalgebra, J. Algebra 177 (1995), 536-568. [26] F. Van Oystaeyen and Y.H. Zhang, Crossed Coproduct Theorem and Galois Cohomology, Israel J. of Mathematics 96 (1996), 579-607.

Multiplication Objects J. ESCORIZA and B. TORRECILLAS,

Dpto. Algebra y Analisis Matematico,

Universidad de Almeria. 04120-Almeria. Spain. l: [email protected] and [email protected]

Abstract

A concept of multiplication object in monoidal categories is given and it is proved that it generalizes that of multiplication module, multiplication module relative to a torsion theory and multiplication graded module. It is proved that the endomorphism ring of an indecomposable multiplication object having the descending chain condition on multiplication subobjects is local in any braided monoidal category.

1

INTRODUCTION

Multiplication modules over a commutative ring are narrowly related to finitely generated, projective or distributive modules, sharing important properties with some of these families. As for commutative multiplication rings, they are of interest in multiplicative ideal theory (see [14]). Hereditary rings and von Neumann regular rings can be studied from the common perspective of being multiplication rings. By defining the concept of multiplication module relative to a torsion theory we can study completely integrally closed domains as multiplication rings with

respect to the canonical torsion theory, i.e., the one induced by the height one prime ideals, and therefore some divisorial properties can be investigated in this setting. The graded concept of multiplication ring (a gr-multiplication ring) is useful in order to characterize graded rings which are multiplication rings (cf. [9]). Moreover, some arithmetically graded rings which are important in Algebraic Geometry such as Dedekind graded domains or generalized Rees rings are examples of gr-multiplication rings (cf. [26]). The study of multiplication objects in monoidal categories provides us with a common framework to consider the concept in the different above-mentioned situations. For example, in order to prove that a strongly graded ring R is a multiplication ring, we can consider the Bade equivalence between the category of graded /^-modules and the category of .Re-modules and then 173

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apply the properties of multiplication objects in these monoidal categories (cf. [9] and [10]). Monoidal categories were introduced by Benabou and Mac Lane in 1963 (cf. [2] and [16]). They are, between others, responsible for the narrow relationship between Quantum Groups and Knot Theory and constitute the adequate framework for Hopf algebra representations (see [13]). The concepts of multiplication module and multiplication module relative to a torsion theory (cf. [5] and [6]) have already been studied from a categorical point of view in [8], where the concept of multiplication object in a commutative Grothendieck category is established. However, the natural definition of multiplication graded module, also investigated in [8, Section 5], is not the particularization of the definition of multiplication object in the category gr-_R of graded right .R-modules and degree preserving homomorphisms. This problem is solved in this work where monoidal categories turn out to be the ideal setting to research into all these concepts, including that of multiplication graded modules. The aim of this paper is to study multiplication objects in monoidal categories and some particular cases such as the category of right C-comodules, C being a cocommutative coalgebra or the category of Doi-Koppinen modules. The paper is organized as follows: Section 2 is devoted to introduce monoidal categories, giving examples of them and explaining their elementary properties. In Section 3, some general properties of multiplication objects are obtained. The study is illustrated with some special cases. In Section 4, we obtain some results on endomorphisms of multiplication objects in braided monoidal categories. We have proved that the endomorphism ring of an indecomposable multiplication object which verifies the descending chain condition on multiplication subobjects is local in any abelian braided monoidal category. Throughout this work every category is abelian.

2

MONOIDAL CATEGORIES

A category C is said to be a monoidal category (some authors call this a tensor category) if there is a bifunctor <8>c : C x C —» C (called tensor product), an object I (called the unit of the monoidal category) and natural isomorphisms

a{A,B,c} • (A ®c B) ®c C -> A ®c (B ®c C), 1A : I ®c A -> A,

rA : A c I -> A,

for each A,B and C in C, verifying the following formulas, which will be briefly denoted by (1), (2), (3) and (4) and where / : A -> A', g : B -> B1 and h : C -> C" represent morphisms in C.

f ®c (9 ®c h) o a{A>B^} = a{A',B',c'} ° (I ®C g) ®c h

1A> o (17 <s>c /) = / o 1A, rAi o (/ ®c \i) = f o rA

(IA ®c IB) ° a{A,i,B} = TA ®c IB a

{A,B,C®cD} ° a{A®cB,C,D} = (^A ®C a{B,C,D}) ° a{A,B®cC,D] ° (a{A,B,C} ®C ID)

Property (3) is known as the triangle axiom and property (4) is called the pentagon axiom. In case the isomorphisms a{A,B,c}>^/i and rA are identities, C is said to

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175

be a strict monoidal category. The category defined above will be denoted by (C,®c,/,a,/,r). It is well-known that in every monoidal category,

(see, for example, [12, Proposition 1.1]). Another important fact for our purposes appears in [13, Proposition XI.2.4], where it is stated that the set of endomorphisms of the unit, End(I), is a commutative monoid for the composition. Moreover, identifying I<S)cI with /, the tensor product coincides, in End(I), with the composition. Given two monoidal categories C and D, with units / and /' respectively, a functor F between them is said to be a monoidal functor if there exists an isomorphism $0 : /' -» FI and a family of isomorphisms $2,A,B : FA ® FB —> F(A ® B), verifying the following equalities, which will be called ful, fu2 and fu3 respectivily: $2,A,B®C ° (Ip A ® $2,B,c)°a{FA,FB,FC}

= F(a{A,B,C})°®2,A®B,C ° (®2,A,B <8> IFC)

rFA = F(rA) o $2,^,7 ° UFA <8> $o)-

W = F(1A) o $ 2 ,/,A o (*o ® IFA). When the functors intervening in an equivalence of categories are monoidal functors, the categories are called monoidal equivalent. Notice that the two tensor products have been denoted in the same way. EXAMPLES. We recollect some examples of monoidal categories, most of them wellknown. The category Vect(K) of vector spaces over a field K, with / = K and the tensor product being that of vector spaces. The maps I and r are defined by 1(1 ® v) = r(v <S) T) = v for all v 6 V (V being an object in the category). The category Mod-K[G] of representations of G over K or, equivalently, of K[G}-modu\es is a submonoidal category of Vect(K) with the structures g(u <8> v) = gu ® gv and gk = k for all g e G, u, v 6 V and k € K. Any monoid, regarded as a discrete category is a strict monoidal category, where the tensor product is the multiplication. The category Ab of abelian groups with the usual tensor product and with 2Z

as the unit. The category of right ^-modules, Mod-/?, when R is commutative, with the usual tensor product and R being the unit. The category of K-algebras with the tensor product of /("-algebras, K being the unit.

The category of all /?-/?-bimodules, R being any ring, with the tensor product over R. The opposite category of any monoidal category. The category ( B-M,®,K,a,l,r) of left B-modules, B being a bialgebra. Let gr-R be the category consisting of all .R-modules graded by a group G, R being a commutative ring and morphisms preserving the degree. Then, the category gr-R is a monoidal category considering R as the unit, defining the gradation for the tensor product of two /?-modules, M and N, by means of (M <S> N)g, which is the additive subgroup generated by the elements x ® y with x £ Mh, y & NI such that Ih = g (see [19, p. 12]) and with maps a, I and r as in Mod-/?.

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If H is a Hopf algebra over a field K, the category of all representations of H is monoidal (cf. [4, Example 5.1.4]). Numerous examples, different form the above ones, can be found in [4] and [28].

A Grothendieck category, i.e., a cocomplete abelian category with exact direct limits and with one generator (see [23, Chapter V]) is called commutative if it has a generator U such that its endomorphism ring, R = Endc(U), is commutative. Such a generator is called endocommutative. By the Gabriel-Popescu theorem, these categories are just the categories equivalent to quotient categories of a category of modules over a commutative ring with respect to a hereditary torsion theory, i.e., the ones equivalent to a category of the form Mod-(R,r), consisting of the r-injective and r-torsion-free /^-modules (cf. [23, Chapter X]). Therefore, they preserve many properties of categories of modules over commutative rings. They have been studied in [1] and the references there. By the Gabriel-Popescu theorem, there is an equivalence between any Grothendieck category C with generator U and Mod-(R, T), where R = Endc(U). Using notations as in Stenstrom's book (cf. [23]), we consider the adjoint functors i : C —> Mod-/? and a : Mod-.fi! —> C. Let T : C -^Mod-R be the functor given by T(C) = Homc(U,C) and the induced functor T' : C —* Mod-(R, T), the latter being an equivalence of categories.

THEOREM 2.1 If C is a commutative Grothendieck category, then there exists a monoidal category of the form (C,<3>c,U,a,l,r), where U is an endocommutative generator of C.

Proof. Let T be the torsion theory which arises through the Gabriel-Popescu theorem applied to C. The tensor product making Mod-(R,r) into a monoidal category will simply be denoted by . Then, for any objects A,B in C, one defines A <8>e B = G(T'(A) ® T'(B)), where G is the inverse equivalence of categories of T1 . The natural isomorphism between (A ®c B) ®c C and A ®c (B <S>c C} will be denoted by O,{A,B,C}- This isomorphism
(A ®c B)®CC

= G(T'(A ®c B) ® T'(C)) = G(T'G(T'(A)

and, therefore, it is defined as

where I^M represents the natural isomorphism between T'G(M) and M. Let IA = TIA ° G(lx'(A)) and rA = t]A ° G(rT>(A))i where 77,4 is the natural isomorphism between GT'(A) and A. The tensor product of morphisms is defined as / ®c 9 =

Now, we point out that the category Mod-(R,r), R being a commutative ring, is a monoidal category. The proof can be found in [15, Proposition 3.2] or in [11, Proposition 1.2.2]. Say that M ® N = a(M ®R N), that the unit is a(R), IM and TM are the ones induced by the multiplication of M and R. The tensor product of morphisms is defined by / ® g = a(f <S>K
and for this reason is omitted (all details can be found in [7]). I

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The following result shows that, because of the form in which the tensor product has been defined, the equivalence T' is a monoidal equivalence.

THEOREM 2.2 Given a commutative Grothendieck category C, equivalent to Mod(R,r) by means of the functor T', then T' is a monoidal equivalence between C (by using the tensor product defined above) and Mod-(R,r). Proof. The isomorphism <&o is IT'(U) and ^2,A,B is ^T^A^T'IB)' that the necessary properties are verified. With our notation,

0<{FA,FB,FC}

Let us check

=

'(C\> ° aT'(A),T'(B),T'(C)}-

By naturality, we have

F(a{A,B,C}} (a{AiB,C}) ° ^ ) } ° a{T'(A),T'(B),T'(C)}

and this expression is equal to the above one by simplifying the last two parentheses. Thus ful is verified. In our case,

T'(rA) o $ 2>y4j/ o (1T,(A} ® $ 0 ) = T'(rA) o

By applying T' to the equality rA = r/A o G(rT'(A)), it follows that

T'(rA) = T'(r,A)oT'G(rT,(A}) = ^T>(A) °T'G(rT,(A)). By naturality, the second member of the last equality is r-j-'(A) ° 4>T' (A)®T> (U) • By substituting, T' (rA) O $ 2ij4i/ o (1T,(A) ® $0) =• rT'(A] ° ^T'(A)®T'(U)^T\A)®T'(U)

= TT'(A)

which proves property fu2. The proof for fu3 is totally analogous and is therefore omitted. I

3

GENERAL PROPERTIES OF MULTIPLICATION OBJECTS

Throughout this section (C,®c,I,a,l,r) (or simply C) stands for a monoidal category. The tensor product <S>c will be substituted by ® when no confusion arises.

DEFINITION 3.1 An object X ofC is a multiplication object if every subobject Y of X can be written as Y = l\ (J <8>c -X") = TX (X ®c H] for some subobjects H, J of I.

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In the proof of most results below only the condition using rx is verified, the condition with lx, being similar. Recall that an object S is called simple if it has no subobjects except 0 (if the category has a zero object) and itself. It is easily seen that the unit of every monoidal category and every simple object are examples of multiplication objects. An easy computation allows us to obtain the next result.

PROPOSITION 3.2 Let (C,<8,/,a, l , r ) be a monoidal category. Assume that X is a multiplication object in C and f : X —> Z is a morphism in C. Then, f ( X ) is a multiplication object in C.

REMARK 3.3 Since a quotient object of X is the image of X by the canonical projection, we have that every quotient object of a multiplication object is also a multiplication object. The following theorem shows that multiplication objects are preserved by monoidal equivalences. THEOREM 3.4 Let F : C —> £> be a monoidal equivalence between two monoi-

dal categories. Then, X is a multiplication object in C if and only if F(X) is a multiplication object in TJ.

Proof. Since F is an equivalence of categories, every subobject of F(X) is of the form F(Y), where Y is an subobject of X. As X is a multiplication object, Y = lx(J <8> X) = rx(X ig> H), for some subobjects J, H of /. By applying F,

we obtain that F(Y) = F(lx)(F(J ® X)) = F(rx)(F(X H ) ) . On the other hand, F(J®X) = $2 jx(FJ ® FX) = $2 jx ° ($o ® lpx)(J ® FX). Therefore, F(Y) = [ F ( l x ) o $2,j,V o ($0 ® 1FX)](J ® FX). By fu3, F(Y) = 1FX(J ® FX). The other condition is proved similarly. I REMARK 3.5 From the proof of Theorem 3.4, it is clear that every dense monoidal functor preserves multiplication objects.

Let C be a commutative Grothendieck category with an endocommutative generator, U, and R = Endc(U). Let X be an object of C. Given a morphism r G R, since C is commutative, it is possible to consider, for X, the morphism induced by the multiplication by r, i.e., r* : T(X) —> T(X), given by r*(a) — a o r for every morphism a from U in X. Since T is a full functor, there exists an endomorphism r** de X, such that T(r**) = r*, i.e., r** oa = aor for every morphism a : U —> X (since T is faithful, this assignment is unique). By taking into account all these considerations, we gave the next definition in [8]. An object X in C is called a multiplication object if for any subobject Y of X there exists a family of morphisms {ri}i£i such that Y = X]ig/ 7 "i*(-^0 ^or some endocommutative generator U. Let R be a commutative ring, T a hereditary torsion theory over R and M an -R-module. The functor a : Mod-R —>Mod-(R,r) maps the .R-module M into its module of quotients with respect to T, the E-module MT (see [23] for further information). The following result shows that the definitions of multiplication object in monoidal categories and commutative Grothendieck categories coincide.

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THEOREM 3.6 Let C be a commutative Grothendieck category with endocommutative generator U and let (C, ®c, U, a, I, r) be the corresponding monoidal category. An object X of C is a multiplication object in C if and only if it is a multiplication object in (C,®c,U,a,l,r). Proof. Let R = Endc(U). Then the Gabriel-Popescu theorem says that C is equivalent to the category Mod-(R,r) via T'. Every object of Mod-(J?, r) is of the form a(M) for some M £ Mod-/?. So, a(M) is a multiplication object in the monoidal category Mod-(R,r) if and only if, for every subobject a(N) of a(M), there exists an ideal A of R such that a(N) = r 0 ( M )(a(M) a(A)) = a(M).a(A). By following the same steps as in the proof of [8, Theorem 4.4], this is equivalent to say that a(M) is a multiplication object in the commutative Grothendieck category Mod-(R,r). Since T' is an equivalence of commutative Grothendieck categories mapping an endocommutative generator to an endocommutative generator, by [8, Proposition 3.6], the object X of C is a multiplication object if and only if T'(X) is in the commutative Grothendieck category Mod-(R,r). On the other hand, T' is also a monoidal equivalence. By Theorem 3.4, X is a multiplication object in the monoidal category C if and only if T'(X) is in Mod-(.R, T). At the beginning of the proof, we noticed that, in Mod-(R,r), the two interpretations of multiplication objects are identical, which finishes the proof. I

As examples, we shall characterize multiplication objects in some particular cases. EXAMPLE 1. Graded R-modules. Recall that if R is a commutative ring, then a graded .R-module M is a multiplication graded module if every graded submodule N can be written as N = AM for some ideal A (which can be taken graded) of R. Multiplication graded modules and rings are studied in [8], [9] and [10] but only in some cases the category of graded _R-modules is a commutative Grothendieck category. Now, we show that this concept coincides in any case with that of multiplication object in the corresponding monoidal category.

PROPOSITION 3.7 An object M, in the monoidal category gr-R, is a multiplication object if and only if M is a multiplication graded (gr-multiplication) R-module. Proof. The object M is a multiplication object in gr-R if for every graded submodule, N, of M (N
N = rM(M ® A) — 1M(A® M).

Since MA = rM(M ® A), this is equivalent

to say that for every N
Given a graded /^-module M, the g-suspension of M is the -R-module M(g) with the gradation defined by (M(g})h = Mhg for every h G G. It is well-known that the functor Tg from gr-R to itself and defined by TgM = M(g) is a monoidal equivalence of categories. As a consequence of Proposition 3.7 and Theorem 3.4,

for any g € G, M is a multiplication graded R-module if and only if M(g) is. If .R is a ring graded by G, we can consider the group ring R[G] with the following gradation: (R[G])g = QheG^-gh-1^ (see [20])- Then, Dade's Theorem yields that the categories Mod-/? and gr-R[G] are equivalent. Moreover, this equivalence is clearly monoidal. By Theorem 3.4, R is a multiplication ring if and only if the ring R[G] is a gr-multiplication ring.

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EXAMPLE 2. Recall from [13, Proposition XV.1.2] that a quasi-bialgebra is a Kalgebra A with algebra morphisms A : A —> A <8> ^4 and e : A -+ K such that the category of left A-modules (A — Mod, <8>^, K, a, l , r ) is monoidal, where ®K is the usual tensor product in K and a,/, and r are the usual morphism families. In [21, Theorem 2], the dual concept appears: a semi-coalgebra is a coalgebra with coalgebra morphisms M : C <8> C —> C and u : K —> C such that the category of right C-comodules (M°,®Ki K,a,l,r) is monoidal (a,l,r are as usual).

PROPOSITION 3.8 If A is a quasi-bialgebra, then an A-module M is a multiplication object in (A — Mod,(g>K,K,a,l,r) if and only if M = 0 or M is simple. If C is a semi-coalgebra, then a C-comodule M is a multiplication object in (M. ,®K,K,a,l,r) if and only if M = 0 or M is a simple comodule, i.e., it has no non-zero proper subcomodules. Proof. Suppose that M is a multiplication object in A-Mod. Then for any Asubmodule N there exist submodules / and /' of K such that N = rj^(M # /) = IM(I1 ® M). But the only possibilities for / and /' are 0 and K. Consequently, M has to be simple. Conversely, every simple module is clearly a multiplication object. The proof for comodules is analogous. I

EXAMPLE 3. Consider a family of groups (G,) i6 jv with G0 = {1} and group morphisms pn,m '• Gn x Gm —> Gn+m for any (n,m) 6 IN x IN. We define n®m —

and potn = pnfl = \Gn f°r

an

Y n,m,p € IN (after natural identification), then the

category (Q, ®, 0,1,1,1) is monoidal, where the class of objects of Q is IN, the set

Homg(i,j) is 0 if i ^ j and Gj if i = j and the composition in Homg(i,i) is the group operation (see [13, XI.3.2]). Since in this category the only subobject of n is n itself, every object is a multiplication object. EXAMPLE 4. Multiplication comodules. Throughout this example (C, A,e) denotes a cocommutative coalgebra over a field K with comultiplication A and counit e. The category of right C-comodules will be denoted by M.c. We follow the notation of [24]. If M e M°, then its structure map will be WM '• M —> M (giC1. The corresponding structure map of M as a left C-comodule will be denoted by w'M. Recall that given two right C-comodules M and N, the cotensor product of M and N, MDCN, is the kernel of the map a = WM®^-N-^M®W'N : M<8>JV -» M®C®N. We write IM,N f°r the injection from MO^N into M N. By the definition of kernels, there exists a unique morphism 99 : M —> MOCC such that iM,N0f — WMMoreover, it is well-known that it is an isomorphism. The inverse isomorphism of tp will be denoted by TM- In the same way, one defines the isomorphism IM '•

CacM -> M. The isomorphisms aL>M,N '• (LOCM)OCN

—> Lnc(MOcN) are

induced by the associativity isomorphisms from (L<8> M) <8> N to L® (M ® N). It is easily seen that (Mc', Oc, C, a, I, r) is a monoidal category. As a consequence, with the above notations and by taking into account the cocommutativity, we are able to give the following definition. Henceforth, for simplicity of notation, we write D instead of DC-

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DEFINITION 3.9 A right C-comodule M is called a multiplication comodule if for every C-subcomodule N there exists a right coideal D ofC such that ru(MOD) — N or, equivalently, MOD =

DEFINITION 3.10 A cocommutative coalgebra C is a multiplication coalgebra if every right coideal is a multiplication C-comodule, with its natural structure.

Any cocommutative coalgebra C is clearly a multiplication C-comodule. A right C-comodule M is called cyclic if it is isomorphic to a quotient of the form C/I for some right coideal / of C. From Remark 3.3, every cyclic right comodule is a

multiplication comodule. A cosemisimple coalgebra is a coalgebra which is the sum of its simple subcoalgebras. Then, by Remark 3.3, every cosemisimple cocommutative coalgebra is a multiplication coalgebra.

REMARK 3.11 There are right comodules which are multiplication objects in Jv{c but they are not multiplication C*-modules.

Let S = {CQ, GI, ....} and C the divided power coalgebra, that is, C = KS with A(CJ) = Y^J-OCJ ® ci-i and £(ci) = ^o.i- C is obviously a cocommutative coalgebra and therefore, C is a multiplication C-comodule. It is well-known that C* is isomorphic to the power series ring ^[[X]]. The C*-modules of C are of the form 0, C and Y^=oKci for some n e ^+ ( cf - I 24 > P- 44 D- Then, the morphism of C*-modules (K([X]}/(xn+1)) = M'n given by

i=0

i=0

is clearly an isomorphism. Therefore c*C = IJngW M'n where M'n C M^+1 for all n e IV. But this is not a multiplication C*-module as there is no ideal / of /C[[X]] TS- r r V"ll such that C = /^J 1 ./. This proves the statement. Recall from [25] that a C-comodule M is invertible if there exists another Ccomodule M' and isomorphisms of C-comodules / : C —> MOM' and g : C —> M'DM such that a) (idMng) o WM = ct{M,M',M} ° (l^idM) o w'M. b) (idM>tJf) o WM> = a{M',M,M'} ° (gOidM,) o w^,. The next proposition provides us with new examples of multiplication comodules. We use the above notation. PROPOSITION 3.12 If M is an invertible C-comodule, then M is a multiplication object in M.c. Proof. Let L be a subcomodule of M. We shall prove that MOg~1(M'OL) and WM(L) are equal by checking that their images under the monomorphism i coincide. Moreover, (idM^g}(MUg~l(M'OL)) = MD(M'OL). We have = a{M,M',M} ° = a{M,M',M}

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By using b), one proves in the same way that w'M(L) = /~ 1 (LDM')DM. I Let M € Mc and take m € M. Then, there exists a minimal subcomodule of M, which is moreover finite-dimensional, containing m (see [18, Theorem 5.1.1]). It will be denoted by < m >. This notation will be used in the sequel.

PROPOSITION 3.13 A C-comodule M is a multiplication comodule if and only if for every m G M there exists a right coideal I of C such that WM(< m >) = Proof. Suppose that for every m G M, WM(< m >) = MD/ m for some right coideal Im of C. Let N be a subcomodule of M. Then, WM(< n >) = MD/ n for every n G N. Let us consider / = X)neAf^ n - Since the sum of subcomodules is a subcomodule, / is a right coideal. We have WM(N) = WM(Y^neN < n >) — ) . The necessity is clear. I EXAMPLE 5. Doi-Koppinen modules. Consider the category A-M(H)C of leftright G-Doi-Koppinen modules and ^-linear (7-colinear homomorphisms, where the triple G = (H, A, C) is a monoidal Doi-Hopf datum over a commutative ring R (see [3] for definitions and further information). By [3, Proposition 2.1], the category ( AM(H)C,®R, R, a, l , r ) is monoidal. Given two Doi-Koppinen modules N C M, we write (N : M) for {r G R;rM C JV}. Then, we have a characterization of multiplication object similar to that of multiplication modules.

PROPOSITION 3.14 A Doi-Koppinen module M is a multiplication module if and only if N = M(N : M) for any Doi-Koppinen submodule N of M . Proof. By following the remarks of [22, §1], it suffices to prove that (N : M) is a Doi-Koppinen submodule of R. It is clear that it is an ideal of R. Let EA be the counit of A, a G A and r G (N : M). Then, a.r.M = £A(a).r.M C £A(a)N C N. It follows that (N : M) is a left ^4-module. Let WR be the structure map for the comodule R, then WR(T) = r ig) 1 for every r G R. Since U>(/V : M) = WR\(N : M), i.e., the restriction of WR to (N : M), (N : M) is also a right C-comodule. The compatibility conditions are verified as every element of (N : M) is an element of

R. I 4

ENDOMORPHISMS OF MULTIPLICATION OBJECTS

DEFINITION 4.1 A monoidal category ( C , ( S ) , I , a , l , r ) is called braided if there exists a family of natural isomorphisms CA,B '• A®B —> B®A verifying the following axioms: (Bl) (g <8> /) o CA,B = CA',B' ° (/ ® A', g : B —> B' . C

A,B®C ° a{A,B,c} = (Is ® CA,C) ° O>{B,A,C} ° (CA,B ® l c )I (S3) ac,A,B ° A®B,C ° aA,B,c = (CA,C ® IB) ° aA,c,B {A,c,B} ° ( A ® CB,C)C

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When the braided category is strict as a monoidal category and, moreover, CA,I — CI,A = IA f°r aH objects A of C, the category C is a strict braided category.

LEMMA 4.2 // (C,®,I,a,r,l,c) is a strict braided category, then rx(X ® A) = lx(A ® X] for every object X of C and every subobject A of I.

Proof. Notice that cx,i(X ® A) = CX,A(X ® A). In fact, let IA be the inclusion

morphism from A into /. By axiom (Bl), (IA ® ^x] ° CX,A = cx,i ° (ix ® M)- By applying both members to X®A, we obtain [(iA®^x)°cx,A\(X®A) and [cx,i o (lx ® iA)\(X ® A) = cx,i(X ® A). Now, we have rx(X®A) = (lx°cx j)(X®A) = lX(cXj(X®A))

A))=lx(A®X).

— cx,A(X®A)

= lx(cx A(X®

I

DEFINITION 4.3 An object X of a category C is called totally invariant if f(Y) C Y for every endomorphism f of X and for every subobject Y of X. PROPOSITION 4.4 If X is a multiplication object in a braided monoidal category C, then X is totally invariant.

Proof. It is well-known that every braided monoidal category is equivalent to a strict braided monoidal category. Moreover, since the properties of being totally invariant and being a multiplication object are preserved by monoidal equivalences, it is enough to prove the result for a strict category C. Let Y < X and let / 6 Endc(X). Since X is a multiplication object, Y = rx(X ® A) for some A < I, I being the unit of C. By Proposition 3.2, f ( X ) = rx(X B) for some B < I. By applying / to the first equality, f ( Y ) = (/ o rx)(X ®A)=-rxo(f® li)(X ® A) = r x ( f ( X ) ® A). By substituting the second equality, f ( Y ) = rx(rx(X ® B) ® A) =

rx((rx ® li)(X ®B)®A) = (rx o rx)((X ® B) ® A] = (rx o rx)(X (B ® A ) ) . By using the fact that C is braided and strict, f ( Y ) = (rx o rx)((X ® A) ® B) =

rX(rx(X ®A)®B)= rx(Y ® B) = rY(Y ®B)CB.I PROPOSITION 4.5 Let (C,®,I,a,l,r,c) be a braided monoidal category and let X be a multiplication object. Then, « / / , < ? € Endc(X), (f o g)(Y) = (g o f ) ( Y ) for every subobject Y of X. Proof. It is clear that it suffices to prove this for a strict category. Thus we suppose that C is strict. Firstly, we deal with the case Y = X. By Proposition 3.2, f ( X ) = rx(X ® A) and g(X) = rx(X ® B) for some subobjects A,B of /. By applying g to the first equality, we have g ( f ( X ) ) = (g o rx)(X g) A) = rxo(g® lx)(X ®A) = rx(g(X) ® A) = rx(rX(X <S> B) ® A). Since C is strict and braided, (g o f ) ( X ) = r\((X ® A) ® B). In a similar way, we have f ( g ( X ) ) = f(rx(X®B)) =rx(f(X}®B) = r'2x((X®A)®B). Let us now consider the general case Y < X. Since X is a multiplication object, Y = rx(X ® C) for some C < I. Then, f(Y)=rx(f(X)®C}. Thus g ( f ( Y ) ) = rx((go f ) ( X ) ® C } . By the previous step, g ( f ( Y ) ) = rx((f o g)(X) ® C) = f ( g ( Y ) ) . I

PROPOSITION 4.6 Let X be a multiplication object in a braided monoidal category C and let (f € Endc(X). Then, ip is an isomorphism if and only if it is an epimorphism.

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Proof. The first step of the proof consists in proving that 9? is an epimorphism if and only if 931 y : Y —> Y is an epimorphism for every subobject Y < X. Notice that by Proposition 4.4, (Y) = rx(X A) = Y. The other implication is clear. Now we shall prove that if 9? is an epimorphism, then it is a monomorphism. In fact, assume that 9? is an epimorphism. By the first step, we have 99(0) = 0 and (Y) = Y ^ 0, for Y 7^ 0. It follows that (p is a monomorphism.

The next result is an analogue of Fitting's Lemma.

PROPOSITION 4.7 Let X be a multiplication object in a braided monoidal category C and let 9? G Endc(X) such that
REMARK 4.8 The condition stated in Proposition 4- 7 is obtained when X has the descending chain condition over multiplication objects because it suffices to consider the chain of multiplication objects X 3 f ( X ) D • • •. Recall that an object X is called indecomposable if X = X\ ® X% (X\,X-2 being subobjects of X) implies X\ = 0 or X% = 0.

COROLLARY 4.9 Let X be an indecomposable multiplication object in a braided monoidal category and let

is an automorphism, so is (pn and hence,

PROPOSITION 4.10 Suppose that X is a multiplication object in a braided monoidal category C and

1. if is a monomorphism. 2. (f> is an epimorphism. 3. Lp is an automorphism.

Proof. Suppose that

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185

Now, assume that (f is an epimorphism. Since X is a multiplication object, Ker A) for some A < I. By applying f, we have

0 = (p o rx(X A) — rx(v(X} ® A) = rx(X ® A) — Ker p. Thus (f is a monomorphism. By Proposition 4.6, under these circumstances, 2 is equivalent to 3. I

THEOREM 4.11 If X is an indecomposable multiplication object in a braided monoidal category that verifies D. C. C. on multiplication subobjects, then the ring End(X) is local. Proof. By using Corollary 4.9, the theorem is proved as in the module case. I

Acknowledgements. Both authors have been supported by grant PB98-1005 from DGES and PAI FQM 0211. The authors are grateful to the referee for his suggestions and comments.

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[4] Chari, V. and Pressley, A., A guide to Quantum Groups, Cambridge University Press, Cambridge, (1994).

[5] Escoriza, J. and Torrecillas, B., Multiplication modules relative to torsion theories. Comm. in Algebra, 23 (11), (1995) 4315-4331. [6] Escoriza, J. and Torrecillas, B., Relative Multiplication and Distributive Modules, Comment. Math. Univ. Carolinae, 38 (2), (1997) 205-221. [7] Escoriza, J., Objetos multiplicacion (Ph. D. thesis), Serv. Publicaciones Univ. Almeria, Almeria, 1997.

[8] Escoriza, J. and Torrecillas, B., Multiplication Objects in Commutative Grothendieck Categories, Comm. in Algebra, 26 (6), (1998) 1867-1883.

[9] Escoriza, J. and Torrecillas, B., Multiplication rings and graded rings, Comm. in Algebra, 27 (12), (1999). [10] Escoriza, J. and Torrecillas, B., Multiplication graded rings, Algebra and Number Theory, Marcel Dekker, (2000) 127-136.

[11] Jeremfas Lopez, A., El grupo fundamental relativo. Teoria de Galois y localizacion, Alxebra 56, Santiago de Compostela, 1991.

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Escoriza and Torrecillas

[12] Joyal, A., Braided Tensor Categories, Advances in Math., 102, (1993) 20-78. [13] Kassel, C., Quantum groups, Graduate texts in Mathematics, 155, Springer, Berlin (1995).

[14] Larsen, M.D. and McCarthy, P.J., Multiplicative Theory of Ideals, Pure and Applied Mathematics 43, Academic Press, New York, 1971. [15] Lopez Lopez, M.P. and Villanueva Novoa, E., The Brauer Group of the Category (R,a)-mod (An Alternative to the Theory of Relative Invariants), Proc. of the First Belgian-Spanish Week on Algebra and Geometry, (1988). [16] Mac Lane, S., Natural associativity and commutativity, Rice Univ. Studies, 49, (1963) 28-46.

[17] Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer, Berlin (1971).

[18] Montgomery, S., Hopf algebras and their actions on rings, CBMS 82, Amer. Math. Soc., Providence, (1993). [19] Nastasescu, C. and Van Oystaeyen, F., Graded rings theory, North-Holland, Amsterdam (1982).

[20] Nastasescu, C., Group rings of graded rings. Applications, J. of Pure and Appl. Math., 33, (1984) 313-335.

[21] Panaite, F. and Stefan, D., When is the category of comodules a braided tensor category?, Rev. Roumaine Math. Pures Appl., 42 (1-2), (1997) 107-119. [22] Smith, P.F., Some remarks on multiplication modules, Arch. Math., 50, (1988) 223-235. [23] Stenstrom, B., Rings of Quotients, Springer, Berlin (1975). [24] Sweedler, M.E., Hopf Algebras, W. A. Benjamin, Inc., New York (1969).

[25] Torrecillas, B. and Zhang, Y.H., The Picard Groups of Coalgebras, Comm. in Algebra, 24 (7), (1996) 2235-2247.

[26] Van Oystaeyen, F., Generalized Rees Rings and Arithmetical Graded Rings, J. of Algebra, 82, (1983) 185-193. [27] Wisbauer, R., Introduction to Coalgebras and Comodules. Preprint.

[28] Yetter, D.N., Quantum groups and representations of monoidal categories, Math. Proc. Camb. Phil. Soc., 108, (1990) 261-290.

Krull-Schmidt Theorem and Semilocal Endomorphism Rings A. FACCHINI, Dipartimento di Matematica Pura e Applicata, Universita di Padova. Via Belzoni 7, 35131 Padova, Italy. E-mail: facchini@math. unipd. it

Abstract

We survey some very recent results on the Krull-Schmidt Theorem. We describe the properties of some classes of modules whose endomorphism ring is semilocal and their direct sum decompositions. In particular, we present results obtained by D. Herbera, the author and R. Wiegand about the KQ of a semilocal ring [FH1 1999, W 1999], an example due to G. Puninski of a

uniserial module that is not quasismall [PI 1999], and some results proved by Corisello, Barioli, Herbera, Raggi, Rfos and the author about homogeneous semilocal rings and the Krull-Schmidt Theorem for modules whose endomorphism ring is homogeneous semilocal [CF 1999, BFRR 1999].

Throughout, ring means associative ring with identity 1 ^ 0 . If .R is a ring, we denote the Jacobson radical of R and the ring of n x n matrices over R by J(R) and Mn(R) respectively. All modules are unital right modules unless otherwise specified. Semigroups are additive and commutative and have a zero element.

1

SEMILOCAL RINGS AND MODULES WHOSE ENDOMORPHISM RING IS SEMILOCAL

A ring R is semilocal if R/J(R) is semisimple artinian. By the Wedderburn-Artin Theorem, this means that R/J(R) is isomorphic to a finite direct product of rings of matrices over division rings. Since a ring is semisimple artinian if and only if it is right artinian and its Jacobson radical is zero, we have that a ring R is semilocal if and only if R/J(R) is right artinian, if and only if R/J(R) is left artinian.

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EXAMPLES 1.1 (1) A commutative ring is semilocal if and only if it has only finitely many maximal ideals, because a commutative ring is semisimple artinian if and only if it is a finite direct product of fields. (2) Every right (or left) artinian ring is semilocal. (3) Every local ring is semilocal. (4) If R is semilocal, the ring Mn(R) of n x n matrices over R is semilocal for every positive integer n. (5) Direct product of finitely many semilocal rings is semilocal. (6) Every homomorphic image of a semilocal ring is semilocal. (7) If R is semilocal, the ring eRe is semilocal for every nonzero idempotent e&R. For these examples and the other basic properties of semilocal rings presented in this section see [F 1998, Ch. 1]. Here is a property of semilocal rings that has been observed only recently by Pere Ara (for a proof see [BFRR 1999, Theorem 2.2]):

PROPOSITION 1.2 Let e be an idempotent element of a ring R and suppose that both the rings eRe and (1 — e)R(l — e) are semilocal. Then R is semilocal. Being semilocal is a finiteness condition on the ring. For instance, if R is a semilocal ring, then every set of orthogonal idempotents of R is finite; there are only finitely many simple _R-modules up to isomorphism; and there is only a finite number of finitely generated indecomposable projective /^-modules up to isomorphism [FS 1975, Theorem 9]. Moreover, as we shall see in Theorem 1.3, semilocal rings are exactly the rings with finite dual Goldie dimension. Since the notion of dual Goldie dimension is not quite standard, we shall briefly recall it here. Let L be a modular lattice with a smallest element 0 and a greatest element 1. A finite subset {xi\i£l}ofL\ {0} is join-independent if Xj A (Vi=y xi) = 0 f°r every j e /. An arbitrary subset of L\ {0} is join-independent if all its finite subsets are join-independent. A modular lattice with 0 and 1 is said to be of infinite Goldie dimension if it contains an infinite join-independent subset. Otherwise it is said to be of finite Goldie dimension. In this case it is possible to prove that { card X \ X is a join-independent subset of L \ {0} }, the set of the cardinalities of all joinindependent subsets of L \ {0}, has a greatest element, which is a non-negative integer dim L, called the Goldie dimension of L (for a proof see, for instance, [F 1998, Theorem 2.36]). Thus every modular lattice with 0 and 1 has a Goldie dimension, which is either a nonnegative integer or oo. For a module AS over any ring S, the lattice Ij(As) of all submodules of AS is a modular lattice with 0 and 1, and the Goldie dimension of L(As) is called the Goldie dimension dim AS of the module AS. If L is a modular lattice with 0 and 1, its dual lattice L*, that is, the set L with the opposite order, is a modular lattice with 0 and 1. The Goldie dimension dimL(^4s)* of the dual lattice of i(As) is called the dual Goldie dimension codimAs of the module AS- For a proof of the following result see, for example, [F 1998,

Proposition 2.43]. THEOREM 1.3 A ring R is semilocal if and only if the right R-module RR has finite dual Goldie dimension, if and only if the left R-module xR has finite dual

Goldie dimension. In this case, codlm(RR) = codim(/jl?) =

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Note that in this statement dlm(R/J(R)) is simply the number of direct summands in any decomposition of the semisimple module R/J(R) as a direct sum of simple modules. We shall be mainly interested in modules whose endomorphism ring is semilocal. Hence, let 5 be an arbitrary ring and let AS be a right S-module, and suppose that End(As) is semilocal. Again, this is a finiteness condition on the module AS- For instance, if AS = ® AeA A\ is a decomposition of AS as a direct sum of nonzero modules A\ and End(/is) is semilocal, then the index set A must be necessarily finite. In the following lemma we have collected three important properties of modules with a semilocal endomorphism ring. Recall that two direct sum decompositions AS = ©AeA ^A = ® €M BH of a module AS are said to be isomorphic if there is a one-to-one correspondence tp: A —> M such that A\ = -B^A) for every A G A. LEMMA 1.4 Let As,Bs,Cs be modules over an arbitrary ring S and suppose End(As) semilocal. The following properties hold: (a): (Cancellation property) If A © B = A © C, then B = C. (b): (n-th root property) If An = Bn for some positive integer n, then A = B. (c): The module AS has only finitely many direct sum decompositions up to isomorphism. By Proposition 1.2, if AS and BS are modules with a semilocal endomorphism ring, the module AS ® BS has a semilocal endomorphism ring. If AS is a module and its endomorphism ring End(.As) is semilocal, then codim(End(j4s)) is finite by Theorem 1.3, codim(End(As)) = n say. The next result, which is due to Dolors Herbera and the author and whose statement has not appeared in the literature yet, says that if AS is isomorphic to a direct summand of a finite direct sum B\ © B^ © . . . © Bm of 5-modules, then AS is isomorphic to a direct summand of a direct sum of at most n of the 5-modules BI, 82,..., Bm. THEOREM 1.5 Let AS be a module over a ring S and suppose codim(End(^4s)) = n < oo. Let Bi,B2,... ,Bm be S-modules such that AS is isomorphic to a direct summand of 0™^ Bi. Then there exists a subset a of {1, 2 , . . . , m} of cardinality

card a < n such that AS is isomorphic to a direct summand of @-€a Bj.

EXAMPLE 1.6 The endomorphism ring of any artinian module is semilocal. This was proved by Camps and Dicks [CD 1993, Corollary 6] answering a question posed by Pere Menal [M 1988, Question 16]. Recall that a module-finite algebra over a commutative ring A: is a ring R with a homomorphism of k into the center of R such that R is a finitely generated fc-module. If R is a module-finite algebra over a semilocal noetherian commutative ring, then R is isomorphic to the endomorphism ring End(As) of an artinian cyclic right module AS over a suitable ring S [FHVL 1995, Corollary 1.3]. This allowed us to show that Krull-Schmidt fails for artinian modules [FHVL 1995]. The problem was the following. If a module M is both artinian and noetherian, then the (classical) Krull-Schmidt Theorem states that the decomposition of M into a direct sum of indecomposable modules is unique up to isomorphism. On the other hand, Krull-Schmidt fails for finitely generated modules over certain subrings of Z® • • - ® Z [L 1983]. Realizing every module-finite algebra over a semilocal noetherian commutative ring as the endomorphism ring of

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a suitable artinian right module, we constructed examples of artinian modules for which Krull-Schmidt fails [FHVL 1995]. Further examples of artinian modules with particular direct sum decompositions were later constructed by Yakovlev [Y 1998] and Pimenov-Yakovlev [PY 1998]. We shall come back to the problem of constructing artinian modules with particular direct sum decompositions at the end of the next section. EXAMPLES 1.7 A number of modules whose endomorphism ring is semilocal were discovered by Herbera and Shamsuddin [HS 1995]. For instance, they showed that: (1) If AS is a module over an arbitrary ring 5 and both dim(As) and codim(As) are finite, then End(^4s) is semilocal and codim(End(As)) < dim(j4s)+codim(A,s). This is the case of serial modules of finite Goldie dimension, that is, direct sums of finitely many uniserial modules (see Section 3), or, more generally, direct sums of finitely many biuniform modules (a module is said to be biuniform if both its Goldie dimension and its dual Goldie dimension are equal to 1). (2) If AS is a module over a ring 5, the Goldie dimension dim(.As) is finite and every injective endomorphism of AS is bijective, then End(^4s) is semilocal and codim(End(yls)) < dim(As). For example, artinian modules satisfy these conditions. (3) Dually, if AS is a module over a ring 5, its dual Goldie dimension codim(.As) is finite and every surjective endomorphism of AS is bijective, then End(As) is semilocal and codim(End(As)) < codim(As). This happens for noetherian modules of finite dual Goldie dimension over any ring S, in particular, noetherian modules over semilocal rings.

2

K0 OF A SEMILOCAL RING

Let AS be a module over an arbitrary ring 5 and suppose R = End(^is) semilocal. If we want to study the direct sum decompositions of AS, it is natural to consider the full subcategory &dd(As) of Mod-5 whose objects are the modules isomorphic to direct summands of direct sums Ag of finitely many copies of Ag. For example, add(5s) is the full subcategory proj-5 of Mod-5 whose objects are all finitely generated projective right 5-modules. The categories add(As) and proj-7? are naturally equivalent via the equivalences Homs(As, —): a,dd(As) —> proj-JS and —<S>p.A: projR —> add(Ag), and in these equivalences the module AS correspond to the module RR. Hence, studying the direct sum decompositions in the category add(Ag) is completely equivalent to studying the direct sum decompositions of finitely generated projective right modules over the semilocal ring R = End(^4g). In particular, studying the direct sum decompositions of AS is equivalent to studying the direct sum decompositions of the module RR. The set of isomorphism classes of finitely generated projective right modules over a ring R can be given the structure of a commutative semigroup with 0 in the following way. For every finitely generated projective right R-module PR, let [PR] denote the isomorphism class of PR, that is, the class of all right _R-modules isomorphic to PR. Let V(R) denote the set of all isomorphism classes of finitely generated projective right ^-modules (V(R) is actually a set). Define an addition in VR by [PR] + [QR] = [PR ® QR\- Then VR is a commutative semigroup whose

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zero is the isomorphism class of the zero module. Here we have defined V(R) using right modules. But the contravariant functor Hom/j(—, R)'- proj-R —> /2-proj is a duality between the full subcategory pmj-R of Mod-R whose objects are all finitely generated projective right .R-modules and the full subcategory .R-proj of R-Mod whose objects are all finitely generated projective left .R-modules. Hence if we define V(R) using left modules instead of right modules, we obtain the same semigroup V(R) up to isomorphism. If PR is a finitely generated projective right module over a semilocal ring R, then the endomorphism ring of PR is semilocal by Examples (4) and (7) of 1.1, so that PR cancels from direct sums (Lemma 1.4(a)). Thus the commutative semigroup V(R) has the cancellation property, i.e., it is contained in an abelian group. Let KQ(R) denote the smallest abelian group that contains V(R), that is,

Ko(R) = { [PR] — [QR] 1 PR,QR finitely generated projective right .R-modules}. Here [PR] - [QR] = [P'R] - \Q'R} if and only if [PR] + [Q'R] = [P'R] + [QR] in V(R), that is, if and only if PR® Q'R = PR®QR- Usually one must construct the group Ko(R) considering not the isomorphism classes [PR] of finitely generated projective modules PR, but the stable isomorphism classes [PR]S, that is, the classes [PR]S of all right -R-modules QR such that there exists a finitely generated projective module XR with PR®XR = QR®XR. But as R is semilocal, the finitely generated projective module XR cancels from direct sums, so that two finitely generated projective /^-modules PR and QR are stably isomorphic if and only if they are isomorphic. The V and the KQ we are considering can be viewed as functors V: Rings —> CSemigrps

and

KQ: Rings —> Ab

from the category Rings of associative rings with identity to the category CSemigrps of commutative semigroups with zero or Ab of abelian groups respectively. If S is a ring morphism, V((p):V(R) —> V(S) is the semigroup morphism defined by V(
PR. Similarly for KQ( K0(S). In passing from V(S) to Ko(S) a lot of information is lost. For example, if S is a ring and TT: S —> S/J(S) denotes the canonical projection of S onto S modulo its Jacobson radical J(S), it can be proved that V(n):V(S) —> V(S/J(S)) is always injective [FH2 1999, Proposition 2.11]. In particular, if R is a semilocal ring, the ring R/J(R) is semisimple artinian, so that each finitely generated R/ J(.R)-module is a finite direct sum of simple modules. Thus V(R/ J(R)) is the free commutative semigroup having the isomorphism classes of simple R/J(.R)-modules as a free set of generators, that is, V(R/J(R)) = N n , where n is the number of simple R/J(R)modules up to isomorphism. It follows that KO(TT): Ko(R) —» Ko(R/J(R)) is an injective group homomorphism of Ko(R) into the free abelian group Ko(R/J(R}) = Zre. Thus Ko(R) also is a free abelian group, i.e., Ko(R) = Zm for some m < n. This shows that the KQ of any semilocal ring is simply a finitely generated free abelian group, and this structure is too poor to describe what we were interested in, that is, the direct sum decompositions of RR. If we want to study the direct sum decompositions of RR in Mod-_R (recall that this is equivalent to describing the direct sum decompositions of AS in Mod-S1), we must know how it is possible to write the element [RR] of V(R) as a sum of elements of V(R). Now V(R) C K0(R),

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so that if we do not want to lose information by passing to Ko(R) we must keep track of the subsemigroup V(R) of Ko(R) and its element [RR]. To do this we shall follow Goodearl [G 1991, Chapter 15] considering on Ko(R) the structure of pre-ordered abelian group with order-unit. If G is an abelian group and < is a relation on the set G that is reflexive, transitive and translation-invariant (i.e., for any a, b, c € G, a < b implies a + c < b + c), the pair (G, <) is called a pre-ordered abelian group and the relation < is called a pre-order on G. An element u > 0 of a pre-ordered abelian group G is an order-unit if for any a & G there exists n € N such that a < rat. The pre-order < on G is completely determined by the positive cone of the pre-ordered abelian group (G, <): if we associate to every translation-invariant pre-order < on G its positive cone G+ = { a g G | 0 < a } , we obtain a one-to-one correspondence between the set of translation-invariant pre-orders on G and the set of subsemigroups of G (Recall that subsemigroups of G contain OQ). A pre-ordered abelian group with order-unit (G, <,«) consists of a translation-invariant pre-order < on an abelian group G and an order-unit u. Pre-ordered abelian groups with order-unit form a category. For instance, the abelian group Ko(R) with the pre-order whose positive cone is Ko(R) + = { [PR] PR a finitely generated projective right -R-module} and with the order-unit [RR] is a pre-ordered abelian group with order-unit, and KQ turns out to be a functor from the category of associative rings with identity into the category of pre-ordered abelian groups with order-unit. Let's go back to the previous example, in which R is a semilocal ring, TT: R —> R/ J(R) denotes the canonical projection of R onto the semisimple artinian ring R/J(R), the mappings

V(TT): V(R) -» V(R/J(R))

and

KO(TT): K0(R) - KQ(R/J(R))

are injective, the semigroup V(R/J(R)) is isomorphic to N n , where n is the number of simple J?/J(/?)-modules up to isomorphism, and Ko(R/J(R)) = Zn. The componentwise order on Zn is the pre-order on Z™ whose positive cone is N™, that is, the partial order defined by (a\,..., an) < ( 6 1 , . . . , bn) if a, < 6$ for every i = 1,... ,n. Thus the pre-ordered abelian group with order-unit (Ko(R/J(R)), <, [R/J(R)]) is isomorphic to the partially ordered abelian group with order-unit (Z n , <,u) for a suitable order-unit u in (Z™, <). The order-units of Z™ with respect to the componentwise order < are the n-tuples u = (ui,U2, • • • ,un) € Z n with Ui > 0 for every i = 1 , 2 , . . . , n. It can be proved [FH1 1999, Lemma 2.2] that the injective mapping KQ(TT): Ko(R) —> Ko(R/J(R)) is an embedding of partially ordered abelian groups with order-unit, that is, KO(T^)([RR]) = [R/J(R)] and for every x,y & Ko(R), x < y in K0(R) if and only if KQ(n)(x) < K0(ir)(y) in K0(R/J(R)). This shows that

THEOREM 2.1 Let R be a semilocal ring and TT: R —> R/J(R) the canonical projection. Then the pre-ordered abelian group with order-unit (Ko(R), <, [^?fi]) is isomorphic to a subgroup of (Ko(R/J(R)), <, [ R / J ( R ) ] ) via the embedding of preordered groups with order-unit KQ(TT): Ko(R) —> Ko(R/J(R)). Moreover, if n is

the number of simple right R-modules up to isomorphism, the pre-ordered abelian group with order-unit (Ko(R/J(R)), <, [R/J(R)]) is isomorphic to (Z™, <,u) for a suitable order-unit u in ( Z ™ , < ) . In particular, (Ko(R),<) is a partially ordered group, i.e., its pre-order relation < is also symmetric.

Conversely,

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THEOREM 2.2 [FH1 1999, Theorem 6.3] Let G be a partially ordered subgroup of (Z n , <) and u an order-unit of Zn such that u 6 G. Let f : G —> Zn denote the embedding. Then there exist a semilocal hereditary ring R and two isomorphisms of partially ordered groups with order-unit g:G^> Ko(R) and h:Zn —> Ko(R/J(R}) such that the diagram

G

-^

g [

K0(R)

Z" [ h

K

]

-^

K0(R/J(R))

commutes. A very interesting variant of this theorem in the setting of commutative noetherian rings has been proved recently by Roger Wiegand [W 1999]. Recall that if M is a module over a commutative ring k and M* = Honifc(M, k) denotes the dual module, the module M is called reflexive if the canonical mapping M —> M** into the bidual is an isomorphism. Wiegand has proved that

THEOREM 2.3 [W 1999, Theorem 4.1] Let G be a partially ordered subgroup of (Z n , <) and u an order-unit of Zn such that u G G. Let f:G —> Z™ denote the embedding. Then there exist a semilocal ring R that is the endomorphism ring of a finitely generated reflexive module M^ over a commutative noetherian local unique factorization domain k of Krull dimension 2 and two isomorphisms of partially ordered groups with order-unit g: G —> Ko(R) and h: Zn —> Ko(R/J(R))

such that

the diagram

G

-^

9l

K0(R)

Z™ [h

K

-^

}

K0(R/J(R}}

commutes. Wiegand's result is particularly interesting for at least two reasons. On the one hand, the endomorphism ring R of a finitely generated module Ak over a commutative noetherian semilocal ring k is a semilocal ring (Example 1.7(3)). As KQ(R}+ = V(R), the partially ordered abelian group with order-unit Ko(R) contains the description of all direct sum decompositions of Ak, so that Theorems 2.1 and 2.3 describe all possible direct sum decompositions of any finitely generated module over a commutative noetherian semilocal ring. On the other hand, the endomorphism ring of a finitely generated module over a commutative noetherian semilocal ring k is a module-finite algebra over k, and we have already remarked in Example 1.6 that every module-finite algebra over a semilocal noetherian commutative ring is isomorphic to the endomorphism ring End(^4s) of an artinian cyclic right module AS over a suitable ring S [FHVL 1995, Corollary 1.3]. Thus Theorems 2.1 and 2.3 describe all possible direct sum decompositions of artinian modules. The method expounded in this section allows us to solve the problem of the existence of modules with particular direct sum decompositions whenever we are considering finitely generated modules over commutative noetherian semilocal rings, or artinian modules over arbitrary rings, or projective modules over semilocal rings,

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etc. The idea is always reducing the problem of the existence of a direct sum decomposition to an arithmetical problem about elements and subgroups (G, <,u) of a partially ordered abelian group with order-unit (Z™, <,u). We shall make some examples. EXAMPLE 2.4 Do there exist two indecomposable finitely generated nonzero mod-

ules A and B over a commutative noetherian semilocal ring R such that A2 = J33 ? If the answer is affirmative, there exist a partially ordered subgroup (G, <,u) of a partially ordered abelian group with order-unit ( Z n , < , u ) and two positive elements a, 6 £ G such that u = 2a = 36 and such that a and b cannot be written as a sum of two positive elements in G. Here and in the rest of this section when we say "a positive element" of G we mean an element of G n N ™ whose components are not all zero. Since the equation 2x = 3y has only the solutions x = 3t, y — 2t in Z (where t ranges in Z), it follows that the only positive elements a, b € Z n such that 2a = 36 are a = 3c, b = 2c with c ranging in the positive elements of (Z n , <, u). But for any subgroup G of Z", if a, 6 6 G, then c = 3c — 2c = a — b 6 G, so that both a = c + c + c and b = c + c are sums of positive elements of G. This contradiction shows that the answer is negative. We would have obtained the same answer considering artinian indecomposable modules over arbitrary rings or projective indecomposable modules over semilocal rings, instead of indecomposable finitely generated modules over commutative noetherian semilocal rings.

EXAMPLE 2.5 Do there exist three indecomposable pairwise nonisomorphic finitely generated modules A, B, C over a commutative noetherian semilocal ring R

such that A2 = B&C? We are looking for a subgroup (G, <, u) of (Z n , <, u) and three distinct positive elements a, b, c
EXAMPLE 2.6 Do there exist three indecomposable pairwise nonisomorphic finitely generated modules A, B, C over a commutative noetherian semilocal ring R such that A3 - B2 ® C ? We must look for a subgroup (G, <, w) of (Z n , <,u) and three distinct positive elements a, b, c € G such that u = 3a — 26+ c and such that a, b, c cannot be written as a sum of two positive elements in G. It is easy to find a solution of u = 3a — 26+c in Z 2 , for instance u = (3,6), a = (1, 2), 6 = (0,3),c = (3,0). If G is the subgroup of Z2 generated by u, a, b, c, then G = { (x, y) & Z2 | x + y is divisible by 3 }. There are no elements of G strictly contained between (0,0) and (1,2), strictly contained

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between (0,0) and (0,3), or strictly contained between (0,0) and (3,0). Therefore a, 6 and c cannot be written as a sum of two positive elements in G. This shows that the answer is again positive. We would have obtained the same positive answer considering artinian indecomposable modules over arbitrary rings, etc. EXAMPLE 2.7 Looking at the previous two Examples 2.5 and 2.6, it is natural to ask: Do there exist three pairwise nonisomorphic finitely generated modules A, B, C over a commutative noetherian semilocal ring R such that A2 = B © C and A3 =

B2®C? If the answer is affirmative, there exist a subgroup (G, <,u) of (Zn,<,u) and three distinct elements a, b, c <5 G such that 2a = b + c and 3a = 26 + c. But then in the abelian group G we would have a — 3a — 2a = (26+c) —(6+c) = b, contradiction because a ^ b. The contradiction shows that the answer is negative in this case. As we have already remarked, this technique describes all possible direct sum decompositions of artinian modules. Compare this method with the examples of artinian modules for which Krull-Schmidt fails due to Facchini-Herbera-LevyVamos [FHVL 1995], Yakovlev [Y 1998] and Pimenov-Yakovlev [PY 1998] mentioned in Example 1.6.

3

UNISERIAL MODULES

A further class of modules whose endomorphism ring is semilocal is the class of uniserial modules. Let 5 be a ring. An 5-module Us is said to be uniserial if the lattice L(As) of all its submodules is linearly ordered, that is, for any submodules

M and N of A we have M C TV or TV C M. If A and B are two uniserial modules, we say that A and B belong to the same monogeny class if there exist a monomorphism A —» B and a monomorphism B —» A. In this case we write [A]m = [B]m. We say that A and B belong to the same epigeny class, and write [A]e = [B]e, if there are an epimorphism A —» B and an epimorphism B —> A. Belonging to the same monogeny class and the same epigeny class are two equivalence relations in the class of all uniserial modules. The next proposition shows why monogeny classes and epigeny classes are important in the context of uniserial modules. PROPOSITION 3.1 Let A and B be uniform modules over a ring S. Then A = B if and only if [A]m = [B]rn and [A}e = [B]e. A module is serial if it is a direct sum of uniserial modules. In particular, a module is serial of finite Goldie dimension if and only if it is a direct sum of finitely many uniserial modules. The theorem that follows describes when two serial modules of finite Goldie dimension are isomorphic.

THEOREM 3.2 (Weak Krull-Schmidt Theorem for uniserial modules, [F 1996]) Let AI, ..., An, BI, ... ,Bt be nonzero uniserial modules over a ring S. Then A\ © . . . © An = BI © . . . ® Bt if and only if n = t and there are two permutations a, T of {1,2,... ,n} such that [Ai}m = [Ba^)}m and [Ai]e = [-BT(i)]e for every i = 1 , 2 , . . . ,n.

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This theorem considers direct sums of finite families of uniserial modules. What can be said about direct sums of arbitrary (possibly infinite) families of uniserial modules? That is, if { Ai \ i 6 / } and { Bj \ j
THEOREM 3.3 // {Ai \ i € /} and {Bj j 6 J} are two families of uniserial modules over a ring S and there exist two bijections J such that [Ai]m = (Ba(i)}m and (Ai e - [BT^)}e for every i € /, then ©ie/ Ai = 0 jeJ Bj. Conversely,

THEOREM 3.4 If {Ai \ i 6 /} and {Bj j e J} are two families of nonzero uniserial modules over a ring S and (J)i6/ f/j = ©,-gj Vj, then there is a bijection a: I —> J such that [Ui}m = [V^.(j)] m for every i 6 /.

We hoped that an analog of Theorem 3.4 could hold for epigeny class too, but this is false, as has been proved very recently by G. Puninski [PI 1999]. His counterexample is the following. Recall that a chain ring is a ring R such that both the modules RR and fiR are uniserial. A chain domain R, that is, an integral domain that is also a chain ring, is called a nearly simple chain domain if it has exactly three two-sided ideals, which must be necessarily the ideals 0 C J(R) C R. For an example of a nearly simple chain domain see [BBT 1990, §6.5]. Right ideals, left ideals and finitely presented uniserial modules of nearly simple chain domains have very particular behaviors. For instance,

PROPOSITION 3.5 Suppose R is a nearly simple chain domain, and a and b are nonzero noninvertible elements of R. Then: (1) the left ideal aR and the right ideal Rb are incomparable;

(2) aR + Rb=J(R); (3) the right R-modules R/aR

and R/bR

are isomorphic.

The proof of this proposition, due to Puninski, can be found in Lemma 4.1, Proposition 6.2 and Corollary 4.3 of [PI 1999]. Property (1) is easy. First of all, Puninski proves that the Jacobson radical J = J(R) of a nearly simple chain domain R is not finitely generated. To see this, note that J ^ 0 in a nearly simple chain domain. If J is finitely generated, then J2 ^ J by Nakayama's Lemma. Now J2 ^ 0 because R is a domain. Thus J2 is a two-sided ideal different from 0, R and J, contradiction. Thus J is not finitely generated. If a and b are nonzero noninvertible elements of R, and the left ideal aR and the right ideal Rb are comparable, for instance, if Rb C aR, then J = RbR C aR, so that J = aR is finitely generated, contradiction. Property (2) is the key property that allows Puninski to construct the counterexample. His proof is based on techniques of model theory. Property (3) says that over a nearly simple chain domain R there are only two indecomposable finitely presented right modules up to isomorphism, RR and R/aR, where a is any nonzero noninvertible element of R. Thus every finitely presented right module over

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a nearly simple chain domain R is isomorphic to R^ ® (R/aR)m for two uniquely determined nonnegative integers n and m. Then Puninski shows that there exists a noninvertible element r € R such that fln>i -^r™ T^ 0- For t ms i ^e chooses an arbitrary nonzero noninvertible element s <E~R. As Rs <2 sR by Proposition 3.5(1), there is t e R such that is ^ sR. Thus there exists a noninvertible element r £ R with isr = s. Then s = tsr = t2sr2 = i3sr3 = . . . e n n > i ^ « . Now that this element r has been chosen, Puninski defines a uniserial module UK via generators and relations as follows: let UK be the right J?-module with a countable set of generators xi,X2,xs, • • • subject to the relations x^r = 0 and xn+ir = xn for every n > 1. Clearly UK is uniserial. Set VK = R/rR. It is possible to show that

[/ft®v fl * o) -v^ o) ,

(i)

where V^ °' denotes the direct sum of countably many copies of VR. Since VR

is cyclic and UR is not cyclic, there is no epimorphism VR —> UR. In particular, [I/fi e ^ [UR]B. Thus the epigeny classes that appear in the two direct sum decompositions UR ® V^ = V^ , i.e., one copy of [UR\e and countably many copies of [Vftje in the decomposition on the left and countably many copies of [VR],, in the decomposition on the right, are different. Thus the problem of describing when two direct sums of uniserial modules ® ie/ Ai and ®,-ej Bj are isomorphic, that is, finding a generalization of theorem 3.2 for arbitrary families of uniserial modules, is still open. With his example Puninski solves a problem of [DF 1997, p. Ill], showing that there exist uniserial modules that are not quasismall (also see [F 1998, Problem 15 on p. 269]; here a module A is said to be quasi-small if for every set { Bi i € / } of .R- modules such that A is isomorphic to a direct summand of ® ie/ B^, there is a finite subset F C / such that A is isomorphic to a direct summand of Q)ieF Bi). Because of isomorphism (1), Puninski's module UR is also an example of a pureprojective uniserial module over a chain domain that is not finitely presented. In particular, not every pure-projective module over a chain domain is a direct sum of finitely presented modules. Puninski's module UR also solves a further problem. In [DF 1998, Proposition 2.6] Nguyen Viet Dung and the author showed that if V is a uniserial module over an arbitrary ring R, I is a, non-empty index set and V^ = A ® B, then either A or B must contain a direct summand isomorphic to V. In [DF 1998, p. 99] we asked whether under these hypotheses both A and B must contain a direct summand isomorphic to V. Puninski's decomposition V^°) = U © V^°) shows that for A = U and B — V^°) the answer is negative. In another wonderful paper [P2 1999], Puninski considers uniserial modules over prime coherent nearly simple chain rings that are not domains. Over such a ring R he is able to construct a pure-projective module that is not a direct sum of indecomposable modules. This shows that not every direct summand of a serial module over a chain ring is serial. Thus he answers the question posed in [F 1998, Problem 10 on p. 268]. It is not known yet whether every direct summand of a serial module of finite Goldie dimension is serial [F 1998, Problem 9 on p. 268]. Puninski's example of a pure-projective /^-module that is not a direct sum of indecomposables answers a further question [F 1998, Problem 11 on p. 269], because it is an example

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of a pure-projective module over a chain ring that is not serial. For an example of a prime coherent nearly simple chain ring R that is not a domain see [D 1994] and [P2 1999, §8].

4

HOMOGENEOUS SEMILOCAL RINGS AND MODULES WHOSE ENDOMORPHISM RING IS HOMOGENEOUS SEMILOCAL

Now we consider the class of modules whose endomorphism ring is a homogeneous semilocal ring. We shall present some results that appear in [CF 1999] and [BFRR 1999], A ring R is homogeneous semilocal if R/ J(R) is simple artinian, that is, if P/J(P) is isomorphic to the ring Mn(D] ofnxn matrices with entries in some division ring D for some positive integer n. For instance, if 5 is a local ring and n is a nonnegative integer, the ring Mn(S) of n x n matrices with entries in 5 is a homogeneous semilocal ring. A commutative ring is homogeneous semilocal if and only if it is local. Every homomorphic image of a homogeneous semilocal ring is homogeneous semilocal. If R is a right noetherian ring and P is a right localizable prime ideal of R, the localization of R with respect to P is a homogeneous semilocal ring. Further examples of homogeneous semilocal rings are given in [CF 1999, §§4 and 5]. A number of properties of local rings extend to homogeneous semilocal rings. For example,

PROPOSITION 4.1 In a homogeneous semilocal ring R the Jacobson radical J(R) is the unique maximal proper two-sided ideal of R, that is, J(R) contains all proper two-sided ideals of R. Conversely, if a semilocal ring R has a unique maximal proper two-sided ideal, then R is homogeneous semilocal. Obviously, every homogeneous semilocal ring R has a unique simple right module up to isomorphism.

THEOREM 4.2 Let R be a homogeneous semilocal ring. Then: (a) There exists a unique indecomposable finitely generated projective R-module P up to isomorphism. (b) Every projective R-module is isomorphic to a direct sum pW for some set X. (c) // X and Y are sets, then pW and p( y ) are isomorphic if and only if X and Y have the same cardinality.

We already know that semilocal rings are exactly the rings of finite dual Goldie dimension (Theorem 1.3). Thus it is possible to associate to each homogeneous semilocal ring R its dual Goldie dimension codim(P), which is a positive integer. It is also possible to attach to each homogeneous semilocal ring R a second numerical invariant, the index of R, denoted index(-R). It is defined as follows. By Theorem 4.2, the finitely generated projective module RR, where PL is a homogeneous semilocal ring, is isomorphic to a finite direct sum P* of copies of the unique indecomposable projective .R-module P, and the positive integer t is uniquely determined. This integer t is called the index of the homogeneous semilocal ring P, denoted index(P).

Semilocal Endormorphism Rings

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PROPOSITION 4.3 Let R be a homogeneous semilocal ring and let n, t be positive

integers. Then: (a) codim(-R) = n if and only if R/J(R) = Mn(D) for some division ring D; (b) index(/2) = t if and only if R = Mf(S) ring S with no nontrivial idempotents.

for some homogeneous semilocal

PROPOSITION 4.4 Let R be a homogeneous semilocal ring and let m be a positive integer. Then:

(a) codim(M m (/?)) = m • codim(R); (b) index(M m (.R)) = m • index(.R). The invariant index(jR) always divides the codimension codim(.R) for every homogeneous semilocal ring R, as the following proposition shows. PROPOSITION 4.5 If P is the indecomposable projective module over a homogeneous semilocal ring R, then codim(.R#) = index(l?) • codim(Pfl). Conversely, for any pair of positive integers t and n with t that divides n, there exists a homogeneous semilocal ring R with t = 'mdex(R) and n = codim(R) [CF 1999, Example 5.1]. Let us consider the "extreme cases" in the equality of Proposition 4.5: THEOREM 4.6 Let R be a homogeneous semilocal ring. Then (a) codim(_R) = 1 if and only if R is a local ring;

(b) index(.R) = 1 if and only if R has no nontrivial idempotents; (c) codim(/?) = index(.R) if and only if R is semiperfect, if and only if R = Mn(S) for some positive integer n and some local ring S. As homogeneous semilocal rings generalize local rings and the Krull-Schmidt Theorem concerns modules whose endomorphism ring is local, it is natural to ask whether the Krull-Schmidt Theorem holds for modules whose endomorphism ring is homogeneous semilocal. The answer is given in our last theorem. THEOREM 4.7 (Krull-Schmidt Theorem for direct sums of modules with homogeneous semilocal endomorphism rings, [BFRR 1999]) Let MR be a module over a ring R. Suppose that MR = MI © . . . © Mt = NI © . . . © Nm are two direct sum decompositions of MR into indecomposable direct summands and that all the endomorphism rings End(Mi) and End(A r j) are all homogeneous semilocal. Then t = m and there is a permutation IT of {1, 2, . . . ,t} such that Mi = Na^ for every It is important to note that in order to get the uniqueness of decompositions, it is not sufficient that in the decompositions MR = MI © . . . © Mt = NI © . . . © Nm only one decomposition MI ffi . . . ® Mt is into indecomposable direct summands Mi whose endomorphism rings End(Mj) are homogeneous semilocal. For instance, for any pair of integers n > I and s > I there exist a ring R and an /^-module AR with two non-isomorphic direct sum decompositions into indecomposable direct

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summands AR = PI ® . . . ® Pn = Qs and the End(Pi)'s homogeneous semilocal for every i = l,...,n ([FH2 1999, Lemma 5.5] and [BFRR 1999, Example 3.4]). We conclude with a generalization of Theorem 4.7 to arbitrary, possibly infinite, direct sums. It is due to Dolors Herbera and the author. Recall that a module AR over a ring R is said to be small if for every family { Bi i € / } of ^-modules and any homomorphism f. AR —> 0i6/ Bi, there is a finite subset F C / such that (Di€^ Bi. For instance, finitely generated modules are small.

THEOREM 4.8 Let MR be a module over a ring R. Suppose that MR = ®ig/ Mi =• (J) - 6 j NJ are two direct sum decompositions of MR into indecomposable small direct summands and that all the endomorphism rings End(Af,) and End(JVj) are homogeneous semilocal. Then there is a one-to-one correspondence tp: I —* J such that Mi = ^
REFERENCES [BFRR 1999] F. Barioli, A. Facchini, F. Raggi and J. Rios, Krull-Schmidt Theorem and homogeneous semilocal rings, preprint, 1999.

[BBT 1990] C. Bessenrodt, H. H. Brungs and G. Torner, "Right chain rings, Part 1", Schriftenreihe des Fachbereichs Math. 181, Universitat Duisburg, 1990. [CD 1993] R. Camps and W. Dicks, On semilocal rings, Israel J. Math. 81 (1993), 203-211.

[CF 1999] R. Corisello and A. Facchini, Homogeneous semilocal rings, preprint, 1999. [D 1994] N. Dubrovin, "The rational closure of group rings in left ordered groups", Schriftenreihe des Fachbereichs Math., Universitat Duisburg, 1994.

[DF 1997] N. V. Dung and A. Facchini, Weak Krull-Schmidt for infinite direct sums ofuniserial modules, J. Algebra 193 (1997), 102-121. [DF 1998] N. V. Dung and A. Facchini, Direct summands of serial modules, J. Pure Appl. Algebra 133 (1998), 93-106. [F 1996] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc. 348 (1996), 4561-4575. [F 1998] A. Facchini, "Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules" , Birkhauser Verlag, Basel, 1998.

[FH1 1999] A. Facchini and D. Herbera, KQ of a semilocal ring, to appear in J. Algebra (1999).

[FH2 1999] A. Facchini and D. Herbera, Projective modules over semilocal rings, to appear in the Proc. of the International Conference on Algebra and its Applications, Athens (Ohio), March 25-28, 1999. [FHVL 1995] A. Facchini, D. Herbera, L. S. Levy and P. Vamos, Krull-Schmidt fails for artinian modules, Proc. Amer. Math. Soc. 123 (1995), 3587-3592.

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[FS 1975] K. R. Fuller and W. A. Shutters, Projective modules over non-commutative semilocal rings, Tohoku Math. J. 27 (1975), 303-311. [G 1991] K. R. Goodearl, "Von Neumann regular rings", Krieger Publishing Company, Malabar, 1991.

[HS 1995] D. Herbera and A. Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), 3593-3600.

[L 1983] L. S. Levy, Krull-Schmidt uniqueness fails dramatically over subrings of Z®Z®---®Z, Rocky Mountain J. Math. 13 (1983), 659-678. [M 1988] P. Menal, Cancellation modules over regular rings, in "Proc. Granada Ring Theory Conference", Lecture Notes in Math. 1328, Springer, Berlin, 1988, pp. 187-208.

[PY 1998] K. I. Pimenov and A. V. Yakovlev, Artinian modules over one matrix ring, to appear in the Proc. of the Euroconference Infinite Length Modules, Bielefeld, September 7-11, 1998.

[PI 1999] G. Puninski, Some model theory over a nearly simple uniserial domain and decompositions of serial modules, preprint, 1999. [P2 1999] G. Puninski, Some model theory over an exceptional uniserial ring with applications to decompositions of serial modules, preprint, 1999. [W 1999] R. Wiegand, Direct-sum decompositions over local rings, preprint, 1999. [Y 1998] A. V. Yakovlev, On direct sum decompositions of Artinian modules, Algebra i Analiz 10 (1998), 229-238.

On Suslin's Stability Theorem for R[XI, ... ,xm] J. GAGO-VARGAS,

Departamento de Algebra, Universidad de Sevilla, Aptdo.

1160, 41080-Sevilla. Spain. Ei-m.a,il:[email protected]

Abstract

Let R be a euclidean domain. Suslin's Stability Theorem asserts that any matrix in SLn(R[xi, . . . ,x m ]), with n > 3,m > 1, can be expressed as product of elementary matrices. We give an algorithmic proof of this theorem, following [9], where the case R a field is treated. Using Grobner bases, we compute generators of maximal ideals in R[x\ , . . . , xm] and give a normalization of unimodular vectors. As a corollary, we obtain a constructive proof of Quillen-Suslin Theorem for R[XI, . . . ,xm}.

1

INTRODUCTION

In this paper, R will denote a euclidean domain, k a field, R[x] will denote the polynomial ring R[XI, . . . , xm and analogously fc[x] will be k[xi, . . . , xm}. We assume that we can factor any element a 6 R and that for any given prime element p G R we can factor in (R/(p})[x\. This assumption is needed because we have to find a primary decomposition of an ideal in R[XI, . . . , xm}. DEFINITION 1.1 For any ring S, an n x n elementary matrix Eij(a) over S is a matrix of the form I + a • eij, where i ^ j , a € S and &ij is the nxn matrix whose (i, j) component is 1 and all other components are zero.

DEFINITION 1.2 A square matrix A over a ring S is called realizable if A can be written as a product of elementary matrices. DEFINITION 1.3 A vector

203

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Gago-Vargas

is called unimodular if (i>i, . . . , vn) = S. Um n (S f ) is the set of unimodular vector with entries in S. Suslin's Stability Theorem can be expressed as

SLn(R[x]) = En(R[x})

for all n > 3

(2)

where En(R[x\) is defined as the subgroup of SLn(R[x\) generated by the elementary matrices. This theorem was first proved by A. A. Suslin in [11], and an algorithmic proof for R = k appeared in [9]. The theorem fails when n = 2, as P.M. Cohn showed in [4], and there exists an algorithm that determines when a given matrix in 6X2 (-^M) is realizable, and in that case, expresses it as product of elementary matrices (see [10]). In this paper, we extend the algorithmic proof of [9] to R\x\. There exist two main difficulties. First, given a proper ideal in R[x], we have to give an effective method to build a set of generators of a maximal ideal that contains it. We reduce the problem to the case fc[x], solved in [7]. Second, there exist steps in the process where it is needed that a certain polynomial be monic in one of the variables. When R = k we can use the Noether Normalization Theorem. For euclidean domains in general we do not have such a tool, but for unimodular vectors it is possible. We give a constructive version of a theorem in [12] that allows us to make such a change of variables. Section 1 is devoted to solve these technical questions. In Section 2 we adapt the proof in [9] through the previous lemmas, pointing out the differences. In the same way as in [9], an algorithmic proof of the Quillen-Suslin Theorem for R[x can be deduced. All required computations can be carried out using Grobner bases, as described in [1]. So, this paper may be considered a contribution to the computational side of algebraic K-Theory, like [6]. As pointed out in [10], these algorithms have potential applications to signal processing.

2 2.1

CONSTRUCTIONS IN R[x] Maximal ideals

THEOREM 2.1 Let k be a field, I = {/1; . . . , fr) C /c[x] a proper ideal. Then it is possible to compute a set of generators of a maximal ideal M. that contains I. PROOF.- We follow [7, Proposition 1]. Denote by k an algebraic closure of k. Because / is not the whole ring, we can find a common zero a = (a 1; . . . , a m ) € km of polynomials /i, . . . , fr. Let M — {g 6 fc[x]|
Then J C M., and J is maximal, because ki = fcj_i[o:j]/Aj. Then, J = M..

D

Suslin's Stability Theorem for R[x,,.. .jcj

205

The same result for the ring R[x] will be reduced to a polynomial ring with coefficients in a field. So we can assume that R is not a field. The following propositions are corollaries of results in [5].

PROPOSITION 2.2 Let I = ( / i , . . . , fr) C R[x] be a proper ideal such that I n R = (s), with s ^ 0. Then it is possible to compute a maximal ideal that contains I. PROOF.- Let p £ R be a prime element that divides s, and s = ps'. Then the ideal J = (p, j\,..., fr) is not the whole ring. If J = R[x] then there would exist ao, a i , . . . , ar £ R[x] such that 1 = a^p + Y^l=i aih- Multiplying each member by s', we get s' & I n R, which is a contradiction.^ Then we consider the ideal J = ( f i , . . . , fr) in R/(p)[x}. It is a proper ideal, and by Theorem 2.1, we can find a maximal ideal M — ( A i , . . . , \m) C (R/(p))[x\ that contains J. Let A, £ R[x], i = 1,... ,m be any lifting of A;. Then the ideal M = (p, \i,..., \m) contains J and is maximal, because R[x]/M — ( R / ( p ) ) [ x } / M , which is a field. D

PROPOSITION 2.3 Let I = {/i,. . . , f r ) C R[x] be a proper ideal such that I n R = (0). Then we can find a maximal ideal that contains I. PROOF.- By [5, Proposition 8.2], there exists a d £ R - {0} such that

I = (l,d)n(lQ(R)[x}nR[x]).

(4)

where Q(R) is the quotient field of R. If the ideal (/, d) is not the whole ring, we are under the hypothesis of Proposition 2.2. Otherwise, we have that / = IQ(R)[x] n R[x\. The ideal IQ(R)[x] is not the whole ring, because InR= (0). Then we can compute a maximal ideal MI C Q(R)[x] such that IQ(R)[x C MILet /i = MI fl.R[x]. /i contains the ideal /, and is not equal to the whole ring, because MI is proper. Let Mi = { A i ( x i ) , A 2 ( x 1 , x 2 ) , . . . , A m ( x i , . . . , x m ) } ,

(5)

with the polynomials Aj built as in 2.1. Then there exist a^ £ R,i = 1,... ,m such that \i = a,i\i € R[x}. Let J\ = ( A 1 ; . . . , Xm} C R[x]. It is clear that Ji C /i, and /x = JlQ(R)[x] n R[x}. The construction of the ideal /i, following [1, pp. 239-241] or [5, cor. 3.8], is done by computing a Grobner basis of Ji, and taking s £ R equal to the least common multiple of the leading coefficients of the polynomial in that basis. We are taking x\ < x-2 < ... < xm. We can assume that the generators of Ji are in the Grobner basis. Then

/i = JiQ(R)[x\r\R[x]

= JiRsnR[x] = (Ji,st- l)R[x,t]n R[x],

(6)

where Rs = 5""1/?, S = {sk} is the localization of R at s. We obtain g\,... ,gn € R[x\ such that /i = ( J i , f i > i , . . . ,#„). For each i = 1 , . . . , n , there exist djj-(x, t) £ R[x, t], j = 0,1,... ,m such that 3i(x) = ai0(x,t)(st - 1) + EJLi a «( x ^)^j( x )- If we put t = 1/s, then ^(x) = Y^jLi a ij( x i l/ s )^j( x )- So each ffi(x) can be expressed as a linear combination of AI, . . . , A m through polynomials with coefficients of the form a/sk, that belong to

R*.

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Let p 6 R be a prime element that does not divide s. Then s is a unit in the field R/(p). Let 99 : Rs —> R/(p) be the morphism defined by (p(a/sk) — as~k, and extend it to the polynomial rings in the natural way. Then, if we consider the images of the polynomials Qi in R/(p)[x], we get that
Ideal of principal coefficients

DEFINITION 2.4 Let I be an ideal in R[t}. Iff = a0+ait+.. .+adtd, with ad ^ 0, we say that a^ is the principal coefficient of f. We denote it by a^ = coef(f). Let J be the set of principal coefficients of polynomials in the ideal /. Then J is an ideal in R, that we call c o e f ( I ) . We want to compute a set of generators of c o e f ( I ) . In the ring R[x, t] we take an elimination order in the variable t. For example, xi < 2:2 < • • • < Xm < t. Let G — {
LEMMA 2.5 The ideal coef(I) is generated by the principal coefficients of the polynomials
= h(x).

PROOF.- The leading term of each gi(x,t) has the form a;(x)i di . Let h(x) € J. Then there exists /(x, t) & I such that coef(f) — h(x). The leading term of /(x, t) has the form h(x)td. Because G is a Grobner basis, there exist hi(x,t),..., hs(x,t) G -R[x][t] such that htd = X^=i hi(x, t)ai(x)tdi. Putting t — 1, we have that h(x) can be expressed as a linear combination of the polynomials aj(x). Now, let h(x) £ J. By the previous paragraph, we can compute polynomials hi(x) € R[x] such that h(x) = j^si=1 /i i (x)a i (x). Let f(x,t) = ^s=1 hi(x)gi(x,t). Because G is a Grobner basis, the principal coefficient of /(x, t) is h(x). D 2.3

Normalization of unimodular vectors

Given an ideal / in R[x], it is not possible, in general, to give a change of variables that allows us to put a generator as a monic polynomial in one of the variables. However, if the ideal generates the whole ring, we can get a normalization. It is a result that appears in [2] and [12]. We give the constructive version for R[x], following [12, Sections 9, 10].

PROPOSITION 2.6 Let I be an ideal in R[x], R' = R[x}/I, and f1J2 £ #[x] with (/i + /, /a + I) — R'• Then we can find h € R[x] such that (/i + ^1/2) + / is not a zero divisor in R'.

Suslin's Stability Theorem for /?[*„. . .jcj

207

PROOF.- Through [5], we can compute the prime ideals associated to / in R[x], and compare among them to extract the maximal sets by inclusion. The proof follows as in [12, Proposition 9.2]. D

COROLLARY 2.7 Let n be a natural number and b = (bi, . . . ,6 n )* e Um n (/?[x]) a unimodular vector. Then there exists an upper triangular matrix B e En(R\x\) with ones on the principal diagonal such that B-b = (di, . , . , cJ n )*, with {eJj, . . . , dn} a regular sequence. PROOF.- This is the proof of [12, Corollary 9.3], noting that in each step we use a ring of the form Rk — R[x]/Ik for some ideal Ik in R[x], where we can apply Proposition 2.6. D The next place where we have to give a construction is in [12, Lemma 10.5],

LEMMA 2.8 Let I be an ideal in R[x], with htR^(I) > 2. Then there exists an invertible change of variables xj, . . . , xm <-> j/i, . . . , j/m of the ring R[x] such that I contains a polynomial monic in the variable y\ .

PROOF.- Let J be the ideal in RQ = R[XI, . . . , x m _i consisting of the principal coefficients of / with respect to xm. By Lemma 2.5, we can compute a set of generators for J. Following [12, Lemma 10.5], we obtain a g £ J, and it is possible to extract / G / such that the principal coefficient of / with respect to xm be g. The proof then follows the same steps as in [12, Lemma 10.5]. D The last result is

LEMMA 2.9 Let n > 3, and b e Um n (.R[x]). Then there exist B € En(R[x]) and a change of variables xi, . . . , xm <-> yi , . . . , ym such that B • b = (cj), where c\ is a monic polynomial in the variable ym.

PROOF.- [12, Lemma 10.6]. 3

D

APPLICATIONS TO K-THEORY

The previous results can be applied to give algorithmic proofs of Suslin's Stability Theorem and the Quillen-Suslin Theorem over the ring R[x\. For that, we follow the proof in [9] for rings k[x]. We give only the specific details. The proof that En is normal in SLn, for n > 3, ([9, Section 2]) remains equal, because the only algorithmic process that we need is to find elements in R[x] that express 1 as a linear combination of the components of a unimodular vector. This can be accomplished through a Grobner basis. As in [9, Section 3], we apply induction over the number of variables. Let R' = Rxi,...,xm-i, X = xm.

THEOREM 3.1 Let A € SLn(R'[X}). ideal M in R' , then A & En(R'(X}).

If AM e En(R'M[X}}

for every maximal

PROOF.- Let ai = (0, . . . ,0) G Rm~l and pi G R a prime element. Then MI = (PI,XI, . . . , x m _ i ) is a maximal ideal in R' . With the same technique as in the proof of [9, Theorem 3.1], we find an element r\ £ MI- By Propositions 2.2 and

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2.3, we can compute a maximal ideal M2 that contains r\. We apply again the process over M2, and find r2 £ M2. In general, given ri, . . . ,r,-_i 6 R', such that (TI, . . . , TJ-I) ^ (1), we can compute a maximal ideal Mj that contains them, and find TJ £ Jvij. After a finite number of steps, we reach a number / such that (TI, . . . ,n) = R'. This condition is verificable through Grobner bases in R'. The rest of the proof remains identical. D

The reduction to SLy, is done through the Elementary Column Property: for n > 2, the group En(R[x\) acts transitively on the set Um n (/?[x]). We review the results used in the proof. The first one is [9, Lemma 4.2]. It remains unaltered, because if /i,/2 e R'[X] and r & R' is their resultant, we can find 51,32 € R'[X] such that r = figi + ([3, 13]). A similar proof to Theorem 3.1 can be applied to [9, Theorem 4.3].

THEOREM 3.2 Let e Um n (£[*])

(7)

a unimodular vector with v\(X) monic in X. Then there exist BI € SL,2(R'[X])

and B2 € En(R'[X\) such that B1B2 • v(X) = v(0). PROOF.- Let a = (0, . . . , 0 ) € Rm"1 and pi e .R a prime element. Let MI = (PI,XI, . . . , x m _i), that is a maximal ideal in /?', and fcj = R'/Mi the residual field. Following the proof of [9, Theorem 4.3], we find an element r\ € R' that does not belong to MI. Applying Propositions 2.2 and 2.3, we can find a maximal ideal M.-2 that contains TI. In the same way, we get an element r<2 £ M2- In general, given ri, . . . , TJ_I € 7?' we can find a maximal ideal Mj such that TI, . . . , TJ_I € .A/fj, and TJ ^ MJ. The process follows just like in the original proof of [9]. D In the proof of the Elementary Column Property ([9, Theorem 4.5]), we have to include a slight variation.

THEOREM 3.3 For n > 3, the group En(R[x]) acts transitively on the set

PROOF.- For m = 0, we apply the euclidean division algorihtm over R. By induction, we may assume the theorem for the ring R' = R[XI, • • • ,x m _i]. Let X — xm and

v(X) =

:

e Umn(R'[X}).

(8)

By Lemma 2.9, there exist a change of variables and a matrix B £ En(R'[X]) such that Bv(X) = w(F), with wi(Y) monic in Y. By Theorem 3.2 we can find

BI e SL2(R'[Y}) and B2 e -Bn(^'[^]) such that B^ • w(7) = w(0) e J?'. By the induction hypothesis, it is possible to compute B' € En(R'} with B' • w(0) = e n . Following the proof of [9, Theorem 4.5], we get that w(F) = (B^1B") • e n , with

Suslin's Stability Theorem for R[x,,.. .,rJ

B^B" e En(R'[Y}).

209

Then, v(X) = (B-1B^1B") • en undoing the change of

variables, and B^B^B" e En(R'\X}}.

D

In the realization algorithm for matrices of the form

SL3(fl[x])

(9)

we first apply Lemma 2.9 to assume that p is monic in xm, because we only add the product by a matrix in £?s(.R[x]). All the theorems in [9, Section 5] are valid, and this completes the process. As in [9], the Elementary Column Property implies the Unimodular Column Property: for n > 2, the group GLn(R[x\) acts transitively on the set Umn(.R[x]). In [8] it is shown that this property is equivalent to the Quillen-Suslin Theorem in .R[x], so we get an algorithmic proof of this theorem for R[x].

REFERENCES [I] W.W. Adams and P. Loustaunau, An Introduction to Grobner Bases, Graduate Studies in Math. Vol. 3, (Amer. Math. Soc., Providence, RI, 1994). [2] H. Bass, Liberation des modules projectifs sur certains anneaux de polynomes, Seminaire Bourbaki 1973/74, Expose 448, Lecture Notes in Math, vol. 431, 228254 (Springer-Verlag, 1975).

[3] D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Undergraduate Texts, in Math., (Springer-Verlag, New York, 1992). [4] P.M. Cohn, On the structure of the GL^ of a ring, Inst. Hautes Etudes Sci. Publ. Math. 30 (1966) 365-413. [5] P. Gianni, B. Trager and G. Zacharias, Grobner bases and primary decomposition of polynomial ideals, J. Symb. Comp. 6 (1988) 149-167. [6] R. Laubenbacher and C. Woodburn, An algorithm for the Quillen-Suslin theorem for monoid rings, J. of Pure and Applied Algebra 117-118 (1997) 395-429.

[7] D. Lazard, Solving Zero-dimensional Algebraic Systems, J. Symb. Comp. 13 (1992) 117-131. [8] A. Logar and B. Sturmfels, Algorithms for the Quillen-Suslin Theorem, J. Algebra 145 (1992) 231-239. [9] H. Park and C. Woodburn, An Algorithmic Proof of Suslin's Stability Theorem for Polynomial Rings, J. Algebral78 (1995) 227-298.

[10] H. Park, A Realization Algorithm for 5L 2 (-R[xi,... ,xm}) over the Euclidean Domain, SIAM J. on Matrix Analysis and App. 21 n. 1 (1999) 178-184. [II] A. A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11 (1977) 221-238.

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[12] L.N. Vaserstein and A.A. Suslin, Serre's Problem on projective modules over polynomial rings and algebraic K-Theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976) 993-1054.

[13] C. Woodburn, An Algorithm for Suslin's Stability Theorem, Ph.D. Thesis, New Mexico State University, 1994.

Characterization of Rings Using Socle-Fine and Radical-Fine Notions C. M. GONZALEZ, Departamento de Algebra, Geometrfa y Topologfa. Universidad de Malaga. Apartado 59, 29080 Malaga. Spain.

A. IDELHADJ and A. YAHYA, Universite Abdelmalek Essaad. Departement de Mathematiques. Faculte des Sciences de Tetouan. B.P. 2121. Tetouan, Morocco.

Abstract A class C of left modules is said to be socle-fine if for every pair M, N in C: M = N <=> soc(M) = soc(N). By Duality, we say that C is radical-fine if for every pair M, N in C: M ^ N <S> -^fm — ^HTM- In this note we wil1 characterize using the socle-fine and radical-fine notion the following rings: Perfect rings, self-injective rings, PF-rings, QF-rings, SV-rings, V-rings and semi-primitive rings.

1

INTRODUCTION

The rings considered in this paper will be associative rings with an identity element. Unless otherwise mentioned all the modules considered will be left unitary modules. The notion of socle-fine class has been introduced by A. Idelhadj and A. Kaidi in [7]. A class C of modules is said to be socle-fine if for every M, N in C we have that M ^ N if and only if soc(M) ^ soc(N). In [8] A. Idelhadj and A. Kaidi give the dual of the notion of socle-fine. This dual notion is called radical-fine. A class D of modules is said to be radical-fine if for every pair M, N in D we have that M = N if and only if r a d M ) = One of the interesting problems that can be posed is that of characterizing rings using these notions. In this way A. Idelhadj, A. Kaidi, D. M. Barquero, C. M. Gonzalez and A. Yahya have proved important results that can be found in [9], [8], [7] and [11]. The socle of a module M, denoted soc(M), is the sum of all the simple sub-modules of M. The radical of M, denoted rad(M), is the sum of all the small 211

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sub-modules of M. The injective cover of a module M will be denoted by E(M). If M has a projective cover we will denote it by P(M). Mod-P denotes the category of all right P-modules. A sub-module N of a module M, is large in M, denoted by N < M, and M is an essential extension of N if N n X ^ 0 for every non zero submodule X of M. By Duality a sub-module N of M is small in -M, denoted by N
(Mod-P) <00 (Mod-P) f

C P

= {M e Mod-P I Gdim(M) < 00} — {M e Mod-P | Gdim(M) < n}, where n e N Class of cyclic P-modules Class of injective P-modules Class of projective P-modules Class of finitely generated projective P-modules

with Gdim(-M) is the Goldie dimension of M.

Socle-Fine and Radical-Fine Notions

2

213

PERFECT RINGS AND PSEUDO-FROBENIUS RINGS

A ring R is said to be right perfect if every right /?-module has a projective cover. We say that R is semi-perfect if every cyclic right (or left) .R-module has a projective cover. We recall that an ideal / is left T-nilpotent provided that for every sequence (<2i)j 6 M* of elements of / there exists n G N such that anan-i....ai = 0. A ring R is said to be left pseudo frobenius (PF) if it is a left injective cogenerator. In [12] it is shown that a ring R is left PF if R is left injective, semi-local with large socle. Recall that R is quasi frobenius if R is artinian injective. In [14] Osofsky has introduced left pseudo frobenius rings as a generalization of quasi-frobenius rings. For examples of PF-rings which are not QF, see [14] and [13].

PROPOSITION 2.1 If R does not contain an infinite set of orthogonal (non zero) idempotents, then the following conditions are equivalent: (i) R is right perfect. (ii) Cs is socle-fine. PROOF.(i) =>• (ii) Let M be a non zero element of Cg. Then M is of the form E(C) for some non zero cyclic left module C. Since R is right perfect, by [4, Theorem 22-29] every non zero left module has non zero socle. It follows that soc(C') is essential in C and hence M is isomorphic to E( 0 5,), where (J) 5$ is iel

i€l

the socle of C. Therefore CE is socle-fine. (ii) =3- (i) Let M be a left ^-module. If soc(M) = 0, then consider an arbitrary element x of M. Therefore soc(Rx) = 0 and hence soc(E(Rx)) = 0. Since E(Rx) and 0 are two elements of CE with the same socle E(Rx) — 0, Rx = 0 and then x = 0. Since x is arbitrary in M, M = 0. By [4, Theorem 22-29] R is right perfect.

PROPOSITION 2.2 For any ring R with left T-nilpotent radical J, the following

assertions are equivalent: (i) R is right perfect. (ii) R is left (or right) semi-perfect. (Hi) The class F of left R-modules of the form

r&l^fM\,

whenever M is quasi-

projective, is socle-fine.

PROOF.- (i) =^> (ii) Obvious. (ii) => (in) Since R is semi-perfect, every module of the form

simple as an 4,-module and by [2, Corollary 2-12] ra^M} ^

s ser

J f rad / M)

is semi-

ni-simple as an R-

soc

module, and we have (^gn?y) — ra d(Af) • ^ follows that F is socle-fine. (Hi) => (i) We have soc(-y) = soc(soc(-j)), and since rad(soc(-j)) = 0 we can

write soc(4) = soc(—*° ,&•<•<)• Since R and soc(4) are nquasi-projective, then J4 v v J ' rad(soc(-j))' ^ J' f J > J and —^° .]}.. are two elements of F with the same socle. Then 4J = soc(4). and v rad(soc(j)) J"

hence ^ is semi-simple. Since J is left T-nilpotent, then it follows that R is right perfect. I By the above propositions we can give the following characterization of perfect rings.

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Gonzalez et al.

THEOREM 2.3 For any ring R the following assertions are equivalent:

1) R is right perfect. 2) F and CE are socle-fine.

PROOF.- 1)=^2) Follows from Proposition 2.1 and Proposition 2.2. 2)=>1) Since CE is socle-fine, from [4, Proposition 22-10A] it follows that J is left T-nilpotent and since F is also socle-fine it follows that R is right perfect. | LEMMA 2.4 A left R-module P is finitely generated and protective if and only if for some R-module P' and non zero positive integers n, there is an R-isomorphism P0P' ^ R<-n\ PROOF.- See [2, Corollary 17.3]

,

THEOREM 2.5 For any ring R the following conditions are equivalent. 1- R is self-injective with large socle. 2- FP U FPE is socle-fine. PROOF.- 1=»2) Let P be a finitely generated left projective .R-module. Then by the previous lemma there is a module N and a non zero positive integer n such that P ® N = PJ"). Since R is injective, P is also injective. Thus JPP coincides with J~PE- On the other hand if PI is a sub-module of P such that PI Pisoc(P) = 0 then soc(Pj) =0. Hence PI nsoc(fl(™>) = 0. Since soc(.R(™>) is large in P>), then P! = 0. It follows that soc(P) is large in P. Therefore P is an injective hull of soc(P). If Q is another element of FP, then we have also Q = E(soc(Q)}. If soc(P) = soc(<5), then clearly P = Q and hence FP U J-PE is socle-fine. 2=>1) Conversely, remark that PC and E(R) are two elements of J-P U J-Pe,

with the same socle, so P = E(R) and hence PC is left injective. Let now I be a left ideal of R such that / n soc(R) = 0, then s o c ( E ( I } ) = 0. On the other hand E(I) is a direct summand of PC . It follows that E(I) is projective and finitely generated. Hence E ( I ) and 0 are two elements of FPuFPE, with the same socle, then E ( I ) = 0 and hence 7 = 0. Therefore soc(R) is large in R. ( COROLLARY 2.6 For any ring R with finitely generated socle, the following conditions are equivalent. 1- R is a pseudo-frobenius ring.

2- FV U FPE is socle-fine. PROOF.- See [4, Proposition 24-32] and Theorem 2.5.

(

LEMMA 2.7 Let M be a left finitely generated R-module. If the injective hull of M is projective then it is also finitely generated.

PROOF.- See [12, Lemma 13-6-6, P: 356].

,

PROPOSITION 2.8 If R is a QF-S artinian ring, then R is QF if and only if the class J-P of finitely generated projective R-modules is socle-fine.

Socle-Fine and Radical-Fine Notions

215

PROOF.- Let P be an element of TV. Since R is QF, P is injective and we will have P ^ E(P). It follows that TV is socle-fine. Conversely, let P be a finitely generated projective module. Since R is QF-Z artinian, E(P) is also projective. It follows from Lemma 2-7 that E(P) is again finitely generated. Hence P and E(P) are two elements of TV with the same socle, so P is isomorphic to its injective hull. Therefore every finitely generated projective module is injective. In particular R is injective. Hence R is QF. |

THEOREM 2.9 For any ring R, the following assertions are equivalent: (i) R is quasi-frobenius. (ii) V U VE is socle-fine. PROOF.- (i)=>(ii) Over a QF ring, every projective module is injective with large socle. Hence V U VE is socle-fine. (ii)=>(i) Let P be an arbitrary projective P-module. Remark that P and E(P) are two elements of V U VE with the same socles. It follows that P is isomorphic to its injective hull. Therefore, R is QF.

3

I

RINGS WHOSE CLASS OF FINITE-DIMENSIONAL MODULES IS SOCLE-FINE

A module M has finite Goldie dimension provided E(M) is a finite direct sum of indecomposable sub-modules. In the literature, a module of finite Goldie dimension is sometimes called a finite-dimensional module. If M is a module of finite Goldie dimension, there exists a non-negative integer n such that E(M) is a direct sum of n indecomposable sub-modules. Moreover, by [6, Lemma 4-12] any other decomposition of E(M) into a direct sum of indecomposable sub-modules has exactely n summands. Thus n is uniquely determined by M, and is called the Goldie dimension of M (denoted by Gdim(M)). A ring is called qfd-ring provided every cyclic module has finite Goldie dimension. The class of qfd-rmg contains all rings with left Krull dimension. So, in particular, every left noetherian ring is left qfd. For further properties of qfd rings, see [1]. We recall that a ring R is called a left 1^-ring if every simple left Pi-module is injective. In [15] Villamayor has proved that R is a, left V-r'mg if and only if every left ideal of R is an intersection of maximal ideals. Hence the Jacobson radical of each F-ring is zero. A ring R is a right semi-artinian if every non zero right PL-module has non zero socle. We recall that P is a right SV-fmg if it is a right semi-artinian V-fmg. SV-rings form a special class of VonNeumann regular rings [3]. In [3] Baccela has proved that R is a right SV-r'mg if and only if every non zero PL-module has non zero injective sub-module. We recall that a ring PL is semi-primitive provided that the intersection of its primitive ideals equals zero. In [10] Jacobson has proved that PL is semi-primitive if it has a zero Jacobson radical. Consequently each V-ring is semi-primitive.

PROPOSITION 3.1 Let R be a qfd-ring. Then the following conditions are equivalent: 1- R is a right SV -ring. 2- (Mod-R)<00 is socle fine.

216

PROOF.-

Gonzalez et al.

1=>2) Let M be a module such that Gdim(M) = n. If n = 0 then

M = 0. Suppose that n ^ 0, hence there are many finitely indecomposable modules Ei,...,En such that E(M) = E\ ® ... ® En. Since R is semi-artinian, for every i € {!,...,n}, soc(E)i ^ 0. Then for each i, there exists a simple module Si n

such that ^ = E(Si). It follows that soc(M) = g) Sj. Since /? is a V-ring then i=l

soc(M) ^ ^

£'(5',). Hence M ^ soc(M) =* £(M). Therefore over an 5^-ring

i=l

every module with finite Goldie dimension is injective and semi-simple. It follows that (Mod-R)<00 is socle-fine. 2=>1) Conversely, let M be an arbitrary non zero .R-module and let x be a non zero element of M. Since R is a qfd-fmg then the cyclic /2-module x/? has finite Goldie dimension. Hence there are many indecomposable R-modules Ei,...,En such that E(xR) = EI ® ... ® En. So, Gdim(>#) = Gdim(E(xR)) = n, hence xR and E(xR) belong to (Mod-R)<00. Since soc(xR) — soc(E(xR)), it follows that xR = E(xR). Therefore xR is a non zero injective sub-module of M. Hence by [3, Theorem 2-7] R is an SV-ring. (

PROPOSITION 3.2 For any ring R, the following statements are equivalent: 1- R is a right SV-ring. 2- J U (Mod-R)<00 is socle fine. PROOF.- 1=>2) By the above Proposition, over an SV-ring every module with finite Goldie dimension is injective. On the other hand, over a semi-artinian ring every injective module is of the form E( @ Si), for some set / and some family ie/ (5j)j 6 / of simple modules. It follows that J U (Mod-.R)<00 is socle-fine. 2=>1) Conversely, let 5 be an arbitrary simple right .R-module,

Gdim(S') = Gdim(E(S)) = 1, then E(S) and 5 are two elements of J U (Mod-/?) <00 . Since they have the same socle, 5 = E(S) and hence R is a right V-ring. Let now M be an arbitrary Rmodule. If soc(M) = 0 then soc(E(M)) - 0. Since 0 and E(M) belong to I, it follows that E(M] = 0 and hence M = 0. Hence R is semi-artinian. (

PROPOSITION 3.3 For any left noetherian ring R, the following statements are equivalent: (i) R is left artinian. (ii) The class (J)<2 of injective R-modules of Goldie dimension < 2, is soclefine. PROOF.- (i) => (ii) Over any artinian ring, for every nonzero element M of (T)<2 there exists a simple module 5 such that M = E(S). Therefore (2")<2 is socle-fine. (M) =£> (i) Let M be an arbitrary injective left .R-module. By [16, Theorem 4-4] there exists a family (Ei)i^i of injective indecomposable modules such that M = 0 Ei. Suppose that there exists i0 6 / such that soc(Eia) = 0. Since iei Gdim(£'j 0 ) = 1, it follows that 0 and Ei0 are elements of (2T}<2 with the same socle.

Socle-Fine and Radical-Fine Notions

217

So Eio = 0 which is not possible, because Eio is indecomposable. Hence for every i £ I,soc(Ei) ^ 0, so for every i 6 / there exists a simple module Si such that Ei ^ E(Si). Therefore M =* 0 £(.%). It follows by [16, Theorem 4-5] that R is i€l

left artinian.

I

PROPOSITION 3.4 For any left noetherian ring R, the following statements are equivalent: (i) R is semi-simple. (M) The class (QI)<2 of quasi-injective R-modules of Goldie dimension < 2, is socle-fine.

PROOF.- (i) => (ii) Obvious. (M) => (z) Since every injective module is quasi-injective, (I)<2 is socle-fine. It follows by the previous Proposition that R is left artinian. Thus it is sufficient to prove that R has a zero radical. Consider an arbitrary simple left /?-module S. We have Gdim(S) = Gdim(£(5)) = 1. Since S and E(S) are quasi-injective they belong to (QJ)<2- On the other hand soc(E(S)) = soc(S), so it follows that S ^ E(S). Therefore R is a left V-ring and from [15, Theorem 2-1] every left fl-module has a zero radical, in particular rad(/?) = 0. Hence R is semi-simple. |

PROPOSITION 3.5 Let R be a noetherian domain with Krull dimension 1. Then any class C of direct sums of injective R-modules, with the same finite Goldie dimension, is socle-fine. PROOF.- Let M be an element of C. Then M is injective and hence there exists a family (/?i)ig/ of indecomposable injective modules such that M = 0 Ei, where i€l

|/| < oo. Since R is commutative noetherian, every indecomposable injective Rmodule is of the form E(-), where p is a prime ideal of R. So, the fact that the Krull dimension of R is 1 implies that p is either maximal or zero ideal of R. It follows that every indecomposable injective /?-module is either of the form E(R) or of the form E(S), where S is simple. Thus there is a family (Si)iei1 of simple modules such that M = 0 E(Si) 0 E(R)(Iz\ where /i and /2 are contained in /. Let N be another ie/i

element of (7, then in the same way as before N ^

0 E(Tj] 0 E(R)(-J2\ for i€J\

some family (Tj}jej1 of simple modules and some set J%. If soc(M) = soc(N) then Sj = Tj. Then there is a bijection a : /i —> Jj such that |/i| = |Ji| - i

j€Ji

and for every i e /j, 5j = T^;). Since M and AT are two elements of C then they have the same finite Goldie dimension. So, Gdim(M) = Gdim(N) implies that I/! | + |/2 1 = [J: + (J 2 | and hence |/2 = |J2|. It follows that E(R^ ^ E(R)(J21 Therefore M = N. ,

COROLLARY 3.6 Over a noetherian domain with Krull dimension 1, any class of finitely generated injective modules with same Goldie dimension is socle-fine.

218

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RADICAL-FINE CHARACTERIZATION OF RINGS

An element r of R is said to be a right zero divisor if there exists a non zero element s of R such that sr = 0. A left /^-module E is said to be divisible if E = rE, whenever r is an element of R which is not a right zero divisor. A Z -module M is divisible if and only if it is injective.

LEMMA 4.1

[16, Proposition 2-6].

Let R be an arbitrary ring. Then every

injective left R-module is divisible. PROOF.- Let E be an injective _R-module, let e e E, and let r be an element of R which is not a right zero divisor. Consider the next diagram:

0

->

Rr

-->

fl

/g

E

R

where / is defined by /(sr) = se (s e R). Note that, because r is not a right zero divisor, if sr = 0 then s = 0 and hence /(sr) = 0, so / is well-defined Rhomomorphism. Hence there exists a homomorphism g : R —> E which agrees with / on Rr. Thus

e = f ( r ) =5(0 = r f f ( l ) which shows that E is divisible.

I

LEMMA 4.2 [16, Lemma 2-4]. Let E be a divisible R-module and let E' be a submodule of E. Then -j^ is also a divisible R-module.

THEOREM 4.3 For any ring R, the following conditions are equivalent: 1) R is a left V-ring.

2) The class TJ of divisible R-modules is radical-fine. PROOF.- 1=>2) This is obvious, because every module over a V-ring has a zero radical (see [15, Theorem 2-1]). 2=>1) Let M be an arbitrary left .R-module. By Lemma 4.1, E(M) is divisible. E(M) It follows from Lemma 4.2 that 'ls a'so divisible. Then TaMgM)) E(M) are two elements of D such that E(M]

^

rad(B(M))

Hence E(M] = r f ) ) , it follows that rad(£(M)) = 0 and hence rad(M) = 0. Since M is arbitrary, it follows from [15, Theorem 2-1] that R is a left V-ring. t

LEMMA 4.4 If

Socle-Fine and Radical-Fine Notions

219

PROOF.- See [2, Proposition 9-15].

(

THEOREM 4.5 For any ring R, the following properties are equivalent: 1- R is semi-primitive. J..,.. -r,!,. f _ _ _ __ 2- The class S of R-modules of the form •£, whenever P is projective and K
x

H->

x+K

Since K is small in P, then tp is a minimal rad(-j^) = ra ^+— . As P is projective rad(P) radical of R. Since R is semi-primitive J = 0 and M has zero radical. Therefore S is radical-fine. 2=^1) Conversely, rad( -j) = 0 implies that -j

epimorphism and by lemma 4-4 = J.P where J is the Jacobson hence rad(P) = O.It follows that = ^, % . . Since 0 and J are two ad(3)-

-*Vi fV>^4small submodules of R, both R and j are elements of S such that

_____

ra rad(fl)

It follows that R = -y and hence J — 0. Therefore R is semi-primitive.

LEMMA 4.6 [2].

If M is a module which has a projective cover P(M). Then

M ~ P(M) rad(M) ~ rad(P(M)) '

The next theorem is the dual version of theorem 2-9.

THEOREM 4.7 For a left perfect ring R, the following statements are equivalent: 1- R is QF. 2- T U Tp is radical-fine.

PROOF.- If R is QF, then every injective module / coincides with its projective cover P(/). Since R is artinian, for every projective Pi-module P, we have rad(P)
REMARK 4.8 // the direct product of any family of projective R-modules is also projective then necessarily R is perfect. However, the converse is not true ( see [4, Theorem 22.3'IB P: 168] ). If we let "P^ be the class of all direct product of projective R-modules, then we have the next proposition. PROPOSITION 4.9 For a perfect ring R, the following conditions are equivalent: 1- The direct product of any family of projective R-modules is projective. 2- PK is radical-fine.

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Gonzalez et al.

PROOF.- l=>2) Follows from [8, Proposition 5-2-1, P: 73]. 2=>1) Let (Pi)i£i be an arbitrary family of projective ^-modules. Putting M = Pi. Since R is perfect then M has a projective cover P(M). By the lemma 4-6 _

R

• It follows that

Since M and P M

( ) are two elements of ^ then M ^ P(M).

Y\ Pi is projective.

i el

REFERENCES [I] A. H. Alhuzaim, S. K. Jain and S. R. Lopez. Permouth. Rings whose cyclics have finite Goldie dimension, J. of Algebra 153, 37-40 (1992). [2] W. Anderson and R. Fuller. Rings and categories of modules, Berlin, Heidelberg, New-York, 1974, Springer-Verlag.

[3] G. Baccella. Semiartinian F-rings and Semi-artinian Von-Neumann regular rings, J. of Algebra 173, 587-612 (1995). [4] C. Faith. Algebra II. Ring Theory, Berlin, Heidelberg, New-York 1976, Springer Verlag.

[5] C. Faith and E. Walker. A direct sum representation of injective modules, J. of Algebra 5, 203-221 (1967). [6] K. R. Goodearl and R. B. Warfield J.R. An introduction to non commutative noetherian rings, London. Math, soc, Student texts Vol 16, Cambridge, U P 1989. [7] A. Idelhadj, A. Yahya and C. M. Gonzalez. Socle-fine characterization of rings over which certain modules are injective. to appear in Algebras, Groups and Geometries. [8] A. Idelhadj, Classification d'anneaux par des proprietes relatives au socle et au radical de leurs modules.These doctorale (1995) Faculte des Sciences de Rabat, Universite Mohammed Morocco. [9] A. Idelhadj and A. Yahya, Socle-fine characterization of Dedekind and VonNeumann regular rings, Lecture notes of pure and applied mathematics. 157163,Volume 208, (1999).

[10] N. Jacobson, The radical and semi-simplicity for arbitrary rings, Amer. J. Math. 67, 300-342, (1945). [II] A. Kaidi, D. M. Barquero and C. M. Gonzalez, Socle-fine characterization of artinian and noetherian rings, Algebras, Groups and Geometries, 10, 191-198, (1993).

[12] F. Kasch, Modules and rings, London, Academic Press, 1982. [13] L. Levy, Commutative rings whose homomorphic image are self injective,Pac. J. Math. 18, 149-153, (1966).

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[14] B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pac. J of Math, 14, 645-650, (1964). [15] G. O. Michler and Q. F. Villamayor, On rings whose simple modules are injective, J of Algebra 25, 185-201, (1973). [16] P. Varnos and D. Sharp, Injective modules, Lecture in pure mathematics, University of Sheffield (1972) .

About Bernstein Algebras S. GONZALEZ and C. MARTINEZ, Dpto. de Matematicas, U. Oviedo. C/ Calvo Sotelo s/n. 33007 Oviedo. Spain. E-m&il:[email protected] and [email protected]

Abstract The aim of this paper is to present an overview of some known results and recent advances on the structure of Bernstein algebras and compare the situation for Bernstein algebras with the general situation for fc tfe order Bernstein algebras.

1

PRELIMINARY RESULTS

A very good review of nonassociative algebras that arise in relation to Genetics can be found in [57]. We will refer the reader to it to understand how nonassociative algebras can be used in Genetics. In this paper we will concentrate on some particular classes of those algebras, the so called Bernstein (nth-order Bernstein) algebras. We will try to give a general overview of the known facts about their structure and the relations with other nonassociative classes such as Jordan algebras, and other algebras relevant in Genetics. Since an exhaustive reference to all articles related to this subject that appeared in the last fifteen years would exceed the admissible length for this article, we will mainly refer to those papers that have been written inside of our research group. Almost all algebras that appear in Genetics are commutative nonassociative algebras. Those with a genetic meaning are real algebras, but in order to do an algebraic study of them if has become usual to consider algebras over an arbitrary field, that in our case this will always be of characteristic different from 2. In general, algebras that appear in this context have a scalar matrix representation, that is, our algebras will be baric algebras. DEFINITION 1.I An algebra A over a field F is called baric if it admits a nonzero

algebra homomorphism uj : A —» F, that is called weight function. 223

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In nonassociative algebras we can define powers of an element in several ways. If x is an element of a commutative nonassociative algebra we will consider principal powers of the element x defined by x, x2,..., xt+l = xlx and plenary powers x, x^l, . . . denned by x^+V = x®xW.

If the algebra A has genetic meaning and an element x £ A represents a population, each element x1 of the sequence of principal powers represents a population obtained mating back the previous population xl~l with the original population x (this is usually done in a laboratory) and the sequence of plenary powers contains the sucessive generations obtained by random mating within the population, beginning with x (and that appears in nature). If {ai, 02, • • • , an} is a basis of A, it can be proved that there exists a polynomial in principal powers that annihilates all elements of A ([62]),

f ( x ) =xr + 9lXr-1 + ••• + <9 r _iz where 0, is a homogeneous polynomial of degree i in the coordinates x, of a generic element x = X)™=i %iaiWe refer to the polynomial /(x) as rank polynomial.

DEFINITION 1.2 A baric algebra A with rank polynomial f ( x ) is a train algebra of rank r if the coefficients &i of f ( x ) are functions ofu>(x), where u represents the weight function. Working in an adequate extension of F, f ( x ) splits into linear factors

/(x) = x(x — AIU>(X))(X — \-2fjJ (x}) • • • (x — A T ._iw(x))

and the elements AI, A2, . . . A r _ j are called principal train roots of A. DEFINITION 1.3 A baric algebra with weight function w is a special train algebra if N = KeruJ is nil-potent (that is, Nm = (0) for some m) and all principal power subalgebras Nl = Nl~lN of N are ideals of A. Every special train algebra is a train algebra. Genetic algebras form a class of algebras between special train and train algebras.

DEFINITION 1.4 A commutative finite dimensional algebra A is genetic (in the sense of Gonshor [14]) if the algebra has a canonical basis {ao, ai, . . . ,a n } with structure constants A^ (a^j = Y^k=o^ijkOQ and k < max(i,j). This definition turned out to be equivalent to the definition given by Schafer of genetic algebras.

DEFINITION 1.5 A baric algebra A with weight function w is a genetic algebra (in the sense of Schafer [59]) if given an arbitrary elementT = aI+f(RXl, . . . , RXn) in the transformation algebraT(A), the characteristic function \XI — T\ is a function Here, as usual, the transformation algebra T(A) denotes the algebra generated by the identity operator 7 and all right multiplications Rx : A —> A, x e A.

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DEFINITION 1.6 A baric algebra A with weight function ut is called a Bernstein algebra (resp. nth- order Bernstein algebraj if the plenary powers of any element x & A satisfy x^ = w(x)2x^ (resp. a>+2l = (w(x)) 2 "xl™ + 1 ]). When a Bernstein (resp. fc'^-order Bernstein) algebra has genetic meaning, elements of weight 1 (those that represent populations) satisfy x'3l = x^ (resp. z [fc+2] _ x{k+i}^ th^ jS) equilibrium is reached after one generation (resp. k generations) of random mating within the population. In what follows, A will denote a Bernstein (or fc^-order Bernstein) algebra, uj will represent its weight function and N will be used to denote the barideal of A,

N = Keruj. 2

IDEMPOTENTS

The existence of an idempotent in an algebra A leads in general to a (Peirce) decomposition, so it has always algebraic interest. But idempotents also have genetic significance. If an element e2 — e e A represents a population, then genetic equilibrium has been achieved after one generation of random mating within the population x. So idempotents play an important role in the study of algebras that appear in Genetics, but their existence is not guaranteed. Clearly, if A is a baric algebra and e2 = e is an idempotent in A, then w(e) = 0 or 1. In the case of .A a Bernstein (resp. fc^-order Bernstein) algebra there always exist idempotent elements. Indeed, if x s A satisfies w(x) = 1, then e = x2 (resp. e = zl fc+1 !) is an idempotent element and w(e) = 1. Hence we have a first decomposition A = Fe + Keruj.

It is well known ([62]) that if A is a Bernstein algebra then Kerui = Ue® Ze, where Ue = {u e Keru>\2eu = u] and Ze — {z 6 Kerui\ez = 0}. Products of elements of Ue and Ze satisfy the following relations:

(B.I) UeUe C Ze, UeZe C Ue and ZeZe C Ue. (B.2) For every u € Ue, z e Ze we have U3 = 0 = u(uz) — (uz)2 = uz2.

The set of idempotent elements of A is I (A) = {e+u+u2 \u£Ue}. Furthermore, if / = e + u + u2 is another idempotent element, then Uf = {x + lux \ x e Ue}, Zf = {z- 2uz - 2u2z z e Ze}.

Consequently dimUe = dimUf and dimZe = dimZf are independent of the particular idempotent element. This allows us to define the type of the Bernstein algebra A as the couple (1 + dimUe,dimZe). In this way type(A) is an invariant that plays an important role in the classification results. If A denotes a /ct/l-order Bernstein algebra, then A = Fe © Keruj and Keru> = Ue + Ze, where Ue = {x e Kerw\1ex = x}, Ze = {z e Keru\R™z = 0}. But it is not known how to get all idempotent elements from a given e2 = e in

A, as it is known in the Bernstein case.

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It was proved in [21] that the dimension of Ue (and so the dimension of Ze) does not depend on the particular idempotent element e. Notice that for k > 2 and e, f two idempotent elements, we do not know how to express the elements of Uf (resp. Zf) in terms of the elements of Ue (resp. Ze), even if we know the expression of / in terms of e. In [33], [34] and [35] idempotent elements of second order Bernstein algebras were studied. In this case, given an idempotent element e, the set {(e + u + u2)2 \ u <E Ue} is contained in I (A), but in general both sets do not coincide. It happens in some particular cases.

Even in the case k = 2 relations satisfied by products of elements in Ue and Ze are more complicated and difficult to use. For instance U2 C Ze is still valid, but nothing is known about UeZe and Z2. Also the relations in (B.2) are no longer valid. In the case k = 2 we have instead the following relations for arbitrary elements u € Ue, z £ Ze

(B.3) e(uz) + u(ez) e Ue, z2(ez] e Ue, ez2 + 2(ez)2 e Ue, eu3 + u(eu2) e C/e, eu3 + u(eu2) = 0 , u3 + 2u(eu2) e Ze, uz 2

= 2e(uz2 + 4(uz)(ez)) + 2u(ez2 + 2(ez)2).

Coming back to Bernstein algebras, some particular classes can be considered.

DEFINITION 2.1 a) A Bernstein algebra A is nuclear if A = A2, or equivalently , U2 = Ze for an arbitrary idempotent e. b) A Bernstein algebra A is called exclusive (or exceptional,) if U2 = 0 for some idempotent element. Then U2 = 0 for every idempotent e. c) A Bernstein algebra is called normal or conservative if x2y = w(x)xy for every x,y G A. Such algebras satisfy Z2 + UeZe = 0 and this gives a characterization of normality. Consequently, the three definitions above do not depend on the particular idempotent e that we consider in the algebra. We know that dimU2 and dim(UeZe + Z2) are invariants of the algebra. This is not the case with dimZ2 and dimU3, which can change when we take different idempotents.

A Bernstein algebra is called orthogonal if Uf = 0 for some idempotent element e. When U3 = 0 holds for every idempotent element, the Bernstein algebra A is called totally orthogonal. It can be proved that A is totally orthogonal if and only if for some idempotent element, we have U3 = 0 qnd U2U2 = 0 (and so the relation is satisfied for every idempotent). In [30] it was proved that UQ = {u € Ue \ uUe = 0} is an ideal of the Bernstein algebra A, which is invariant under derivations and does not depend of the particular idempotent element e. Indeed, UQ = Hee/M) ^e' ^ *he algebra A is nuclear, then = 0. This ideal has been very helpful in the study of Bernstein algebras.

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227

RELATIONS WITH OTHER CLASSES OF ALGEBRAS

Initially a baric algebra may have more than one weight function. However it is known (see [62]) that if Keruj is nil, then the weight function is unique.

If A is & Bernstein algebra then N = Kerui is not necessarily nil, understanding that a (nonassociative) algebra is nil if some principal power of every element is zero. For instance, A = Fe + Fu + Fz with products e2 — e, leu = u, ez = 0, u2 = Q,uz = z2 = u is a, Bernstein algebra with N = Fu + Fz, and zn = u for every n > 2.

However every element x e Kerut in a Bernstein algebra satisfies (x 2 ) 2 = 0 and it can easily be seen that this suffices to prove the uniqueness of the weight function. This is also the case for fc^-order Bernstein algebras, since elements in Keruj satisfy x^+^ = 0.

Since the ideal N = Kerui of a Bernstein algebra is not nil, clearly it is not nilpotent in general. Of course this is the case if A is a Jordan-Bernstein algebra, that is, a Bernstein algebra that satisfies also the Jordan identity x2(yx) = (x2y)x, Vx, y e A. In the case of Jordan-Bernstein algebras it is not necessary to distinguish between principal and plenary powers, since every Jordan algebra is power-associative, that is, the subalgebra generated by one element is associative. Of course the converse is not true and there are commutative power-associative algebras that are not Jordan. In the case of Bernstein algebras we have

THEOREM 3.1 (see [30]) Let A = Fe + Ue + Ze be a Bernstein algebra. Then the following statements for A are equivalent 1. A is power-associative, 2. A is a Jordan algebra, 3. Z2 =0 and (uz)z = 0 Vu S Ue, z e Ze, 4- Z2 — 0 for every idempotent element f2 — f € I (A). Furthermore, it is proved that the above mentioned ideal UQ satisfies that is a Jordan-Bernstein algebra. In every Jordan-Bernstein algebra the identity x3 = 0 for elements of N is satisfied. The converse is not true, as is shown by the algebra B — Fe + Fu + Fz with e2 = e, eu = ^u, ez = 0, u2 = 0 = uz, z2 — u, that is not Jordan, although x3 = 0 for all x e N = Fu + Fz. The previous fact becomes a useful tool. If UQ = 0 in a Bernstein algebra A, then A is Jordan (so genetic). If UQ ^ 0, then the quotient A/Uo is a Jordan-Bernstein algebra of smaller dimension. This is used in [1] to prove that, under the assumption chF ^ 2,3,5 the square of the barideal iV2 is nilpotent (the algebra is not assumed to be finite-dimensional). The proof uses two well known facts: i) If J is a special Jordan algebra without elements of additive order < 2n and xn = 0, then J is solvable ([60]), ii) If J is a solvable Jordan algebra, then J2 is nilpotent ([55]).

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However Theorem 3.1 is no longer true if we consider A^-order Bernstein algebras, k > 1. Power associative and Jordan second order Bernstein algebras are studied in [35]. Using the counterexample by Suttles of a 5-dimensional nil algebra, solvable but not nilpotent, we construct an example of a power-associative second order Bernstein algebra that is not Jordan.

Nuclear Bernstein algebras have more significance from a genetic point of view. It is known that the barideal of a finite dimensional nuclear Bernstein algebra is nilpotent. Using the ideal UQ it is easy to prove that x4 = 0 for all x e N. Clearly (Keru>)n = 0 if n = dirnA and the product of arbitrary 2n — 1 elements of Kerw is zero. Griskhov conjectured that if the nuclear Bernstein algebra A is generated by r elements, then (Keru)2r+2 = 0. It can be shown that a nuclear Bernstein algebra can be generated by r elements if and only if dimUe — dimU3, < r. We introduce (see [32], [50]) the notion of free nuclear Jordan-Bernstein (and free Jordan-Bernstein) algebra which allows us to prove the above mentioned conjecture and which has been useful in the study of these algebras. In [4] the above bound is improved, proving that if A is a nuclear Bernstein algebra generated by r elements then (Keruj)r+'i = 0 if r is even and (Kerui)r+3 = 0 if r is odd. Nilpotency of the barideal of a Bernstein algebra has been studied in several papers. Since it is known that the powers of the barideal N are again ideals of the Bernstein algebra A, nilpotency of N suffices to assure that the algebra A is genetic. For instance, in [56] Engel conditions are studied. If we consider a Bernstein algebra A = Fe + Ue + Ze, the operator Lu : Kerui —> Keruj for u G Ue is always nilpotent. Indeed, it satisfies L3 = 0. However Lz : Kerui —> Kerui, z € Ze does not need to be nilpotent. Bayara, Micali and Outtara prove that a Bernstein algebra that satisfies (L z |j/) 3 = 0 for every z € Ze is genetic, that is, the barideal Kerw is nilpotent. In [12] some relations between Bernstein (resp. fc t/l -order Bernstein) algebras and train algebras are studied. It is known that if A is a train algebra of rank 3 then the following three statements are equivalent. 1. A is Jordan, 2. A is power-associative, 3. The train equation of A is either x3 — w(x)x 2 or x3 — 2u>(x)x2 + w(x) 2 x. In [12] it is proved that there are no Bernstein algebras of rank 3 with train identity x3 = w(x)2x. Indeed, if the baric algebra A satisfies the train identity x3 = w(x) 2 x, then to be Jordan, to be power-associative and to be Bernstein are equivalent conditions for the algebra. Furthermore, any of the three equivalent conditions above is equivalent to the train identity x 2 = w(x)x in A. It is known (see [61]) that a Bernstein algebra A is a train algebra of rank 3 if and only if it is Jordan. In such case the train equation of A is x3 = w(x)x 2 . In [12] the following generalization (if chF = 0) is proved:

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THEOREM 3.2 IfchF = 0, A = Fe + U + Z is a Bernstein algebra the following conditions are equivalent.

1. A is a train algebra of rank less than or equal to k, k > 4. 2. The train equation of A is

where \(k} =^(^( 3.

k

V

l ~z ^ +

J

k

\

~l

1

J

) for i = I , 2, . . . , k - 2 and \{k} =

k z

~1 = 0 = z ( z - • • ( z ( z u ) } • • •)) for every u e Ue, z e Ze. k-l

k

l

4- z ~ = 0 for every z € Zf and for every f e I (A). This characterization resembles the one obtained for Jordan-Bernstein algebras in [30] and suggests that the condition about products in Z is related to the fact of being a train algebra and the characterization for Jordan-Bernstein algebras is a consequence of the fact that they are exactly the Bernstein algebras of rank 3. This point of view has another confirmation in the following result, also proved in [12].

PROPOSITION 3.1 If A is a Bernstein algebra and k > 4 then (Keru)1*-1 n C/o is an ideal of A, that is invariant under derivations and the quotient algebra A/((Kerio}k~l n C/Q) is a train algebra of rank less than or equal to k.

In the same paper some relations between train algebras and second order Bernstein algebras are studied. It is proved that a second order Bernstein algebra A can not satisfy a train identity of degree less than 4. If it is a train algebra of rank 4, then the train equation is x4 = ui(x)x3 and if it is a train algebra of rank 5, then the train equation is either xb = ui(x)x4 or x5 — |w(o;)x4 — ~uj(x23 x As we have already mentioned, power-associativity and Jordan conditions are not equivalent in a second order Bernstein algebra (as it is the case in Bernstein algebras). However it is proved in [12] that a second order Bernstein algebra A is Jordan if and only if it is both power-associative and a train algebra of rank 4. Notice again the difference with the situation in Bernstein algebras. Second order Bernstein Jordan algebras are not characterized as the second order Bernstein algebras that are train algebras of rank 4, since there are second order Bernstein algebras satisfying z4 = w(x)x 3 that are not power-associative.

4

BERNSTEIN PROBLEM

A real algebra A has genetic realization if it has a basis {ai , . . . , an} and a multiplication table didj = Y^ik=i lijko-k such that the structure constants satisfy 0 < 7^ < 1 for all i,j,k and X)fe=i 7ijfc = 1-

Such a basis is called a natural basis or stochastic basis and algebras with genetic realization are the ones that can have genetic significance. The class of algebras

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with genetic realization is very big and includes the class of baric algebras, since the map u : A —> ffi, w(£iai + • • • + £ ra a n ) = £1 + ••• + £„ is a weight function.

In the 20's S. Bernstein ([5],[6]) studied a quadratic evolutionary operator ^ that maps the simplex of genetic frequency distributions An = into itself and represents the passage of one generation to the next. The operator * is called a Bernstein (or stationary) operator if it satisfies 'I'2 = $, which indicates that the population is in equilibrium after one generation. Associated to every Bernstein operator \& we have a real Bernstein algebra obtained by considering the real vector space A over a basis {ap, a j , . . . , a n } with the multiplication xy = ^ ( t y ( x + y) — ^(x) — $(y) for elements x,y & An and in general xy = w(x)w(y)xy, if x - u(x)x, y - u(y)y, x,y e A re . In this way the study of Bernstein operators becomes equivalent to the classification of real Bernstein algebras having a stochastic basis (that is, with genetic realization). This problem, known as the Bernstein problem has been for years one of the main problems in this area. This problem was studied by Lyubich (see [46]) who gives an explicit description of Bernstein operators in the regular and exceptional cases. This implies that the problem (already solved by Bernstein for dimensions 2 and 3) was totally solved for Bernstein algebras of dimension < 4.

In [22] the Bernstein problem in dimension 5 was considered. Since a Bernstein algebra of type (2,n — 2) or (n — 1,1) is either exceptional or regular, only type (3,2) has to be considered. It was proved that (up to isomorphism) there are six nonregular, nonexceptional Bernstein algebras of type (3,2). The Bernstein problem for type (n — 2,2), dimension 6 and type (3, n — 3) was studied in [36], [37] and [38] respectively. Each time that the dimension of the Bernstein algebra increases a little, the difficulty of the problem increases a lot. Due to space limits we do not enter in details of proofs, that are quite technical. In [40] we proved that a nuclear Bernstein algebra having a stochastic basis is regular. In this way we answer in a positive way a conjecture posed by Lyubich and give an important step to the final solution of the Bernstein problem, that has been given in [39], where the explicit form of all nonregular, nonexceptional stationary operators can be found.

5

AUTOMORPHISMS AND DERIVATIONS

Another way to approach the structure of Bernstein algebras is via the group of automorphisms and the Lie algebra of derivations. Since we would like a classification up to isomorphism of Bernstein algebras, it seems natural to attempt a characterization of isomorphisms that allows to treat them in a simpler way. So in [49] the following theorem was proved

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THEOREM 5.1 If A and A' are Bernstein algebras and 9 : A —> A' is a bijective linear map, then 9 is an isomorphism of Bernstein algebras if and only if the following two conditions are satisfied:

i) An element e € A is an idem/potent if and only if 0(e) € A' is an idempotent, ii) xy = 0 in A if and only if 9(x)9(y) = 0 in A'. Since a general classification, up to isomorphism, has not been possible except for small dimension ([8],[23]), some other classification attempts have been made. In [16] the homotope algebra of a Bernstein algebra (according to the general definition given by Teddy for nonassociative algebras) was studied. It was proved that the homotope of a Bernstein algebra by an element a is again a Bernstein algebra, whenever w(a) ^= 0. However a natural notion of isotopy does not exist (see [26]). In [19], taking into account relations between an algebra A and some homotope, the notion of quasiisomorphism (weaker than isomorphism) is defined and studied. If A = Fe + Ue + Ze is a Bernstein algebra, then a derivation D of A is characterized by a triple (u, /, Ue, g : Ze —» Ze are linear maps that satisfy the following three conditions: 1. g(uu') = f(u)u' + u f ( u ' ) , 2. f ( u z ) = f ( u } z + ug(z) + 2(uu)z, 3. f ( z z ' ) = g(z}z' + zg(z') - 1((uz)z' + (uz')z) for every u,u' 6 Ue, z,z' 6 Ze.

So dim Der(A) < r + r2 + s2, where type(A) = (r + 1, s). The upper bound can be reached exactly for trivial Bernstein algebras, that is, algebras with (fcerw) 2 = 0. Clearly, there is also a natural lower bound of dim Der(A), that is 0. And again this bound can be reached, as was proved in [31] where some examples of Bernstein algebras having zero derivation algebra are given. In the same paper some necessary conditions for a Bernstein algebra to have zero derivation algebra were obtained, but sufficient conditions were not found. The fact that the derivation algebra of a Bernstein algebra may be zero or have the maximal admissible dimension shows that the structure of Bernstein algebras changes in a very big range and justifies that, up to the moment, there is no structure theory for those algebras. Inner derivations of Bernstein algebras have been studied in [17]. A derivation is inner if it lies in the Lie transformation algebra generated by the (left and) right multiplications. If A is a Jordan algebra this Lie transformation algebra is known

to be C(A) = R(A) + [R(A),R(A)}. Jacobson proved that if A is a semisimple associative (Lie, alternative or Jordan) algebra over a field _F (ch F =£ 2) then all derivations of A are inner. Schenkman proved that every nilpotent Lie algebra over F has a derivation which is not inner. In [48] inner derivations of a Jordan-Bernstein algebra are studied. In this case the Lie derivation algebra is a 2-graded algebra with its homogeneous component of degree 1 isomorphic (as vector space) to Ue and consisting of inner derivations. In particular, a Jordan-Bernstein algebra always has nonzero derivation algebra. In many cases, for instance if UZ = 0 or if U2 = 0, derivations that are not inner are constructed in an explicit way.

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We conjectured (the problem is still open) that in a Jordan-Bernstein algebras there always exist derivations that are not inner. It is proved that a JordanBernstein algebra can be embedded as ideal of codimension 1 in another JordanBernstein algebra that has non-inner derivations. In [13] derivations in a second order Bernstein algebras are studied. In this case (and in general for a fc t/l -order Bernstein algebra) every derivation is still defined uniquely by a triple ( u , f , g ) , but relations that must be satisfied are sensibly more complicated. Lower and upper bounds of dim Der(A) are found and an explicit form of inner derivations in power-associative and Jordan second order Bernstein algebras is given.

6

SOME OTHER ASPECTS

A standard way to approach a structure is via the study of the substructures. In this direction, the knowledge of minimal and maximal subalgebras (ideals) of a Bernstein algebra becomes interesting. In [18] one-dimensional subalgebras and minimal subalgebras of a Bernstein algebra have been studied. It is also a usual problem to study conditions under which an isomorphism of the lattice of subalgebras (resp. ideals) of two algebras A and A' implies an isomorphism between those algebras and also to study properties of an algebra that are preserved under a lattice isomorphism. Bertrand [7] proved that if A, A' are Bernstein algebras and 6 : JC(A) —> C(A'} is a lattice isomorphism, then dimA = dimA' and dimS* = dimS' for every subalgebra S < A. Furthermore, if type(^l) = (1 + r, 0) or (1,1), then A and A' are isomorphic.

In [9] it was proved that the existence of a lattice isomorphism 9 : C(A) —> C-(A') implies (if type(A) ^ (1,1)) the existence of an idempotent e2 = e € A such that 0(< e >) =< e' >, with e'2 = e' e A'. In general a lattice isomorphism 6 : C(A) —» C(A') does not satisfy 9(Kerui) = Kerui'. If the barideal of A is not mapped into the barideal of A', then the set of onedimensional subalgebras of A generated by one idempotent element is not applied in the analogous set of subalgebras of A'. The existence of such isomorphism imposes strict restrictions on the structure of the Bernstein algebra A. Indeed, in [51] it was proved that the existence of a lattice isomorphism 9 : £(A) —> C(A') that does not preserve the nucleus implies that both algebras A and A' are exclusive and type(^4) = type(j4') = (1 + r, s), with s = 0 or 1. Furthermore, if type(A) ^ (1,1), then 9(A2) = A'2. In general the isomorphism of the algebras A and A' does not follow from the existence of the isomorphism 0, but it happens under some additional assumptions given in the paper. If two Bernstein algebras A and A' have isomorphic lattices of subalgebras we may always assume a lattice isomorphism 9 : C(A) —> C(A') that applies the barideal of A to the barideal of A', as it is proved in [52]. The type of a Bernstein algebra and some properties, like the fact of being exclusive, normal, genetic, Jordan or totally orthogonal are preserved under a lattice isomorphism.

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In [58] Bernstein algebras having a lattice of subalgebras that is distributive (resp. complemented) are characterized. Although the isomorphism of the algebras does not follow from the isomorphism of the lattice of subalgebras, it is proved in [58] that (under the assumption dimU > 1 and existence of square roots in the field

F) the existence of a lattice isomorphism implies the existence of an automorphism if) : F —» F and a semilinear map <j>, <j) : A2 —> A'2 (of automorphism if>) between the square of the algebras A and A'. A similar study based on the lattice of ideals has been made. In [53] Bernstein algebras having a linear lattice of ideals are considered. Only two algebras satisfy this condition: 1. A-i = F e i + F u i ,

2. A2 = Fe + Fu + Fu2, where products are given in the obvious way. In [54] Bernstein algebras with a distributive lattice of ideals are studied and characterized. It is proved that a Jordan (resp. nuclear) Bernstein algebra has a distributive lattice of ideals if and only if it is either normal or exclusive. Using this fact we can prove that a Bernstein algebra that has a distributive lattice of ideals is trivial or its lattice is linear or it is exclusive. So the problem has been reduced to the exclusive case that can be totally characterized. If it is not trivial then dim Z = 1. The following result can be proved:

THEOREM 6.1 Given a vector space U, a nonzero endornorphism f G Endp(U) with coinciding minimal and characteristic polynomials and a fixed element u\ G U

there exists an exclusive Bernstein algebra B = B( U. Furthermore, B(ip,U,ui] ^B(^',U',u{) if and only if dim U =dim U', ]
Some properties, like the fact of being trivial, normal or nuclear, can be characterized via the lattice of ideals, but in general nonisomorphic Bernstein algebras may have isomorphic lattices of ideals and it is not known if exclusivity is preserved by isomorphisms between the lattice of ideals. Identities Polynomial identities are also important tools in the structure theory of algebras.

If A is an arbitrary algebra over the field F, we can associate an ideal T(A) to A that is an ideal of F < X >, the free nonassociative algebra over an infinite set of generators, that consists of all identities of A, that is, polynomials f(x\,... ,xn) G F < X > such that f(a\,..., a n ) = 0 V a i , . . . , an 6 A. In [2] generators of the ideal of identities satisfied by regular Bernstein algebras and exclusive Bernstein algebras are found. If (A, <jj) is a baric algebra we can consider the baric T-ideal T(A, w) that consists of all polynomials in R < X > which vanish under substitution of elements a € A with w(a) = 1

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Using the weight function w an identity of a baric algebra (A,ui) can be transformed into another identity, involving w, that holds for any values of variables in A. It suffices to replace each variable x by x/u>(x] and multiply by the common denominator. In [2] the following result is proved.

THEOREM 6.2 If (A,w) is a Jordan-Bernstein or nuclear algebra, then the ideal T(A, (jj) has a finite set of generators. Superalgebras The origin of superalgebras lies in Physics and they are, at the present moment, an important part of Mathematics. Their use has become an important tool, as can be shown by the techniques developed by Kemer to study identities in associative algebras in relation to the speach property. The problem is to prove the existence of a finite basis of the ideal of identities of a subvariety. If V is a variety of algebras, the usual way to define a V-superalgebra is:

DEFINITION 6.1 A 2-graded algebra A — AQ + A1 is a V-superalgebra if the Grassmann envelope algebra G(A) = G(V)o ® AO + G(V)i AI belongs to V. This definition can not be directly applied in the case of Bernstein algebras, since they do not form a variety. This fact opens several ways to "extend" the notion of Bernstein algebra to a "natural" notion of Bernstein superalgebra.

One way, that works well if we are interested in the study of identities, is the following. We may consider a baric algebra (A, uj) as an algebra with an additional multiplication a * b — aiu(b) that satisfies i) a * (be) = (a * 6) * c, ii) (a&) * c = (a * c)b = a(b * c) , iii) a * (6 * c) = (a * b) * c

Conversely, if A is a commutative algebra with an additional multiplication * satisfying conditions i) to iii), we can consider u(a) e EndpA denned by u>(a) : x —-> x* a and the linear mapping from A to Endp(A), a —» u>(a). Condition i) assures that it is a homomorphism and by ii) the image is an associative commutative subalgebra K of the centroid of A, Y(A) = { e EndF(A) Va, b 6 A (a)b = a(b) = (ab)(j>} = {(j) € Endp(A) Va & A (j)Rb = Rb4>}- By iii) a; is a homomorphism over K. DEFINITION 6.2 A generalized baric algebra is a commutative algebra with an additional multiplication * that satisfies i) to iii) above.

Let M be a variety of generalized baric algebras defined by multilinear identities {fi i € a;}. A Z2-graded generalized baric algebra A is called .A/f-superalgebra if its Grassmann envelope G(A) belongs to .M. In this way, .M-superalgebras are defined by the identities obtained from {/j} in the usual way followed for superalgebras. In [3] we prove:

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THEOREM 6.3 If (A,w) is a Jordan or nuclear Bernstein algebra, then the ideal T(A) is equal to the ideal of identities of the Grassmann envelope of a finitely generated baric superalgebra. Another way of considering superalgebras is to concentrate on the barideal of a Bernstein algebra. In this way the notion of (U, Z)-Bernstein algebra appears.

DEFINITION 6.3 Let B = U + Z be a commutative algebra over F that is a direct sum of two vector spaces U, Z that satisfy U2 C Z, UZ C U, Z2 C U. B is called a (U, Z)-Bernstein algebra (or graded Bernstein algebra) if it satisfies for all u G [/, z 6 Z 1. u3 = 0,

2. u(uz) = u, 3. uz2 = 0,

4- («22)2 = o, 5 (uz) = 0.

Notice that if A = Fe + Ue + Ze is a Bernstein algebra, then B = Kerw = Ue + Ze is a (Ue, Ze)-Bernstein algebra. Conversely, if B is a (U, Z)-Bernstein algebra, then A = Fe + B is a Bernstein algebra in which U = Ue and Z = Ze. So a graded Bernstein algebra defines uniquely a Bernstein algebra, however a Bernstein algebra can define several graded Bernstein algebras, depending of the chosen idempotent element (and the associated Peirce decomposition). Considering graded Bernstein algebras instead of Bernstein algebras, we have a variety of graded algebras and we can define the notion of Bernstein superalgebra.

DEFINITION 6.4 LetB = BQ+Bi be a superalgebra over the field F. LetG(B) = GQ BO + GI ® BI the Grassmann envelope algebra of B. Then B is a Bernstein superalgebra if G(B) is a graded Bernstein algebra. The notions of Bernstein module and Bernstein supermodule can now be given in the usual way. In [27] and [28] Bernstein superalgebras of low dimension are classified, up to isomorphism of superalgebras and up to isomorphism of Bernstein superalgebras. Bernstein supermodules are studied in [29] and the classification of the faithful irreducible ones is obtained. It is possible to have a graded Bernstein algebra that is also Jordan, but whose associated Bernstein algebra is not Jordan. In [45] the speciality of Bernstein Jordan algebras is considered. It is proved that every graded Jordan Bernstein algebra satisfies the Glennie identity G&. The authors of [45] construct a graded Bernstein Jordan algebra whose associated Bernstein algebra is also Jordan, but does not satisfy G§ and, consequently, is not special. There are many subjects and different directions that have been followed in the study of Bernstein algebras. For instance, in [10] exceptional Bernstein algebras are associated to some graphs, and properties of the Bernstein algebra are related with properties of the associated graph. As we mentioned at the beginning, our

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aim has not been to present an exhaustive overview of all the work in the area and we have chosen the mentioned topics by a "proximity" reason, since they have been obtained by people in the same research group. For the same reason we include only references of the mentioned topics.

REFERENCES [1] J. Bernad, A. Iltyakov and C. Martinez, Bernstein representations, Kluwer Ac. Publishers, 39-45, (1994). [2] J. Bernad, S. Gonzalez, A. Iltyakov and C. Martinez, On identities of baric algebras and superalgebras, J. of Algebra 197, 385-408, (1997). [3] J. Bernad, S. Gonzalez, A. Iltyakov and C. Martinez, Polynomial identities of Bernstein algebras of small dimension, J. of Algebra 207, 664-681, (1998).

[4] J. Bernad, S. Gonzalez and C. Martinez, On the nilpotency of the barideal of a Bernstein algebra, Comm. in Algebra 25(9), 2967-2985, (1997). [5] S. Bernstein, Principe de stationarite et generalisation de la loi de Mendel, Comptes Rendus Acad. Sci. Paris, 177, 528-531, (1923). [6] S. Bernstein, Demonstration mathematique de la loi d'herdite de Mendel, Comptes Rendus Acad. Sci. Paris, 177, 581-584, (1923). [7] M. Bertrand, Algebres non-associatives et algebres genetiques, Memorial des Sciences Mathematiques, CLXII, Gauthier-Villars Editeur, (1966) [8] T. Cortes, Classification of 4-dimensional Bernstein algebras, Comm. in Algebra 19 (5), 1429-1443, (1991). [9] T. Cortes, Lattice isomorphisms and isomorphisms, Nonassociative Algebraic Models, Ed. Nova Science Publish., 69-91, (1992). [10] R. Costa and H. Guzzo Jr., A class of exceptional Bernstein algebras associated to Graphs, Comm. in Algebra 25 No. 7, 2129-2139, (1997). [11] M.A Garcia Muniz, Ph.D. thesis, University of Oviedo, Spain (1998). [12] M.A Garcia Muniz and S. Gonzalez, Baric, Bernstein and Jordan algebras, Comm. in Algebra, 26(3),913-930, (1998). [13] M.A Garcia Muniz and C. Martinez, Derivations in second order Bernstein algebras, Nonassociative Algebra and its Applications, Marcel Dekker, 105-124, (2000). [14] H. Gonshor, Special train algebras arising in genetics, Proc. Edinburgh Math. Soc. (2) 12, 41-53, (1960). [15] H. Gonshor, Contributions to genetic algebras, Proc. Edinburgh Math. Soc. (2) 17, 289-298, (1971).

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[16] S. Gonzalez, Homotope algebra of a Bernstein algebra, Hadronic Mechanics and Nonpotential Interactions, Nova Science Publish., New York, 186-200, (1992). [17] S. Gonzalez, Inner derivations of Bernstein algebras, Linear Algebra and its Applications, 170, 206-210, (1992).

[18] S. Gonzalez, One dimensional subalgebras of a Bernstein algebra, Algebra and Logic, 30(4), 310-327, (1991). [19] S. Gonzalez, Quasiisomorphisms of Bernstein algebras, Comm. in Algebra, 21(11), 4153-4166, (1992). [20] S. Gonzalez, A. Grishkov and C. Martinez, A general splitting theorem for Bernstein algebras, Comm. in Algebra 26(8), 2529-2542, (1998). [21] S. Gonzalez, J.C. Gutierrez and C. Martfnez, On Bernstein algebras ofnthorder, Kluwer Ac. Publishers, 158-163, (1994). [22] S. Gonzalez, J.C. Gutierrez and C. Martinez, The Bernstein problem in dimension 5, J. of Algebra 177, 676-697, (1995). [23] S. Gonzalez, J.C. Gutierrez and C. Martinez, Classification of Bernstein algebras of type (3,n-3), Comm. in Algebra 23(1), 201-213, (1995).

[24] S. Gonzalez, J.C. Gutierrez and C. Martinez, Second order Bernstein algebras of dimension 4, Linear Algebra and its Applications 233, 243-273, (1996). [25] S. Gonzalez, J.C. Gutierrez and C. Martinez, On regular Bernstein algebras, Linear Algebra and its Applications 241, 389-400, (1996).

[26] S. Gonzalez and J. Laliena, Bernstein algebras and quantum mutation, Hadronic Mechanics and Nonpotential Interactions, Nova Science Publish., New York, 201-211, (1992). [27] S. Gonzalez, C. Lopez-Dfaz and C. Martinez, Bernstein superalgebras of low dimension, Comm. in Algebra 27(9), 4477-4492, (1999). [28] S. Gonzalez, C. Lopez-Dfaz and C. Martinez, Bernstein superalgebras of dimension 4, Nonassociative Algebra and its Applications, Marcel Dekker, Inc , 189-203, (2000). [29] S. Gonzalez, C. Lopez-Dfaz, C. Martinez and I. Shestakov, Bernstein superalgebras and their supermodules, J. of Algebra 212, 119-131, (1999).

[30] S. Gonzalez and C. Martinez, Idempotent elements in a Bernstein algebra, J. London Math. Soc.(2) 42, 430-436, (1991).

[31] S. Gonzalez and C. Martfnez, Bernstein algebras with zero derivation algebra, Linear Algebra and its Applications, 191, 235-245, (1993). [32] S. Gonzalez and C. Martfnez, On Bernstein algebras, Kluwer Ac. Publishers, 164-170, (1994).

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[33] S. Gonzalez, C. Martinez and P. Vicente, Power-Associative and Jordan 2ndorder Bernstein algebras, Nova Journal of Algebra and Geometry, Vol. 2, No. 4, 367-381 (1993).

[34] S. Gonzalez, C. Martinez and P. Vicente, Some special classes of 1nd-order Bernstein algebras, J. of Algebra 167, No. 3, 855-868, (1994). [35] S. Gonzalez, C. Martinez and P. Vicente, Idempotent elements in a 2nd-order Bernstein algebras, Comm. in Algebra 22 (2), 595-610, (1994).

[36] J.C. Gutierrez, The Bernstein problem for the type (n-2,2), J. of Algebra 181, 613-627, (1996). [37] J.C. Gutierrez, The Bernstein problem in dimension 6, J. of Algebra 185, 420-439, (1996).

[38] J.C. Gutierrez, Structure of Bernstein population of type (3,n-S), Linear Algebra Appl. 268, 17-32, (1998).

[39] J.C. Gutierrez, Solution of the Bernstein problem in the Non-regular Case, J. of Algebra 223, 109-132, (2000). [40] J.C. Gutierrez and C. Martinez, Nuclear Bernstein algebras with stochastic basis, J. of Algebra 217, 300-311, (1999). [41] H. Guzzo Jr. and P. Vicente, Train algebras of rank n which are Bernstein or Power-associative algebras, Nova J. Math., Game Theory and Algebra 6, No. 2-3, 103-112, (1997). [42] H. Guzzo Jr. and P. Vicente, On Bernstein and train algebras of rank 3 , Comm. in Algebra 26, No. 7, 2021-2032, (1998). [43] H. Guzzo Jr. and P. Vicente, Classification of 5-dimensional Power-associative 2nd-order Bernstein algebras, Submitted.

[44] P. Holgate, Genetic algebras satisfying Bernstein's stationarityPrinciple, J. London Math. Soc. (2) 9,613-623, (1975). [45] C. Lopez-Diaz, I. P. Shestakov and S. N. Sverchkov , On speciality of Bernstein Jordan algebras, Comm. in Algebra. To appear. [46] Yu. I. Lyubich, Mathematical structures in population genetics, in "Biomathematics", Vol 22, Springer-Verlag, Berlin/Heidelberg, (1992).

[47] C. Mallol, A propos del algebres de Bernstein, Ph.D. Thesis, Universite des Sciences et Techniques du Languedoc, Montpellier II (1989).

[48] C. Martinez , Inner Derivations in Jordan- Bernstein algebras, Hadronic Mechanics and Nonpotential Interactions, Nova Science Publish., New York, 217228, (1992). [49] C. Martinez, Isomorphisms of Bernstein algebras, J. of Algebra 160, 419-423, (1993).

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[50] C. Martfnez, Free nuclear Jordan-Bernstein algebras, J. of Algebra, 177, 676697, (1995). [51] C. Martfnez and J.A. Sanchez-Nadal, Bernstein algebras with a lattice isomorphism that does not preserve the nucleus, Comm. in Algebra 22, 4781-4792, (1994). [52] C. Martfnez and J.A. Sanchez-Nadal, Lattice isomorphism of Bernstein algebras, Kluwer Ac. Publishers, 269-274, (1994). [53] C. Martfnez and J. Seto, Bernstein algebras whose lattice of ideals is linear, Kluwer Ac. Publishers, 175-178, (1994). [54] C. Martfnez and J. Seto, Bernstein algebras whose lattice of ideals is distributive, Nonassociative Algebra and its Applications, Marcel Dekker, Inc , 357-364, (2000). [55] Yu. A. Medvedev and E. I. Zelmanov, Solvable Jordan algebras, Comm. in ALgebra 13(6), 1389-1414, (1985). [56] M. Ouattara, J. Bayara and A. Micali, Autour de la condition d'Engel dans les lebres de Bernstein, Comm. in Algebra 28(1), 363-373, (2000). [57] M. Lynn Reed, Algebraic structures of genetic inheritance, Bull. Amer. Math. Soc. 34, 107-131, (1997). [58] J.A. Sanchez-Nadal Isomorfismos de reticulos en algebras de Bernstein, Doctoral Dissertation, University of Oviedo, (1994) [59] R. D. Schafer, Structure of Genetic Algebras, American J. of Mathematics, 71,121-135, (1949). [60] V. G. Skosyrskii and E.I. Zelmanov, Special Jordan nilalgebras of bounded index, Algebra i Logica 22 (6), 626-635, (1985). [61] S. Walcher, Bernstein algebras which are Jordan algebras, Arch. Math. 50,

(1988) 218-222. [62] A. Worz-Busekros, Algebras in Genetics, Lecture Notes in Biomathematics, vol. 36, Springer-Verlag, New York, (1980). [63] A. Worz-Busekros, Bernstein Algebras, Arch. Math., 48, 388-398, (1987).

About an Algorithm of T. Oaku M. I. HARTILLO-HERMOSO, Departamento de Matematicas, Universidad de Cadiz. C/ Por-Vera, 54. 11403-Jerez de la Frontera. Cadiz Spain. E-mail: Isabel. hartillo@uca. es

Abstract

We give a new version of an algorithm of T. Oaku computing the global Bernstein polynomial associated to a polynomial / e C[xi, . . . ,xn\. The great difference is in the homogenization technique we use.

1

INTRODUCTION

Let An(C) be the Weyl algebra in n variables, that is the algebra of differential operators in n variables with coefficients in C[zi, . . . , xn}. This algebra is generated by the elements Zj, di i = 1, . . . ,n with relations [zj,Zj] = [<9j,<9j] = 0, [di,Xj] = Sij. The elements of >4 n (C) can be written in a unique way as a finite sum P = !Cae«" aa(x}da where aa(x) e C[x] with x = ( z i , . . . , z n ) and d - (di,...,dn). We shall denote this ring by An, instead of An(C) to simplify the notation. Let /(z) € C[z] and s a new and consider the ring of polynomials -4n[s]. Then by the Bernstein theorem (see [3]) there exists a b(s] £ C[s] with b(s) ^ 0 and there exists a P(s) 6 An[s] such that:

b(k}fk = p(k)(fk+l]

v^ez.

(1)

Polynomials b(s) G C[s] such that there exists a P(s) 6 An[s] verifying the equation (1) form an ideal Bf of C[s]. The Bernstein polynomial associated to / is the monic generator of that ideal. We denote this polynomial by 6/(s). Let MI be the free C[z, s][/~1]-module of rank one and denote its free generator by /s. We can define an action by An[s] on MI as follows:

241

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Hartillo-Hermoso

fk Let M be the submodule of MI generated by / s , that is M = An[s}fs- We write J={P&An[s}\P(f")=0}. We have that C^sH/" 1 ]/ 8 ~ M[f~1}. Now we define a structure of C[i]module, where t is a new variable. Assume g(s) <E C[x, sjf/" 1 ], then we define the action: Generally, given a
Let us point out that this action does not commute with the action of An s} over M[f~1}. That is, if P e An and 9? e C[t] then [P, yj] = 0, but (s + l)t = ts. In general, we have

Since this action is a bijection over M[/~ 1 ], we can consider t~l . This leads to define a connection over the C[i]-module M[f~1} in the following way:

Let fc e Z. We can define

Then we have

M = M0,

MfecMfc_i,

On the other hand, the Mk are stable under the action of t. This action is defined for P(s) e An[s by Then we have that tMk = Mfc+1. That is, 1

} = (J Mfc =

Note that A_d_M^ C Mk-i although this sequence is not stable by this action. dt

Let An+i = C[x,t](d,dt).

We define the structure of ^4n+1-module on M[f~1}.

For di we define the action as before, due to the formal derivation. We define the action of dt as the action of A_d_. Given g(s) e C[x, slf/" 1 ], we have: dt

d t ( g ( S ) f s ) = -sg(s - I)/ s-l

An Algorithm of T. Oaku

243

For f = X]l=o ak(x)tk (:

[x,t] we define:

i = ^Tak(x)g(s-

s+k

Let N = An+ifs, it holds M c N C M[r1} = M[f-1}. Hence, it is clear that

+

(t ~ f ) f ' = 0,

-dt

fs = 0.

Even more, these elements generate the annihilator of fs in An+i-

LEMMA 1.1. Let be f & C[x], where x = (z 1; . . . ,z n ). Then the left ideal

is maximal in An+iPROOF.- Let P e An+i, P ^ J. Then we must prove that (P) + J = An+iWe write

Using the generators of J, we can write:

i=0

for some m g N and a^(x) e C[x]. Then, for an element in this form, we have:

(P) If we are in the case ai(x) — 0 for all i ^ 0, we are in a simplest case which will be solved later. Otherwise, we have that / € C[x]. Then it commutes with dt, and ao(x) G C[x], which commutes with t. Hence: , t] = i=0

.

.i=0

.

, t]

i=0

and, once more as ai(x) e C[x] commutes with t:

We have reduced the degree of dt in our element, and then we have an element a(x) € (P) + J, with a(x) ^ 0 (this fact comes from P g J). If a(x) e C we have

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finished. Let us suppose the degree of a(x) to be greater than 1. Then there is an index i which verifies that the degree of a(x) with respect to Xj is greater than 1. Using that a(x) <E C[x] commutes with dt as well as with the polynomials of C[x]:

+ 7^Tdt. afc) \=(di' a(x)} = -4^-Then

da(x] ,„. T —~- e (P) + J

, /da(x] and deg ' ^ '

we have finally found an element c(x) € (P) + J which deg(c(x)) = 0, that is c(x) e C and we have finished the proof. D

In An+i we can consider the Kashiwara-Malgrange filtration, which comes from the vector subspaces:

Vm(An+1) = P =

aw,«,f}xat"dfi% e A•Wi

= Oifz/ —

We use this filtration and the above results to calculate the Bernstein polynomial of/. PROPOSITION 1.2. Lei / e C[x]. T/iere exisi P(s) e An[s] and b(s) e C[s], b(s) ^ 0 swc/i f/iai P(s)/ s+1 = b ( s ) f s if and only if there exist Q G V-i(A.n+i) and c(s) e C[s], c(s) ^ 0 swc/i that c(-dtt) -Q e J.

PROOF.- Let us consider

P(s)fs+1 = 6(s)/s

with

P(s)6^ n [s]

and 6(s) e C[s b(s) £ 0.

Using the action defined before we have

p(-dtt)tfs = b(-dtt)fs

^ (b(-dtt) - p(-dtt}t)fs = o,

from which it immediately follows that P(—dtt)t € V-i(An+i)Conversely, let us consider c(—dtt) — Q € J with Q € V_i(An+i). Then we can write Q = ^T=i Qi(tdt}t\ Take the element

and it is now clear that

D

With these results we have reduced the computation of the Bernstein polynomial to finding a special element of the ideal J, this element belongs to Vo(^4 n +i). Moreover it is the element of smallest degree because it is polynomical in —dtt.

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2

245

HOMOGENIZATION OF DIFFERENTIAL OPERATORS

The greatest drawback we find in the algorithm of T. Oaku [10] for the computation of the Bernstein polynomial is the calculation of a Grobner basis with respect to a monomial ordering which considers firstly the Kashiwara-Malgrange filtration. Precisely for this reason, a monomial ordering with this restriction can not be a well ordering, and we can not use the algorithms in the Weyl algebra which parallel Buchberger's one. For a deep development of these results the reader may consult [6]. The goal of computing a basis of this form is obtaining the elements of degree 0, that is bf(-dtt). Then T. Oaku gives, in a natural way, an homogenization procedure for differential operators. The use of this technique preserves the Kashiwara-Malgrange filtration. The following step is to consider an ideal in .4n[s]. It is generated by the principal symbol, for the V-filtration, of the elements of F-degree 0 of /. This process is very similar to graded ring theory. In [7], we find a general technique which begins with an "admissible filtration" in the Weyl algebra. Then it introduces an homogenization in the Rees algebra with respect to the Bernstein filtration. This leads us to a standard basis in the Weyl algebra which preserves the "admissible filtration". The Kashiwara-Malgrange filtration is an admissible filtration, that is why we use the new technique. We give in this section the principal results we use in the calculation of the Bernstein polynomial. Any element of An+i is of the form: P= Y 'W.a./jz"^^ ^,i/a,/3

We consider the Kashiwara-Malgrange filtration, that is:

Vm = {P € An+i


if v - fj. > m}.

For P s An+i we define ordy(P) as the minimum m such that P € Vm. We define also the principal symbol of P as W(P)

=

v — n=ordv(P)

Let -< be a monomial well ordering. We define the following ordering in p^ 2rl + 2 :

v - n < v' — fj,' _ M = ^' _ M' or •{ and v

The monomial ordering -
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system of generators of /. But given a ^/-standard basis of /, {G1; . . . , G;}, the family {ov(Gi), . . . , ov(G;)} is a system of generators of giy(/) [7]. We even know that the family (ov(Gi), . . . , <7y(G;)} is a Grobner basis of giy(7) for the well ordering -<. To compute a F-standard basis we must use homogenization techniques. The great difference we find between [7] and [10] is that the ring in which the Weyl 3ra is embedded in order to homogenizing the operators is

with relations

[x0,Xi] = [x0,t] = {x0,di} = [x0,dt] =

j] = [xi,t] = [di,dj} = [di,dt] = 0,

This is a graded algebra, with the degree of the monomial XQXat^d^d^ being k + \a + n + \(3\ + v. This algebra is isomorphic to the Rees algebra associated to the Bernstein filtration of An+iWe define in a natural way the homogenization of the operator P £ An+i. Given P we denote by ord T (P) its total order

ord T (P) = max{|a + n + \/3\ + v \ aa^^tV ^ 0} and we define the homogenization of P, denoted by h(P), as:

Now we define a well ordering -
' k + \a + fj. + \/3\ + v < k' + a' + // + |/3'| + v1

k + \a\ + fj. + \J3\ + v = k' + \a'\ + // + \j3'\ + v1 or { and Starting from a system of generators {Pi, . . . ,Pr} of an ideal / of An+i, we just consider the ideal / in the Rees algebra generated by {h(Pi), . . . , h(Pr}}. Using an algorithm which resembles Buchberger's, we obtain a standard basis of / for ~
An Algorithm of T. Oaku

3

247

COMPUTATION OF THE BERNSTEIN POLYNOMIAL

First, we fix the notations and the orderings we shall use later. Let -< be a monomial well ordering in N2n+2 such that it is an elimination order for x and d. For example, we may consider a Ith elimination order of Bayer and Stillman [2]. We define the mapping

An+i f

—>

gr°v(An+i) -

£ OT <MP)p -or

jf .f

ordy(P) > 0

Every P & gTy(An+i) can be written in an unique way as a polynomial in —dtt. Let us write in this case P(—dtt). We substitute formally —dtt by s. We write in this case Q(P) for P(s). Then, we consider the ring homomorphism:

P I—> 0(P). Then © is a ring isomorphism.

THEOREM 3.1. Let be f ( x ) e C[x] where x = (xl,. . . ,xn). Let J be the left ideal of An+i defined by:

Let Q = {GI, . . . ,Gr} be a V -standard basis of J with respect to the ordering previously fixed. Define Q' by

G' = {av(Gi), . . . ,av(Gr)} n CM,] = {Gi, . . . , G'J. / / ? P SPf

is a system of generators of the ideal Bf ofC[s]. REMARK 3.2. T. Oaku gave a complete computational solution to the open problem of calculating the Bernstein polynomial in [10]. His work was based, as well as ours, on the Grobner basis theory for rings of differential operators but some important differences between his algorithm and ours must be emphasized. First of all, Oaku goes from J C An+i to a homogenized ideal Jh C An+i[zo] but, neither is Jh the homogenization we used, nor is ,4ra+i[xo] the Rees algebra of An+i with respect to the Bernstein filtration. Afterwards, his extremely specific homogenization process leads to the necessity of performing two Grobner basis computations in order to obtain the Bernstein polynomial. This is due to the fact that his method for obtaining a Grobner basis of a(^(J)) (an analog for gry(J)) does not allow him to eliminate the variables xi, ...,xn, and this final step requires then another Grobner basis (now in a polynomial ring) to be computed. As it has been stated in theorem 3.1 our algorithm

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needs only one Grobner basis computation to calculate the Bernstein polynomial of /. NOTE 3.3. We have obtained a system of generators of an ideal of C[s], which is a principal ideal domain. To compute the Bernstein polynomial, we just need to compute the greatest common divisor of these elements. PROOF.- We shall use proposition 1.2 to prove this result. The mapping 0 just takes —dtt to s. We do not use the action of 0, and we shall consider the Bernstein polynomial b(—dtt) using the fact that 0 is an isomorphism. We have, using 1.2, that an element g € C[— dtt] belongs to B/ if and only if there is an element Q G V^i(An+i) such that g — Q e J. Equivalently, it suffices that g <E gry(J)nC[—dtt]. Because if we have an element g — Q g J in the previous way, then ffv(g — Q) = 9 G gr1/(J). Besides oidy(g) = 0 and it is V- homogeneous. Hence g 6 gr^(J) n C[-dti\. Conversely, if we have an element g € gr^(J) n C[— dtt], then g € C[t,<9t] and g & gr^(J) and therefore g 6 C[— dtt]. We have g e gry(J), so there is an element F e J such that ay(F) = g, then F = g -Q, where Q e V-i(An+i)Then, in order to prove this theorem it suffices to prove that Q((p(Q')) is a system of generators of gr^(J) n C[— dtt]. We have that the principal symbol ov(CJ) = {ov(Gi), . . . ,ov(G r )} is astandard basis of gry (J) for -<. If we choose -< to be an elimination order, then we have that Q' is a system of generators of gr^(J) n C [ t , d t ] , and these generators result to be ^-homogeneous.

Let us see that
H — y ^HjGj. i=i The element H is F-homogeneous and its F-order is equal to 0. The elements G't are ^-homogeneous and they lie in C[t,dt]. We shall denote by m^ its l^-order. It is clear that we can always choose Hi ^-homogeneous with y-order — m^ and in We write Hi as: 1

~

[d~m*

if if

mi mi

>0 < 0.

This leads to

where the H[ are F-homogeneous of K-order 0 and belong to C[t,dt}. Therefore H- 6 C[-dtt]. We have proved that ip(Q') generates gr^( J) n C[-dtt}. D EXAMPLES 3.4. Using a program in COMMON LISP due to prof. Jose Maria Ucha we have computed the algorithm described before. We find the Bernstein

An Algorithm of T. Oaku

249

polynomial in these cases: bf(s)

x2 + xy3 + y4

x2 + xy2 + y3

(x

x(x + y)(xz •

y(x-

s+1

(* + !)

|)(s + i)3

xy(x + y ) ( x z •

REFERENCES [1] A. Assi, F. J. Castro Jimenez, J. M. Granger. How to calculate the slopes of a P-module. Compositio Mathematica, 104, 107-123, 1996. [2] D. Bayer and M. Stillman. A theorem on refining division orders by the reverse lexicographic order. Duke J. Math. 55, 321-328, 1987. [3] I. N. Bernstein. The analytic continuation of generalized functions with respect to a parameter. Functional Anal. Appl. 6, 273-285, 1972. [4] J.-E. Bjork. Rings of Differential

Operators. North-Holland, 1979.

[5] A. Borel et al. Algebraic D-modules. Academic Press, Boston, 1987. [6] F. J. Castro Jimenez. These de Seme cycle Universite Paris VII, 1984. [7] F. J. Castro Jimenez and L. Narvaez Macarro. Homogenising differential operators. Prepublicaciones de la Facultad de Matematicas, Universidad de Sevilla, 36, 1997. [8] M. Kashiwara. Vanishing cycle sheaves and holonomic system of differential equations. Lecture Notes in Math. 1016, Springer-Verlag, 134-142, 1983. [9] B. Malgrange. Le polynome de Bernstein d'une singularite isolee, Lecture Notes in Math. 459, Springer-Verlag, 98-119, 1975.

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[10]

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T. Oaku. An algorithm of computing 6-functions. Duke Mathematical Journal, 87, vol. 1, 115-132, 1997.

Minimal Injective Resolutions: Old and New M. P. MALLIAVIN, Universite Paris VI. 10 Rue St. Louis, L'Ile-75004, Paris. France. E-mail: [email protected]

I

INTRODUCTION

It is well know that if R is a commutative noetherian ring and if 0 —->R —>I

is a minimal injective resolution of R than

where E(-) denotes an injective envelope of a fl-module, /^(p) is the cardinal

dim fc and fc(p) is the residual field at p, i.e. fc(p) = (_R/p) p .

If inj dim R < oo and R is commutative then, by H. Bass, each indecomposable injective modules E ( R / f ) appears exactly in I1 for i = htp. More precisely H. Bass proved that /Mj(p) = 1 if htp = i and /Zj(p) = 0 if not, so 7* is the direct sum of injective hulls of R/ip, where p runs in the set of prime ideals of height i in R. To obtain analogous results for non commutative rings, we need alternative hypotheses. Let A be a left and right noetherian ring. It is known for a long time (by [13]) that the injective dimension of the module AA is equal to the injective dimension of A -A if the two are finite and in this case we note inj dim(A) = d this common dimension. One of the main tools in the theory of finite injective dimensional noetherian rings is the spectral sequence of Ischebeck (~ 75'): If A is a noetherian and 251

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Malliavin

injdim/i = d and if M is a left noetherian A-module, there exists a spectral sequence

E%q(M) = ExtpA(ExtqA(M,A),A) where W>~q(M} = 0 if p ^ q and H°(M) = M. 2

=> mp~q(M)

GRADE AND AUSLANDER-GORENSTEIN RINGS

Let A a noetherian ring and M a left .A-module. We name grade of M denoted j ( M ) the natural number or +00 defined as follow

Of course j(Q) = +00. If inj dim A = d < oo, one proves that j^(M) < d for every non zero A-module

M. DEFINITION 2.1 Let A be a noetherian ring. A left (or right) finitely generated A-module M satisfies the Auslander condition if for any q > 0 one has j(N) > q for every A-submodule N of finite type of ExtgA(M,A), DEFINITION 2.2 The noetherian ring A is said Auslander— Gorenstein of dimension d if

1. mjdimA = d < oo, 2. every finitely generated right (resp.

left) A-module satisfies the Auslander

condition. EXAMPLES 2.3 Commutative Gorenstein rings; quasi-frobenius artinian rings, for instance kG for any field and any finite group, any enveloping algebra of finite dimensional Lie algebra, iterated Ore polynomial algebras

k[Xi}[X2;T2,S-2} . . . [Xm;Tm,6m} where TJ is an automorphism and 6j a Tj-derivation of k[Xi][. . .][Xj',Tj,6j],

([2],

[4])REMARK 2.4 If in 1. we assume gidimA < oo we say that A is Auslander regular. All the examples above (except perhaps commutative Gorenstein rings and Q.F.artinian rings) are Auslander regular.

3

COHEN-MACAULAY CONDITION

Recall that if A is an algebra over a field k

GK dim A = supv lim ——;——— Ign where V is a dimensional finite subspace of A with I & V and if M is an A-module finitely generated i.e. M = AF such that dim^ F < oo, then _ . , , . . —— lgdimT/"F GK dim A = lira ——-———— Ign

if A is finitely generated by a finite dimensional space V .

Minimal Injective Resolutions

253

DEFINITION 3.1 Let B be a k-algebra noetherian with GKdimB = w 6 N. One says that B is Cohen— Macaulay (CM in short) if for every non zero finitely generated B -module one has

GKdimM +j(M) =u. EXAMPLES 3.2 1. If B is a commutative a/fine k-algebra, B is CM iff B is a equidimensional Cohen- Macaulay ring. 2. If k = C and B is an iterated skew Ore extension, B is CM by [6].

4

RESULTS

DEFINITION 4.1 Let B be a noetherian ring and p a completely prime ideal of B. We denote /^(p, B} the dimension over the skew field F r ( B / f ) of the vector space

Fr(B/p)®B/fExti(B/v,B). REMARK 4.2 /^j(p,5) ^ 0 iff E(B/y) injective resolution of

appears in P, the ith module of minimal

THEOREM 4.3 (See [8] and the bibliography given there). Let B be a k-algebra over a field k, Auslander-Gorestein satisfying CM, ancJp a completely prime ideal of B. Let h be the grade of the left B-module B/p. Then /x/,(p, B) ^ 0 andju,(p, B) = 0 This theorem applies to the enveloping algebras of solvable finite dimensional Lie algebras over a field of characteristic 0 or to the Weyl algebra An over a field of characteristic 0. Also it applies to many recent algebras, the so called g-skew iterated polynomial algebra over the field C say, ([6]), and when q 6 C is not a root of unity in C. In fact K. R. Goodearl and E. S. Letzter have proved that each prime ideal is completely prime.

As another example there is the C- algebra Oq(MnC), C- algebra of coordinates of the quantic matrices n x n (q ^ \/l), so is the quantum algebra of coordinates of SL(n).

CONJECTURE 4.4 This should be true for all the semi-simple quantum groups for a parameter q not a root of I . PROBLEM 4.5 If h = j(B/p) does Mp,B) = l? Le- does E(B/*>} appear in the appropriate term of the minimal injective resolution %B one and only one time? This is the case for B — U(£) where £ is a solvable finite dimensional Lie algebra over a field of characteristic 0 and for the Weyl algebra over a field of characteristic 0. I want to present you other cases for this works. Let Uq(M+) be the quantized enveloping algebra of a maximal nilpotent subalgebra A/"+ of a dimensional simple complex Lie algebra Q. We assume Q of type A, D, or E or of type B%.

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Malliavin

For the three first cases Ringel proved that Uq(J\f+) is a g-skew iterated Ore extension and for the last case I proved it using a result of [12]. This result is not known neither for the general case of a semi-simple Lie algebra nor for other types of simple Lie algebra (Bn, n > 3, Cn, F, G). Under the above hypothesis T. Lenagan and K. Goodearl proved that Ucl(Af+), q =£ VT, q € C, is an affine noetherian C-algebra with finite GK-dimension which is Auslander regular, Cohen-Macaulay and (thanks to C. Ringel and to M. Takeuchi) each of its prime ideals is completely prime.

PROPOSITION 4.6 For Q simple over C of type A2 ~ sl(3,k) or B2 ^ so(5,k}, any prime ideal ofUq(J\f+) q ^ \/I is a completely prime ideal. Iff is one of these ideals with d = j ( B / p ) then /^ d (p,T) = 1 and /ij(p,T) = 0 if i ^ d. PROOF: For A2 this is proved in [9] and it is easy because each (completely) prime ideal is generated by a regular normalizing sequence (zi,z2, . . . , zn) with n < 3. Using an old result of [3] there is a [/,j(A/"+)-isomorphism

For 82 the algebra is B = C[2,ei i3 ][ei,2,0-i,2][e 2 ,3, 02,3, 02,3] where z is central 0-1,2(0) = z, 0x2(61,3) = g~ 2 e 1]3 , 0- 2 , 3 (z) = z, cr 2 , 3 (ei, 2 ) = 2 , 02,3(61,3) =
Then the grade of Uq(J\f+}/'p is one and l l dimFr(-)®Ext ~ B(~,B) V P P

l l = dim Fr(S~ ~11-) Ext (--, S~~11B] = 1. / V P P /

n PROPOSITION 4.7 For type A2 (resp. B2) if B = Uq(M+), the minimal injective resolution gB is

where 1° = Fr(B) and for i > 1 where n = 3 (resp. n = 4) l?i ON Z?i 1n — — &-[ & 12/2

where p6Spec(T),htp=i

Minimal Infective Resolutions

255

and E\ is the injective envelope of a sum of critical modules, each of them being torsion modulo their annihilator. Moreover 1% = (0) and

where J runs over the family of 2 -sided ideals of finite codimension of B. In general, there is an antiautomorphism of U+ , u —> ul where u € U+ such that e\ = ej, {ei} being the Serre basis of A/"+. Take V = Cg [N] the restricted dual of U+ where N is the unipotent group with Lie N = A/"+. Then V is a left U+ -module under

(xf)(u)

=f(xlu)

each 6j acting locally nilpotently. Applying this to the case A% and B% gives with notations as in Proposition 4.7.

PROPOSITION 4.8 In is isomorphic to the restricted dual of B. PROOF: It is the same as in [1].

D

PROBLEM 4.9 The same result should be true for every Uq(J\f+) where A/"+ is a maximal nilpotent subalgebra of a semi-simple finite dimensional Lie algebra Q.

REFERENCES [1] Barou G., Malliavin, M. P.: Sur la resolution injective minimale de 1' algebre enveloppante d'une algebre de Lie resoluble. J. of Pure and Applied Algebra, 37, 1985, 1-25. [2] Bjork, J. E.: The Auslander condition on noetherian rings. Seminaire DubreilMalliavin 1987-88, Lecture Notes in Math., 1404, Springer Verlag, 1989, 137173. [3] Brown K. A., Levasseur T.: Cohomology of bimodules over enveloping algebras. Math. Z., 189, 1985, 393-413. [4] Ekstrom E.K.: The Auslander condition on graded and filtered noetherian rings. Seminaire Dubreil-Malliavin 1987-88, Lecture Notes in Math., 1404, Springer Verlag, 1989, 220-245. [5] Goodearl K.R., Lenagan T. H.: Catenarity in quantum algebras. J. of Pure and Applied Algebra, 111, 1996, 123-142.

[6] Goodearl K.R., Letzter E.S.: Prime ideals in skew and q-skew polynomials rings. Mem. Amer. Math. Soc., 109, 1994. [7] Levasseur T., Stafford J. T.: The quantum coordinate ring of the special linear group. J. of Pure and Applied Algebra, 86, 1993, 181-186. [8] Malliavin M. P.: Algebre de Bass et extensions de Ore iterees. Collect. Math., 45, 1994, 85-99.

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[9] Malliavin M. P.: La catenarite de la partie positive de 1'algebre enveloppante quantified de 1'algebre de Lie simple de type B%. Beitrdge zur Algebra und Geometme, 35, 1994, 73-83.

[10]

Ringel C. M.:PBW-basis of quantum groups. J. reine angew. Math., 470, 1996, 51-88.

[11]

Takeuchi M.: The g-bracket product and quantum enveloping algebras of classical types. J. Math. Soc. Japan, 42, 1990, 605-629.

[12]

Zaks A.: Injective dimension of semiprimary rings. J. of Algebra, 13, 1969, 73-89.

[13]

Zelevinsky A.V.: Parametrization of canonical bases via the generalized determinantal calculus. Summer school on Hall Algebras and quantum groups, Hesselberg, August 1999.

Special Divisors of Blowup Algebras S. E. MOREY, Department of Mathematics, Southwest Texas State University, San Marcos. Texas 78666-4616. USA. Ei-mail:[email protected] W. V. VASCONCELOS1, Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway. New Jersey 08854-8019. USA. E-ma,il:[email protected]

Abstract

The nature of the divisors of a Noetherian ring A provides a window to examine its arithmetical and geometric properties. Here we examine the divisors of Rees algebras of ideals-the so called blowup algebras-from the perspective of the operation of shifting. It permits the organization of some previous results in a more structured manner and at the same time predicts the occurrence of several new divisors. Of these, we single out the fundamental divisor that plays a more basic role than the canonical module.

1

INTRODUCTION

The set of divisors of a commutative Noetherian ring A is the class of rank one A-modules that satisfy the condition 52 of Serre. It is a notion based on that of unmixed height 1 ideals of a normal domain. The latter is rich in additional structures such as the divisor class group. The approach we will follow here is to define the divisors of A through the intervention of a finite homornorphism tp : S —> A, with 5 admitting a canonical module ws- The divisors of A are those A-modules arising in the following manner: Suppose dim A = d, dim S = d + n, and let L be a rank 1 A-module. We will call

Martially supported by an NSF grant

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the canonical module of L, and the set of divisors is made up of all w/,, more precisely, of their isomorphism classes, Div(yi). A distinguished divisor of A is its canonical module Another divisor is A itself, when it already satisfies the property 52. It is a consequence of the theory of the canonical module that Div(A) is independent of the homomorphism

A, a centerpiece of the cohomology theory of A; (ii) the divisor of Serre IR[It}; and (iii) its dual,

which will play an even more critical role than U>A- It will be labeled the fundamental divisor and one of our aims is to explore its role in the arithmetical study of the Rees algebra U = R[It}. We will emphasize the study of graded divisors for the following reason that we illustrate with the canonical module of Ti. Suppose that R is a Noetherian local ring and 7 is an ideal of positive height. We shall assume that R has a canonical module U>R. (As a consequence both ~R. — R[It] and Q = grj(R) also have canonical modules.) If / G / is a regular element of R, then / will also be a regular element on uj-fr, so that From this one has that the a-invariant of Tl is — 1. Thus the graded module (and T>(I) likewise) has a special representation

un = Dit + D2t2 + ••• + Dntn + • • • , where IV s are fractionary ideals of R. A basic property of these components — shared by many other graded divisors in general and all graded divisors when R is an integral domain — is that they define a decreasing filtration D\ D D% D • • •. This peculiar property will be exploited systematically. It defines an operation on the set Divh(ft) of such divisors, S : Divh(ft) H-* Divh(ft)

and a corresponding construction of a graded module over

= L/S(L). We shall call S the shifting operator of Divh(7£), and one of our goals is to determine conditions for a divisor L to be an end, that is not to lie in the image of S. This is connected to the notion of prolongation of a divisor, meaning from a given D what conditions are required to solve the equation S(L) = D, and what is the structure of the set of solutions. One of the results will show that under general conditions a Cohen-Macaulay divisor L can hardly be in the image of Sn for n large (determined by dim R) . On

Special Divisors of Blowup Algebras

259

the other hand, there remains the issue of the existence of ends and whether they are always finite in number (up to isomorphism). Applying the construction to the module T>(I) yields G(7). The structures of G(w-j^) and of G(/) as grj(/?)-modules will be used to describe very explicitly in many cases of interest the canonical modules of 72 and Q, but also will shed light on when the Cohen-Macaulayness of Q forces 72 to be Cohen-Macaulay. Another goal is to identify Cohen-Macaulay divisors and to examine their relationship to the reduction number of the ideal. Additionally, we are interested in finding the divisors that carry information about the Briangon-Skoda number of /. This paper is partly a review of the literature on the divisors of a blowup algebras but is organized around the new shifting operation and the introduction of a new kind of divisor, the fundamental divisor of a Rees algebra. Its contents are distributed as follows. Section 2 outlines our approach to the study of the divisors of a ring A. The definition and the construction of a pairing of divisors require that A be a homomorphic image of a Gorenstein ring. In case A is a Rees algebra, the other novel definition is that of the shifting operator

S : Divh(A) ^ Divh(^). In the next section we give an exposition of the divisor class group of a normal Rees algebra. Section 4 is an application of the theory of the shifting operator to describe a class of canonical modules of Rees algebras that have a well-packaged form. For a Cohen-Macaulay Rees algebra R[It] over a Gorenstein ring R, they correspond to ideals for which g i f ( R ) are Gorenstein. The more novel material here begins with the introduction of the fundamental divisor T>(I) of a Rees algebra 72 = R[It] in Section 5. It has the property <S(P(/)) = ui-fc, so in some fashion is a more primitive divisor than the canonical module of 72. The components of T)(I),

are important carriers of information about the Cohen-Macaulayness of 72, including the reduction number of I. We single out the class of ideals for which D\ ~ WR. We will say that such divisors have the 'expected' form. It will follow from a result of Lipman that ideals of regular local rings always have this property. We will identify other classes of ideals with similar behavior. In the brief section 6, we give a calculation of how the Cohen-Macaulayness of T>(I) also bounds the reduction number of I. Finally, for a Cohen-Macaulay local ring (R,m) of dimension d > 0, where OT = (m, /£), we examine several instances of the exact cohomology sequence

0 ^ HdM(I^}-i -

HdM(n}0 -^ Hdm(R) -^ HffVK)-!

-+ Hd^(n)0 = 0

which is one of the most significant exact sequences in the cohomology of Rees algebras. The vanishing of is equivalent to V(I) having the expected form. In several classes of ideals (ideals of near linear type, equimultiple) we identify such events.

260

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DIVISORS

We shall use various sources for general results on the theory of Cohen-Macaulay rings: [4], [6] for the general theory and [32] for special properties of Rees algebras. Unexplained terminology or notation can be found in [32]. Let R be a Noetherian ring and / C R be an ideal, / = (/i, • . • , /n)- We assume that R has finite Krull dimension, dim R = d, that its total ring of quotients is Artinian, and height / > 0. If R has a canonical module MR, we can use for the presentation morphism the standard one, S = R{Ti,...,Tn] —> R [ I t ] , when ws = WRS[—n}. The semigroup of divisors To obtain divisors we must supply fodder to the process outlined above. When A is the ring of polynomials R[t], some divisors are simply extensions of the ideals of R, JR[t\. If A is however a Rees algebra R [ I t ] , there are constructions like J(l,t)m C R[t], or more generally (Jit, . . . , Jmtm). Let us begin with the formal definition of divisors.

DEFINITION 2.1 Let A be a Noetherian ring. An ,4-module L is a divisor if it satisfies the following conditions: (i) L satisfies the condition (£2) of Serre; (ii) If K is the total ring of fractions of A then K ®A L ~ K. An equivalent way to describe a divisor is: If A has an Artinian total ring of quotients, L is a, module isomorphic to an ideal H C A all of whose primary components have codimension one. We begin by listing some elementary properties of divisors as defined here. We emphasize that a module L over the Noetherian ring A has rank 1 if it is torsionfree and L ®A K ~ K, where K is the total ring of quotients of A.

PROPOSITION 2.2 Let A be a Noetherian ring with an Artinian total ring of quotients that admits a presentation S —> A as above. The following hold:

(i) For any fractionary ideal L the module

satisfies the condition 5*2 • (ii) There is a natural isomorphism

D -» Ext on modules D = U>L.

(iii) // DI and D2 are divisors, the operation

defines a monoid structure on the elements of Div(A) which are principal in codimension one.

Special Divisors of Blowup Algebras

261

(iv) If A satisfies the condition 82 and is Gorenstein in codimension one, then

UJL ~ Hom^(Hom(L, A), A) is an isomorphism for every rank one module L. (v) If A is an integrally closed domain then for every divisor L, LoHom / i(L, A) ~ A. As a consequence, the isomorphism classes of divisors form a group, the so-called divisor class group of A. (vi) If A is an integral domain with a canonical module UA, the smallest rational extension of A that has the property 82 is called its 82 -ification, 0-> A —> A —>C ->0.

(C in the graded case is the so-called Hartshorne-Rao's module of A). It can be obtained as

^

A = RomA(u>A,LL>A) = (vii) A and A have the same sets of divisors, Div(A) — Div(A). Homogeneous divisors and the shifting operation

The homogeneous divisors of 72. = R[It] are those of the form

n>r

The role of the components Dn is its distinguishing feature. We begin with an observation which will be used in several constructions. Note also its extensions to more general nitrations such as the integral closure filtration, the Ratliff-Rush filtration and some symbolic powers. There are other divisors of a Rees algebra R[It] with the property that for every nonzero divisor / € /, L/ ~ JfR[t] f°r some ideal J C R— that is Lf is 'extended' from R. This is, for instance, the case for all divisors if R is integrally closed. There are also examples of homogeneous ideals in more general cases— and we will consider one later.

DEFINITION 2.3 For each graded, divisorial ideal of R[It]

n>r

the subideal n>r

is the shifting of L.

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Note that in view of the exact sequence 1]) —> L —> L r f - > 0 ,

r = inf{n Ln ^ 0},

S(L) is also a divisor, as asserted in the introduction. It guarantees that the graded module over gij(R)

0 -> S(L) —> L —> G(L) -> 0 has positive depth. The following shows that the shifting operation is well behaved in all integral domains.

PROPOSITION 2.4 Let R be a Noetherian integral domain with a canonical module and let I be a nonzero ideal. Then for every homogeneous divisor L of R [ I t ] ,

S(L) c L. PROOF: It will suffice to show that the conductor ideal L : <S(L) has codimension at least 2. From the construction, It is contained in the conductor L : S(L). On the other hand, passing to the field of fractions K of R, we obtain KL = KS(L) = K[t]tr , which shows that the conductor contains a nonzero element x of R. This means that it contains (It,x) which is an ideal of height at least two as ( I t ) is a prime ideal. Since L has the condition (52) of Serre, S(L) C L. D Prolongation of a divisor The process of shifting can run backwards under some conditions. Let

L = L1t + L2t2 + ••• be a divisor. By a prolongation we mean another divisor

D = L0t + L^2 + • • • , that is S(D) = L. As a prerequisite we must have that ILo C LI but more conditions are to be met. We consider a case when prolongation can be achieved.

PROPOSITION 2.5 Let R be a Cohen-Macaulay domain with canonical ideal MR and let I be an ideal of height at least 2. Let

L = Lit + L2t2 + ••• ,

IujRcLiCuR

be a divisor of R[It}. Consider the ideal D defined by 0 -> Lt —> D —> u>Rt -> 0.

Then D is divisorial. PROOF: As before we are still assuming that locally R is a homomorphic image of a Gorenstein ring. We may assume R is local of dimension d and write A = R[It]

Special Divisors of Blowup Algebras

263

as a homomorphic image of a graded Gorenstein domain 5 of dimension d+1. The condition £2 f°r an ideal such as D is equivalent to height (ann (Extls(D, u> s ))) > 2 + i,

i > 0.

Consider the usual Ext sequence 0 —> Hom(wRi,o;s) = 0

D,w s ) -> Ext l (Lt,w s ) -» Ext 2 (w fl ^,w s ) = 0, and the isomorphisms

L^ws

i > 1.

For z > 1 the isomorphism just above says that height (ann (Extl(D, ws))) > 2 + z. For i = 1, note that .Ri"1 is already annihilated by It. What is needed is for the cokernel of the map

Let us work out a more general approach to prolongations. Suppose L = Sn>i Lntn is a divisor of R[It] that we seek to prolong into D = 5Zn>i Ln-itnThere are at least two requirements on LQ:

LI I-Lo

C C

L0 LI,

plus some divisoriality condition on LQ yet to be fully determined. There is however at least one situation when it leads to a unique "solution" for LQ.

PROPOSITION 2.7 Suppose that height / > 2 and LI is an ideal with the condition 83 of Serre. Then LQ = LI and D is a prolongation of L.

PROOF: Consider the exact sequence 0 -> / —>R —» R/I -> 0,

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and apply Honift(-,Li):

0 -> EomR(R/I, Li) = 0 — » EomR(R, LI) — > Hom R (/, LI) Since 7 has height at least two and LI satisfies the condition 82 of Serre, / contains two elements forming a regular sequence on LI; thus Ext 1R (R/I, LI) = 0. This means that LI = Hom^/, LI). Now note that LQ C Honifl(7, LQ), which with the other containment shows that LQ = LI. One can now repeat the proof of Proposition 2.5. Using the sequence

Ext z s (L x ,w s ) —-> Ext*s(I>,ws) — > Extls(L,ws) one sees that the codimension of the module in the middle must be at least i + 2. (This is the place where we used the condition £3 on LI.)

D

The process of shifting cannot be indefinitely applied without changing key properties of the divisor. Consider for instance the case of a divisor

L = Rt + L2t2 + --- + Lntn + • • • ,

Ln+1cLn.

The exact sequence 0 -> L n > 2 —> L —> #t -> 0, shows that if R and L are both Cohen-Macaulay then both <S(L) and G(L) will also be Cohen-Macaulay. This follows simply from the standard device of examining the relationship between the Cohen-Macaulayness of Tl and g r j ( R ) in terms of the following exact sequences (first paired in [15]):

Q

(1)

Q^It- R[It] —> R[It] —> R -> 0,

(2)

with the naive isomorphism It • R[It] ~ / • /?[/£] playing a pivotal role. Another general property of Rees algebras that we shall make use of is the following observation of Valla:

PROPOSITION 2.8 Suppose that the residue field of R is infinite. Then (a) R[It] has a system of parameters of the form

{xi,x2 + ait, £3 + ait, . . . ,xd + ad-i where {xi, . . . , Xd] is a system of parameters of R and (ai, . . . , a^) is a reduction of I .

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265

(b) (Valla) // / is m-primary then one can choose the a» 's to form an arbitrary minimal reduction of I and set Xi = a,i, Vi. The following shows that some restriction must be placed on iterated applications of shifting. For simplicity we use that the early components are isomorphic to R but using Cohen-Macaulay ideals of height one would serve the same purpose.

PROPOSITION 2.9 Let R be a Cohen-Macaulay ring and let I be an ideal of height at least 2. Suppose

L = Rt + • • • + Rtr + Lr+itr+l + ••• is a Cohen-Macaulay divisor. Then r < height /. PROOF: We may assume that (R, m) is a local ring of dimension d > 2 and that / is m-primary. Let J = ( a i , . . . , a < i ) be a minimal reduction of /. According to Proposition 2.8, the elements aj, a2 — ait,..., ad — ad—it, ajt form a system of parameters for R[It] and therefore will form a regular sequence on the CohenMacaulay L. (More properly, on LM where M is the maximal homogeneous ideal of R[It}.) Observe the result of the action of this system of parameters on Rt + • • • + Rtr: after reduction modulo aj, multiplication by a2 — a-\t on R/(ai)t + • • • + R/(ai)tr~1 has the same effect as multiplication by a2 only so that

L/(ai,a2 -ait)L = R/(ai,a2)t + • • • + R/(al,a2)tr~1 + • • • . Repeating the argument up to the element a^ — ad-it, gives a module

L/(ai,...,ad-ad-it)L

= R/(alt..., ad)t + • • • + R/(alt... ,ad)tr~d+1 +higher components.

It is clear that (with r > d) adt is not regular on this module.

D

Dimension one

Let R be an integral domain of Krull dimension 1 and let / be a nonzero ideal. Set R = U n>1 Hom#(/ ra , /") and / = IR. Note that if UJR is the canonical module of R, then UJR = UJR. The module / is a locally free ideal of R and the Rees algebra 72. = R[It] is the 52~ification of 72. = R[It] (see [26]). The canonical and fundamental divisors of 72. are now easy to describe: ui-j^ = uifttR, and 'D(I) c± uj^. The other divisors are more difficult to describe, but the following observation is useful. For simplicity suppose that R is a local ring and J = (a) is a minimal reduction of /. The homogeneous divisors of 72. are the fractionary ideals of 72. = R[at] on which {a, at} is a regular sequence. Let J = ZX>i Jntn be such a divisor.

We claim that J = JitR. Using the shifting operator repeatedly it will suffice to show that J2 = aJi. Suppose that bt2 € J; since J2 C Ji, we have the relation with coefficients in J,

at-bt = a- bt2. Thus bt2 = at • ct, with ct G Jit.

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DIVISOR CLASS GROUP

The divisors of a Rees algebra 72 — R[It] are easier to control when R is an integrally closed domain and / is a normal ideal. We will assume these conditions hold in this section and give a general description of the divisor class group Cl(72). There are two approaches to the calculation of Cl(72). One can apply to 72 general localization properties that isolate distinguished batches of divisors (see [7]), or make use of specific properties of Rees algebras. The second method will also be useful for certain algebras of symbolic powers and for the integral closure of ordinary Rees algebras. In the main, our discussion is a merge of several sources: [10], [11], [12], [28], [34] and [35].

THEOREM 3.1 Let R be an integrally closed Noetherian domain and let I be a normal ideal of height at least two. There is an exact sequence of divisor class groups 0 -> H —> Cl(ft) -£+ Cl(fl) -> 0,

where H is a free abelian group generated by the classes [P] defined by the minimal primes of 772. PROOF: Since 72 is a graded ring, to construct Cl(72), it will suffice to consider

homogeneous divisors. Let P = X^n>o Pntn be a height one prime of 71. Define a function from this set of divisors into the group CI(_R) by putting

f ( P ) = [PolThis map clearly defines a homomorphism of divisor class groups, which we denote by the same symbol. The following properties are easy to verify, (i) For each divisorial prime ideal p of R, T(p) = 72 n p72p is a divisorial prime of 'R, and
ai[Qi] + ••• + ar[Qr] = ar+1[Qr+1} + ••• + as[Qs], where the a^ 's are non-negative integers. For each Qi denote by q; its component of degree 0. The q^ are prime ideals of height at least two. Converted to a relation in 72, this means that there exist homogeneous elements / = atm, g = btn of 72, such that the ideals

f-f[Q?,

9- H QT

i=l

i=r+l

have the same value at each valuation of 72.. In particular for a valuation vf defined by T(p) or arising from a valuation of K[t],

Special Divisors of Blowup Algebras

267

In particular a = 6, m = n, so we can cancel / and g, which leads to a contradiction as Oi=i QT nas a nonzero value at the valuation denned by Q\. D REMARK 3.2 The same description of divisor class groups applies equally to other filtrations. For example, if R is a normal domain, / is an ideal of height at least two and the integral closure 72. of 72. is Noetherian, then Cl(72.) has a presentation as above. Similar comments apply to Noetherian algebras of symbolic powers.

COROLLARY 3.3 Let R be a regular local ring and let I be a prime ideal of height two. If S is either the integral closure of Ti = R[It] or the symbolic power algebra 72,j = ^n>o I tn (assumed Noetherian), then <S is quasi- Gorenstein. PROOF: By the theorem and remark, S is a Krull domain whose divisor class group is generated by the classes of the minimal primes of IS. To ascertain that S is quasi-Gorenstein, we localize at I and get in both cases Rj[Ijt], which has a unique such prime, say P. Thus [u>s] = o\P\. But the localization is a Gorenstein ring since // is generated by a regular sequence of two elements. Thus a = 0, which shows that ws = tS. D

To identify the set {Qi, . . . , Qs} of minimal primes of 7.72. is usually very difficult. We consider a few cases of interest. (i) Suppose that I is an unmixed normal ideal of height one. The inclusion R '—> 72 defines an embedding of divisor class groups

C\(R) -^ 0(71),

[p] ^ [T(p)]

which is an isomorphism.

(ii) Let R be a factorial domain and let / be a prime ideal that is generically a complete intersection and is such that the associated graded ring gr/(.R) is an integral domain (in this case 72 is automatically normal, see the comment below). In this case, C\(R) = 0 and C!(72) = Ideals of linear type

Let / be a (normal) ideal of linear type, that is 72. is the symmetric algebra SR(!) (see [32, p. 138]). Let

H* JU Rn — » / _> 0 be a presentation of /. One can determine the minimal primes of 772 using the matrix tp. For a prime ideal p C R there is an associated prime ideal

Assume that R is universally catenary in order to avoid dwelling into technicalities. In this case, if in addition S ( I ) is equidimensional, one sees that height T(p) = 1 if and only if v ( I y ) = height p. We also observe that unless p is a height one prime itself, / is not free if T(p) is to have height one.

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PROPOSITION 3.4 Let R be a universally catenary Noetherian ring and let I be

an ideal such that 5(7) is a domain. Then the set (T(p)

height p > 2 and height T(p) = 1}

is finite. More precisely this set is in bijection with

{p C R

7p not free, p 6 Min(R/It(ip))

and height p = rank(^) - t + 2},

where 1 < t < rank(^).

PROOF: Given a prime p C R, set t = n—z/(7 p ). If height p > 2 and if height T(p) = 1 then, since rank(V') = n — 1, by the preceding remarks we have height p = n- I - t + 2 and p D / t (VO \ /t-i(VO-

On the other hand, since 5(7) is a domain, ^(7 p ) < height p for each nonzero prime ideal. Therefore, one has height 7 t (i/>) > rank(^) — t + 2. It follows that p G Mm(R/It('4>)). The converse is similar. D Reducedness and normality

There are very few criteria of normality for Rees algebras outside of general Jacobian tests, which do not take into account the nature of a Rees algebra.

We begin with the following observation:

PROPOSITION 3.5 Let R be a Noetherian normal domain and let I be an ideal such that gr/(J?) is a reduced ring. Then 72 = R[It] is integrally closed. PROOF: Consider the extended Rees algebra, 72.e = R[It,t~1]. Since Q = 72, e /(t~ 1 ) is reduced it follows that 72e satisfies the (52) and (Ri) conditions of Serre. Thus 72e is normal and since 72, — 72e n R[t], 72 is also integrally closed. D Under some additional conditions on the ideal 7 one obtains a very explicit description of the divisor class group of such algebras.

THEOREM 3.6 ([16]) Let (R,m) be a quasi-unmixed local ring and let I be an ideal of finite projective dimension. If Q = gr/(7?) is a reduced ring then Q is a torsionfree R/1-algebra. Moreover, if I is a prime ideal then Q is a domain. PROOF: We may assume that the residue field of R is infinite. Let P be a minimal prime of 772; then P72.p = 772p. We want to relate these primes to the associated primes of 7. Since Q is reduced, 7 is a radical ideal. Let { p i , . . . ,p s } be the set of minimal primes of 7. For a prime p = pj, set

T(p) = kernel (7e^(72/p7e) p ). Since 7p = p7?p has finite projective dimension, 7?p is a regular local ring and therefore T(p) will be a prime ideal.

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269

The claim is equivalent to saying that any minimal prime of ITL is one of the T(pi), i.e., that PdR — pi for some i. We suppose otherwise, set q = P(~}R, localize at q, change notation and assume that P n R = m is the maximal ideal of R. Two observations arise: (i) / (£_ m2 and (ii)

dim/? = dimft/P < dimTl/mU = 1(1) < dimR. By induction on g — height /, we are going to show they are untenable. We use an argument of [20], that treats the more general condition (Rk) of Q. If g = 0, then 7 = 0 as an ideal of finite projective dimension over a local ring must contain regular elements. Let a\,..., an be a set of generators of 7, let Xi,..., Xn be independent variables and set R = R(Xi,..., Xn), TO = m7? and Q = Q ®R R. Further, set x = £"=i a^, x' = x + 72 <E <51; and 7 = T/(x) C 7? = R/(x). Since 7 <£_ m2 and g > 0, x 0 m2 is 7?-regular, being a generic element of 7. By [21, p. 130], 7 is an ideal of finite projective dimension of 7?. Finally, x' is a generic element of Q+. It follows that a; is a superficial element of 7, and G/(x') is still reduced at its minimal primes since for every Q G Proj (G/(x')), (G/(x'))q is the localization of a polynomial ring over Q. Since a; is a superficial element of 7, gry(7?) is reduced on Proj (gTj(R)). By the induction hypothesis, 1(1) < dim7? — 1 = dim7? — 2. Finally, setting K = R/fh, we have

e(I)

=

dim Proj (grj(R) <8)^ K) + I = dim Proj (G/(x') ®^ K) + 1

(3)

=

dim(G®KK)/(x')>e(I)-l,

(4)

to complete the proof.

D 2

EXAMPLE 3.7 Let R = k [ { x , y , z ] ] / ( y - xz), k a field, and let p = (y,x)R. Since 7? is a domain and (y) : x — (y) : x2 it follows that y, x is a d-sequence so that gr p (7?) ~ 5/j/ p (p/p 2 ), the symmetric algebra of p/p 2 . We then obtain that gr p (7?) = k[z,u,v]/(uv) which is reduced but not a domain. Note however that proj. dim.^p = oo.

COROLLARY 3.8 Let R be a quasi-unmixed normal domain and let I be an ideal such that Q — grj(R) is reduced. Denote by { p i , . . . ,p s } the minimal primes of I and by T(ipi) the corresponding minimal primes of I1 77 —* Cl(ft) -£+ Cl(7?) -4 0,

where 77 is freely generated by the [T(pj)].

4

THE EXPECTED CANONICAL MODULE

There is an obvious need for standard models for canonical modules of Rees algebras. One that surfaced early is given by the following example.

EXAMPLE 4.1 If the ideal 7 is generated by a regular sequence / 1 ( . . . , fg, g > 2, the equations of 71 = R[It] are nice:

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In other words, the defining equations of the algebra are generated by the Koszul relations of the /j's.

Knowing this description of Tl leads immediately to its canonical module ([3])

There are however many other instances of Rees algebras whose canonical modules have this form. It warrants the following:

DEFINITION 4.2 Let R be a Noetherian local ring with a canonical module U>R and let / be an ideal of positive grade. The canonical module of Tl = R[It] is said to have the expected form if

for some integer 6.

The connection with the canonical module of gij(R) is provided in the following result ([10], [37]).

THEOREM 4.3 Let (R,m) be a Noetherian local ring with a canonical module u> — UJR and let I be an ideal of positive grade. Assume that Tt = R[It] is Cohen-

Macaulay and set a = —a(Q) for Q = gij(R). Then the following are equivalent: (a) (b)

PROOF: Denote by ui-j^ the canonical module 72,. Applying Hom-j^(-, w-^) to the sequences

o -> /ft —>n —>g -> o

and

o -> itn —> 7^ —> R -* o,

we obtain and 0-

Suppose that (b) holds. From the last sequence we obtain <^>Itji = w#(l, t)a~1Tl. Feeding this into the previous sequence, given that w^ = tu>It-j^, we obtain that the canonical module of Q is as expected. For the converse it is convenient to express these exact sequences of canonical modules in the following manner: n _

. /V-

and

_ ^—^ \ r) j-n __ /

j

"I

E

_ X—^ \ c1 +n __ , I /v

/

_j

''•

^—>

n

itR- ~ / ^

_ X—> \ u1 fTi _ n C/

/

j

Tl

_ 1

n

* •

5

Special Divisors of Blowup Algebras

271

From which we get (noting that a > 0)

El En

En En/Dn

= LJ = D n _!

= Dn = Fn

n>2

1 a.

It follows that Dn = LO for n < a. For n = a, we have Ea/Da = A.-1/A, = <"/Da ^ W/IU.

On the other hand, IDn C -Dn+i for all n, gives epimorphisms UJ/ILJ —» u>/ Da -» w/I(jj.

It follows that Da = 7u>. The remainder of the argument is similar.

n

REMARK 4.4 For an ideal I in the situation of this theorem, one has a(gr 7 (/?)) = a(gr/ ( R f ) ) for any minimal prime p of /. Thus if / is a generic complete intersection, a(grj(R)) = —height /. More generally, a(gTI(R}) = r(/ p ) — height /. Gorenstein algebras

The arrangement of the proof above can be used in the proof of the following result of Ikeda ([18]).

THEOREM 4.5 Let (R, m) be a Noetherian local ring and let I be an ideal of grade at least 2. If R[It] is Cohen-Macaulay the following conditions are equivalent:

(a) R[It] is Gorenstein. (b) UR~R and w gr/(fi) ~ gr / ( J R)[-2]. PROOF: (a) => (b): Setting K - R[H] and uj-j^ = ]Cn>i Dntn, we have ) +n+l

The embedding 0 -» S P /

=u

—> P J —> G P /

=w

-> 0

shows that if w^ ~ 72. then R = DI C U>R and / • w/j C -R; it follows that D\ — R = <JJR since w^ lies in the total ring of quotients of R and grade / > 2. As Dn = In~l for n > 2, the canonical module of £/ is as asserted.

(b) =^ (a): w^ = ujRt(l, t}a~2K ~ 7^ by the hypothesis and Theorem 4.3.

D

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THE FUNDAMENTAL DIVISOR

Our main purpose is to introduce a divisorial ideal associated to a Rees algebra and sketch out some of its applications. It helps to explain old puzzles while at the same time providing quite direct proofs of earlier results. The reader will note that it is a mirror image of local cohomology modules of Rees algebras. Its Noetherian character however permits a control of computation that is not always possible with Artinian modules.

Let (R, m) be a Cohen-Macaulay local ring of dimension d, with a canonical module w. Let / = (/i, . . . , fn) be an ideal of positive height. Fix a presentation of 72, B = R[Ti, . . . ,Tn] —> 72. We set UB = w ®R B[—n\ as the canonical module of the polynomial ring B. The canonical module of 72 is the module

This means that £>(/) can be written as

D2t2 where each Di is an J?-submodule of K, the total ring of fractions of R. We fix this representation of T>(I) from a given projective resolution of / • 72. and the computation of the cohomology. This divisorial ideal carries more information than the canonical module w-^. Indeed, we can view T>(I) as made up of two parts,

D(I) = Drf + D2t2 + D3t3 + • • • • (0

(")

DI will be called the leading part of 7J>(7), while (ii) is called its canonical part in agreement with the next observation.

A consequence of the proof of Theorem 4.3 is the following relationship between the canonical and fundamental divisors of a Rees algebra.

PROPOSITION 5.1 Let R be a Noetherian ring with a canonical module and let I be an ideal of positive grade. Then the graded components ofujj^ and ofD(I) are related in the following manner:

n>2

that is, T>(I) is a prolongation

DEFINITION 5.2 Let R be a Noetherian local ring with a canonical module UIR and let / be an ideal of positive grade. The fundamental divisor of 72. = R[It] is said to have the expected form if DI ~ U>R.

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273

Let us phrase the condition DI ~ WR in terms of cohomology. We assume that R is a Cohen-Macaulay local ring of dimension d > 0 and set 7£ = R[It}. From the exact sequence 0 -» in[-l] —> Tl —> R -> 0, we have

0 - HdM(IT^)-i -^ HdM(U)0 -^ Hdm(R) -^ H%\IK)^ -^ H%l(U)0 = 0. PROPOSITION 5.3 Let D be the fundamental divisor of U. Then DI ~ LOR if and only if f = 0. One issue is what is the most approachable mapping to examine. Of course the most sensible move thus far has been to prove directly that Hj^(TV)o = 0; we will follow this route in the last section. Now we must look at the mapping