On the Homology of the Schur Complex

JOURNAL OF ALGEBRA ARTICLE NO.

182, 274]286 Ž1996.

0171

On the Homology of the Schur Complex Imtiaz A. ManjiU Departamento de Matematicas, Uni¨ ersidad Simon ´ ´ Bolı´¨ ar, Valle de Sartenejas, Caracas, Venezuela

and † Rafael Sanchez ´

Departmento de Matematicas, Institut Venezolano de In¨ estigaciones Cientıficas, ´ ´ Apartado 21827, Caracas, 1020-A, Venezuela Communicated by D. A. Buchsbaum Received May 2, 1995

INTRODUCTION The theory of Schur complexes in a characteristic free setting has shown, since its appearance wABWx, to be useful in the study of finite free resolutions of determinantal ideals. Its development has gone beyond the original goal and now we have a collection of very interesting problems on characteristic-free representation theory of the general linear group wABx. As a result of this, all the efforts made to develop the theory have a representation theoretic flavor, and besides the results on acyclicity of Schur complexes given in wABWx, there is not much else in the literature. In this paper we study the vanishing behavior of the homology of the Schur complexes Ll w , associated to a partition l and an R-linear map w : G ª F between finitely generated free R-modules, where rank G s n F m s rank F and R is a commutative noetherian ring. In this sense Boffi has studied the homology of LŽ1m . f, and LŽ m. f, where f is a generic alternating map wBx. * E-mail: [email protected]. † E-mail: [email protected]. 274 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Section 1 of this paper is dedicated to quoting, without proofs, the definition of and basic results on Schur complexes from wABWx. In Section 2, we establish an acyclicity criterion that generalizes the one given in wABWx and that includes, as a special case, a well-known result of Avramov in case of the Koszul complex attached to the map w wAvrx. In Section 3 we study the vanishing behavior of the homology of Ll w , and we give two results in the same vein as that in wMx, in which the first author studied the vanishing behavior of the homology of the Koszul complex attached to an R-linear map w : G ª F when rank G G rank F. In this paper, as mentioned before, we consider the case rank G F rank F.

1. BACKGROUND In this section we recall, without proofs, some basic facts about Schur complexes. The reader interested in an in-depth treatment of this topic should consult wABWx. Let R be a commutative ring, and F and G free R-modules of ranks m, n, respectively. Given w g Hom R Ž G, F . we associate to it an element Cw g F mR GU , via the canonical isomorphism Hom R Ž G, F . ( F mR GU . To see this explicitly, we fix bases  g j 4 ,  f i 4 for G and F, respectively, and set g j 4 equal to the dual basis of  g j 4 . The element Cw is then just the sum Ý w i j f i m g j , where Ž w i j . is the m = n matrix of w with respect to the bases  g j 4 ,  f i 4 . In order to define the Schur complexes we need some preliminary concepts. Consider the R-bialgebras SF, H F, HG, and DG, where S is used for symmetric, H for exterior, and D for divided power algebras. We ˙ define two R-bialgebras Sw and H w by means of SF m HG and H F m DG, respectively, where the first is the ordinary tensor product of the R-bialgebras SF and HG and the second is the antisymmetric tensor ˙ . of the R-bialgebras H F and DG. We denote the multiplicaproduct Žm tion, comultiplication, and commutation maps by m s w , D s w , Ts w in the first case, and by m H w , D H w , T˙H w in the second. Regard Cw g F m GU as an element of SF m HGU and also as an ˙ SGU . Sw can be viewed as a complex with respect to the element of H F m action of Cw g SF m HGU on SF m HG. Similarly, H w can be viewed as ˙ SGU on H F m ˙ DG. complex with respect to the action of Cw g H F m For example, in case of Sw , Ž S w . j s Ý iG 0 Si F m Hj G is the jth degree component of the complex, S w , and the differential on Sw is the R-linear map

­ Sw : Ž Sw . j ª Ž Sw . jy1

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given by xmy¬

Ý wi j fi x m g j Ž y .

for x m y g Si F m Hj G, i G 0. DEFINITION 1.1. Ža. Sk w is the subcomplex of Sw given by 0 ª Hk G ª F m Hky 1 G ª ??? ª Sky j F m Hj G ª ??? ª S k F ª 0, where the jth degree component is Ž Sk w . j s Skyj F m Hj G. Žb. Hkw is the subcomplex of H w given by 0 ª D k G ª F m D ky1G ª ??? ª Hky j F m Dj G ª ??? ª Hk F ª 0, where the jth degree component is ŽHkw . j s Hky j F m Dj G. Remark. Sw s Ý k G 0 Sk w , and H w s Ý k G 0 Hkw as a direct sum of complexes. As in wABWx we will denote by N the set of natural numbers Ži.e., non-negative integers., and by N` the set of sequences of elements of N of finite support. We will not distinguish between the element Ž l1 , . . . , l p . in N p and the element Ž l1 , . . . , l p , 0, . . . . in N` . ˜ of l If l s Ž l1 , l2 , . . . . is an element of N` , we define the transpose l ˜ s Ž l˜1 , l˜2 , . . . . of N`, where l˜j is the number of terms to be the element l of l which are greater than or equal to j. DEFINITION 1.2. A partition is an element l s Ž l1 , l2 , . . . . of N` such that l1 G l2 G ??? . The weight of the partition l, denoted by < l <, is the sum Ý l i . If < l < s n, l is said to be a partition of n. The number of non-zero terms of l is called the length of l.

˜ is a partition, and It is easy to see that if l is a partition, then l ˜ < l < s < l <. DEFINITION 1.3. Let l s Ž l1 , . . . , l q ., m s Ž m 1 , . . . , m q . be partitions such that m : l, i.e., m i F l i for all i. We define Hl r m w to be the tensor product of complexes Hl1ym 1 w m ??? m Hl qym q w , and Sl r m w to be the tensor product of complexes Sl1y m 1 w m ??? m Sl qy m q w . Now we are ready to define the Schur complex on w .

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DEFINITION 1.4. Let l s Ž l1 , . . . , l q ., m s Ž m 1 , . . . , m q . be partitions ˜ s Ž l˜1 , . . . , l˜t ., m such that m : l. Let l ˜sŽm ˜ 1, . . . , m ˜ t . be their transposes. Let Ž a i j . the the q = t matrix defined by

ai j s

½

1 0

if m i q 1 F j F l i otherwise.

We define the Schur map dl r mŽ w .: Hl r m w ª Sl˜ r m˜ w to be the composition Hl1ym 1 w m ??? m Hl qym q w x Ž Ha11 w m ??? m Ha1 t w . m ??? m Ž Ha q1 w m ??? m Ha q t w . x Ž S a11 w m ??? m S a1 t w . m ??? m Ž A a q1 w m ??? m S a q t w . x S w m ??? m S w m ??? m Ž S a1 t w m ??? m S a q t w . Ž a11 a q1 . x Sl˜1y m˜ 1 w m ??? m Sl˜ty m˜ t w where the first map is the tensor product of the maps D H w : Hliym i w ª Ha i1 w m ??? m Ha i t w Ž i s 1, . . . , q ., the second map is the tensor product of the canonical isomorphisms Ha i j w ª S a i j w Ž a i j s 0, 1., the third map is the tensor product of the commutation maps, and the fourth map is t multiplication in the algebra m Sw s Sw m ??? m S w t-times. Then the Schur complex is defined as the image of dl r mŽ w .. We may also express the Schur complex as the cokernel of certain map. In order to do that we give the following: DEFINITION 1.5. Define the map of complexes IHw : Hw m Hw ª Hw m Hw to be the composition Hw m Hw m Hw

1mm H w

Hw m Hw.

6

D H w m1

6

Hw m Hw

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We define Ll r mŽ w . to be the cokernel of the map of complexes qy1

l kq1 y m kq1

Ý

Ý

Hl1y m 1 w m ??? m Hl ky m kqlw

ks1 ls m ky m kq1 q1

mHlkq 1y m kq 1ylw m ??? m Hl qy m q w qy 1

Ý 1 m ??? m I

k

m ??? m 1

6

ks 1

Hl1y m 1 w m ??? m Hl qy m q w where I k is the map l kq1 y m kq1

Hlky m kqlw m Hl kq 1y m kq 1ylw ª Hl ky m k w m Hl kq 1y m kq 1 w

Ý

ls m ky m kq1 q1

given by I H w . The relationship between Ll r m w and Ll r m w is given by THEOREM 1.1. There is a natural isomorphism Ll r m w ª Ll r m w of complexes. Consequently, Ll r m w is uni¨ ersally free. Remark. Ža. If G s 0, then w : 0 ª F, so ŽHl r m w . 0 s Hl1ym 1 F m ??? m Hl qym q F, and ŽHl r m w . j s 0 for j ) 0. On the other hand Ž Sl˜ r m˜ w . 0 s Sl˜ y m˜ F m ??? m Sl˜ y m˜ F, and Ž Sl˜ r m˜ w . j s 0 for j ) 0. In this 1 1 t t case, Ll r m w is denoted by Ll r m F, and is called the Schur module corresponding to the pair of partitions m : l Žin wABWx it is called the Schur functor.. Žb. If F s 0, then w : G ª 0, so ŽHl r m w . < l
l kq1 y m kq1

Ý

Ý

Hl1y m 1 M m ??? m Hl ky m kql M

ks1 ls m ky m kq1 q1

mHlkq 1y m kq 1yl M m ??? m Hl qy m q M qy 1

Ý 1 m ??? m I

6

ks 1

k

Hl1y m 1 M m ??? m Hl qy m q M

m ??? m 1 s I

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where I k is the map l kq1 y m kq1

Ý

Hlky m kq 1ql M m Hl kq 1y m kq 1yl M

ls m ky m kq1 q1

ª Hlkym k M m Hlkq 1ym kq 1 M given by the composition Dm1

Hl ky m k M m Hl M m Hl kq 1y m kq 1yl

1mm

Hlky m k M m Hl kq 1y m kq 1

6

Hlky m kql M m Hl kq 1y m kq 1yl M

6

f or each k s 1, . . . , q y 1. Then Ll r m M s CokerŽI.. Besides Theorem 1.1, which establishes the free universality of Ll r m w , and gives a standard basis for it, we have the following decomposition results on Schur complexes. PROPOSITION 1.1. If w s w 1 [ w 2 , then there is a natural filtration of Ll r m w by complexes whose associated graded object is Ým : s : l Ls r m w 1 m Ll r s w 2 . An important consequence of this is the following: COROLLARY 1.1. Ža. Let Ž Ll r mŽ w .. j be the jth degree component of the Ll r m w . There is a natural filtration on Ž Ll r m w . j whose associated graded object is

Ý

m: s : l < l
Ls r m F m Kl r s G.

Žb. If w : G ª F is a split injection, then Ll r m w is acyclic and H0 Ž Ll r m w . s Ll r m ŽCoker w .. Žc. If w s w 1 [ w 2 and w 1 is an isomorphism, then Ll r m w is homotopically equi¨ alent to Ll r m w 2 . We conclude this section with a result about the acyclicity of Ll w in terms of the grade of the ideal generated by the j = j minors of a matrix of w , I j Ž w .. We will generalize this result in Section 2. PROPOSITION 1.2. Let R be a commutati¨ e noetherian ring and w : G ª F an R-map. Let m s rank F G n s rank G and l s Ž l1 , . . . , l q . be a partition with l1 F m y n q 1. If gradeŽ I j Ž w .. G Ž m y n q 1.Ž n y j q 1. for j s 1, . . . , n, then Ll w is acyclic.

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2. THE ACYCLICITY THEOREM We concluded the first section by stating a well known criterion for the acyclicity of a Schur complex Ll w Žcf. Prop. 1.2.. This criterion has proved to be useful in constructing finite free resolutions for certain determinantal ideals. Our objective here is to show that Ll w is acyclic under weaker conditions. Let R be a commutative noetherian ring, and F and G free R-modules of ranks m and n, respectively, with n F m. Let w g Hom R Ž G, F .. We have the following: THEOREM 2.1. Let l s Ž l1 , . . . . be a partition. If gradeŽ I j Ž w .. G l1 q ??? ql ny jq1 , for j s 1, . . . , n, then Ll w is acyclic. Proof. We proceed by induction on rank G. If rank G s 0, the results follow trivially. Since the length of Ll w is at most l1 q ??? ql n , to prove Ll w is acyclic it suffices to prove the statement locally for each prime ideal p with grade smaller than l1 q ??? ql n . Suppose grade Žp. - l1 q ??? ql n ; then by hypothesis I1Ž w . o p. Therefore over R p , wp decomposes into w X [ IR , where w X is a map between free R p-modules GX and F X with rank GX s n y 1. Now I j Ž w X . s I jq1Ž wp . s Ž I jq1Ž w .. p . Therefore, grade Ž I j Ž w X . . s grade Ž I jq1 Ž w . . p G grade Ž I jq1 Ž w . . s l1 q ??? ql nyj , i.e., grade Ž I j Ž w X . . G l1 q ??? qlŽ ny1.yjq1 . By the induction hypothesis LlŽ w X . is acyclic, and by Corollary 1.1, LlŽ wp . s Ž Ll w . p is acyclic. 3. THE VANISHING OF H#Ž Ll w . Let w g Hom R Ž G, F . be as before, with rank G s n F m s rank F, and let l s Ž l1 , . . . . be a partition. We introduce the following: DEFINITION 3.1. A sequence r s Ž r 0 , r 1 , . . . , rn . of n q 1-integers is Ž w , l.-admissible if Ži. r 0 s 0; Žii. grade Ž Iny i Ž w .. G riq1 , i s 0, . . . , n y 1; Žiii. riq1 s ri or riq1 s ri q l iq1yk , where k i is the number of equalii ties occurring amongst r 0 F ??? F ri .

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THEOREM 3.1. Assume w is an R-linear map as abo¨ e, with n F m. If r s Ž r 0 , r 1 , . . . , rn . is a Ž w , l.-admissible sequence, then H# Ž Ll w . s 0

for ) ) l1 q ??? ql k ,

where k is the number of equalities occurring amongst r 0 F ??? F rn . Proof. We proceed by induction on k. If k s 0, then r 0 - ??? - rn . In this case rtq1 s l1 q ??? ql tq1 , and therefore grade Ž Iny t Ž w . . G l1 q ??? ql tq1 for t s 0, . . . , n y 1. Equivalently, gradeŽ I j Ž w .. G l1 q ??? ql nyjq1 , for j s 1, . . . , n. By Theorem 2.1, H# Ž Ll w . s 0

for ) ) 0.

This proves the base case of the induction. Let N G 1 be a fixed integer. We suppose by induction that the theorem is true whenever k F N y 1. Let r s Ž r 0 , . . . , rn . be a Ž w , l.-admissible sequence and suppose that there are exactly N equalities occurring amongst r 0 F ??? F rn . In particular, N F n, and n G 1. For n s N or n s 1, our conclusion states that H#Ž Ll w . s 0 if ) ) l1 q ??? ql n . But this is obviously true since the complex Ll w has length at most l1 q ??? ql n . Thus, we can henceforth assume N - n and n G 2. In this case rn s l1 q ??? ql nyN . Set js

½

0

if rny 1 s rn

max  t : rny t - ??? - rn 4 ,

otherwise.

We shall prove that the theorem holds for k s N, by inducting on j while keeping N fixed. Suppose j s 0; let p be a prime ideal in R with gradeŽp. F l1 q ??? ql nyN y 1.

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Since gradeŽ I1Ž w .. G rn , there is some 1-minor of w which does not lie in p, hence we obtain a matrix representation over R p of the localization wp of w at p, which has the form

ž

1 0

0 wpX ,

/

where wpX is an Ž m y 1. = Ž n y 1. matrix. Since It Ž wpX . s Itq1Ž wp . for t s 1, . . . , n y 1, it follows that grade Ž It Ž wpX . . s grad Ž Itq1 Ž wp . . G grade Ž Itq1 Ž w . . G rny t . It is now easy to see that the sequence r X s Ž r 0 , . . . , rny1 . is a Ž wpX , l.-admissible sequence. Since j s 0, there are exactly N y 1 equalities occurring amongst r 0 F ??? F rny 1. Applying the induction hypothesis to the Ž wpX , l.-admissible sequence Ž r 0 , . . . , rny1 . one has H# Ž Ll wpX . s 0 whenever ) ) l1 q ??? ql Ny1. However, Corollary 1.1Žc. implies H# Ž Ll wpX . ( H# Ž Ll wp . ( H# Ž Ll w . p . Thus, H# Ž Ll w . p s 0 whenever ) ) l1 q ??? ql Ny1. In particular, H#Ž Ll w . p s 0, if ) ) l1 q ??? ql N . Consequently, the complex obtained by truncating Ll w at degree l1 q ??? ql N Ži.e., dropping the modules in degree - l1 q ??? ql N . is acyclic, locally at each prime ideal p with grade Ž p . F l1 q ??? ql nyN y 1. Since the length of Ll w is at most l1 q ??? ql n , the length of the truncated complex is at most

Ž l1 q ??? ql n . y Ž l1 q ??? ql N . s l Nq1 q ??? ql n F l1 q ??? ql nyN .

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Hence, by the Acyclicity Lemma, the truncated complex is acyclic. In other words, H# Ž Ll w . s 0

for ) ) l1 q ??? ql N .

By induction, we now suppose that the theorem holds for each Ž w , l.-admissible sequence with exactly N equalities occurring amongst r 0 F ??? F rn , and having j s M y 1, for a fixed M G 1. Suppose r is a Ž w , l.-admissible sequence with exactly N equalities occurring amongst r 0 F ??? F rn , and having j s M. It remains to prove by induction that the theorem holds for r. Let p be a prime ideal in R with grade Ž p . F l1 q ??? ql nyN y 1. Since gradeŽ I1Ž w .. G rn , there exists some 1-minor of w not in p. As before, over R p , wp has a matrix representation of the form

ž

1 0

0 wpX ,

/

where wpX is an Ž m y 1. = Ž n y 1. matrix. It follows as above that the sequence r X s Ž r 0 , . . . , rny1 . is a Ž wpX , l.-admissible sequence. Since M G 1, r X has exactly N equalities occurring amongst r 0 F r 1 F ??? F rny1 , applying the second inductive hypothesis with j s M y 1 and k s N to the Ž wpX , l.-admissible sequence r X gives H# Ž Ll wpX . s 0 whenever ) ) l1 q ??? ql N . Hence H#ŽŽ Ll w . p . s 0 whenever ) ) l1 q ??? ql N . Consequently, the complex obtained by truncating Ll w at l1 q ??? ql N is acyclic locally at each prime ideal p in R, with grade Žp. F l1 q ??? ql ny N y 1.

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As we saw earlier the truncated complex has length at most l1 q ??? ql ny N , and therefore it is acyclic, i.e., H# Ž Ll w . s 0

if ) ) l1 q ??? ql N .

This finishes the induction. Remark. The following construction yields a Ž w , l.-admissible sequence: Set r 0 s 0, and define riq1 s

½

ri q l iq1yk i ,

if grade Ž Inyi Ž w . . G ri q l iq1yk i

ri ,

otherwise.

We call the resulting sequence r s Ž r 0 , . . . , rn . the canonical Ž w , l.-sequence. This sequence is optimal amongst all Ž w , l.-admissible sequences in the sense that it has the least number of equalities occurring amongst r 0 F ??? F rn , and thus yields the best result for our theorem. We finish by studying the local behavior of the homology of Ll w . Let w g Hom R Ž G, F . as above and let r s Ž r 0 , . . . , rn . be the canonical Ž w , l.-sequence. DEFINITION 3.2. Let k be the number of equalities in r 0 F ??? F rn . For every i s 1, . . . , k, jŽ i . is the position of the ith equality in r 0 F ??? F rn . EXAMPLE. The sequence r 0 s r 1 - r 2 - r 3 s r4 s r5 - r6 , has three equalities in positions 0, 3, 4, respectively. So jŽ1. s 0, jŽ2. s 3, jŽ3. s 4. THEOREM 3.2.

Let w be as abo¨ e with n s rank G F m s rank F. Let r s Ž r 0 , . . . , rn .

be the canonical Ž w , l.-admissible sequence. Ži. If 1 F i F k Supp Ž H# Ž Ll w . . : V Ž InyjŽ i. Ž w . . when l1 q ??? ql iy1 - ) F l1 q ??? ql i . Žii. SuppŽ H0 Ž Ll w .. : VŽ InŽ w ...

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285

Proof. Set j s jŽ i .. Let p be a prime ideal in R, and assume p W Iny j Ž w .. Then we have a matrix representation of wp which has the form

ž

I 0

0 wpX ,

/

where I is the Ž n y j . = Ž n y j . identity matrix, and wpX is some Ž m y n q j . = j matrix. We allow j s 0, in which case i s 1 and then p W InŽ w ., so InŽ w . R p s R p and wp has the form I

ž / 0

.

Hence, H#Ž Ll wp . s 0 for ) G 0, i.e., H#Ž Ll w . mR R p s 0 for ) G 0. So SuppŽ H#Ž Ll w .. : VŽ InŽ w .. for all ) G 0, and therefore in particular for ) F l1. This proves the theorem for j s 0. Now assume j s jŽ i . / 0. Since gradeŽ It Ž wpX .. s gradeŽ Itqnyj Ž wp .. G gradeŽ Itqnyj Ž w .. G r jytq1 , the sequence r X s Ž r 0 , . . . , r j . is a Ž wpU , l.-admissible sequence. As there are exactly i y 1 equalities occurring amongst r 0 F ??? F r j , it follows by Theorem 3.1 that H# Ž Ll wpX . s 0

for ) ) l1 q ??? ql iy1 .

But H#Ž Ll wpX . ( H#Ž Ll wp . ( H#ŽŽ Ll w . p . ( H#Ž Ll w . mR R p . Therefore, H# Ž Ll w . mR R p s 0

if ) ) l1 q ??? ql iy1 .

In particular, H# Ž Ll w . mR R p s 0 whenever

l1 q ??? ql iy1 - ) F l1 q ??? ql i . Hence, suppŽ H#Ž Ll w .. : VŽ Inyj w . for l1 q ??? ql iy1 - ) F l1 q ??? ql i . This proves part Ži.. For Žii., let p be a prime ideal in R and suppose p W InŽ w .. Then InŽ w . R p s rp ; therefore H# Ž Ll Ž wp . . s 0

for ) G 0.

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It follows that H# Ž Ll w . mR R p s 0,

for ) G 0;

therefore suppŽ H0 Ž Ll w .. : VŽ InŽ w ... REFERENCES wABx

K. Akin and D. Buchsbaum, Characteristic-free representation theory of the general linear group, Ad¨ . in Math. 58 Ž1985., 149]200. wABWx K. Akin, D. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Ad¨ . in Math. 44 Ž1982., 207]278. wAvrx L. L. Avramov, Complete intersections and symmetric algebras, J. Algebra 73 Ž1981., 248]263. wBx G. Boffi, Remarks on some Schur Complexes of a generic alternating map, in ‘‘Seminari di Geometria 1991]1993’’ Ža cura di S. Coen., Dip. to di Matematica della Universita di Bologna press Tecnoprint, Bologna, 1994. wMx I. A. Manji, Vanishing of the homology modules of a Koszul complex, J. Pure Appl. Algebra 102, No. 1 Ž1995., 61]66.