A Duality Theorem for Generalized Local Cohomology

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 8, August 2008, Pages 2749–2754 S 0002-9939(08)09400-8 Article electronically published on February 28, 2008

A DUALITY THEOREM FOR GENERALIZED LOCAL COHOMOLOGY

MARC CHARDIN AND KAMRAN DIVAANI-AAZAR

(Communicated by Bernd Ulrich)

Abstract. We prove a duality theorem for graded algebras over a ﬁeld that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

1. Introduction Several duality results for local cohomology (notably Grothendieck and Serre du- alities) and for generalized local cohomology (Suzuki duality) are classical results commonly used in commutative algebra. A more recent duality theorem was estab- lished by Herzog and Rahimi in the context of bigraded algebras. We here present a duality result that encapsulates all these duality statements. Before stating this theorem, we need to set some notation. Let Γ be an abelian group and R be a Γ-graded polynomial ring over a commutative Noetherian ring R0. For a Γ-graded R-ideal I, we recall from [CCHS] that the grading of R is I-sharp i ∈ if HI (R)γ is a finitely generated R0-module for any γ Γandanyi.Ifthisisthe i case, then HI (M)γ is also a finitely generated R0-module for any finitely generated graded R-module M,anyγ ∈ Γandanyi. InthecasewhereS = S0[x1,...,xm,y1,...,yn] is a graded quotient of R = R0[X1,...,Xm,Y1,...,Yn], p := (x1,...,xm)andq := (y1,...,yn), we define the grading of S to be p-sharp (equivalently q-sharp) if the following condition is sat- isfied: for all γ ∈ Γ, |{ ∈ m × n }| ∞ (α, β) Z≥0 Z≥0 : αi deg(xi)=γ + βj deg(yj) < . i j This condition on the degrees of the variables is equivalent to the fact that the grading of R is (X1,...,Xm)-sharp (equivalently (Y1,...,Yn)-sharp) by [CCHS, 1.2]. ∗ We set HomR(M,R0)forthegradedR0-dual of a graded R-module M.This ∗ module is graded and HomR(M,R0)γ =HomR0 (M−γ,R0). The notion of generalized local cohomology was given by Herzog in his Habil- itationsschrift [H]. For an ideal I in a commutative Noetherian ring R,andM

Received by the editors May 21, 2007 and, in revised form, September 13, 2007. 2000 Mathematics Subject Classiﬁcation. Primary 13D45, 14B15, 13D07, 13C14. Key words and phrases. Generalized local cohomology, duality.

c 2008 American Mathematical Society Reverts to public domain 28 years from publication 2749

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and N two R-modules, the i-th generalized local cohomology module of M and i N with respect to I is denoted by HI (M,N). One has a natural isomorphism i i i j ⇒ i+j HI (R, N) HI (N) and a spectral sequence HI (ExtR(M,N)) HI (M,N). Our main result is the following duality result.

Theorem 1.1. Let S = k[x1,...,xm,y1,...,yn] be a Cohen-Macaulay Γ-graded algebra over a field k.Letp := (x1,...,xm) and q := (y1,...,yn). Assume that the grading of S is p-sharp (equivalently q-sharp), and M and N are finitely generated graded S-modules such that either M has finite projective dimension or N has finite S projective dimension and Tori (M,ωS)=0for i>0;then i ∗ dim S−i ⊗ Hp(M,N) HomS(Hq (N,M S ωS ),k). Several previously known duality results are particular cases of this theorem. Namely, (i) graded local duality in the case that the base ring is a field corresponds to the case where q =0andM = S, (ii) Serre duality follows from the case where p =0andM = S, via the spectral i j ⇒ i+j sequence Hq(ExtS(N,ωS)) Hq (N,ωS), (iii) Suzuki duality in the context of graded algebras over a field (see [CD, 3.1]) corresponds to the case where q =0, (iv) the Herzog-Rahimi spectral sequence corresponds to the case where M = S (see [HR] for the standrard bigraded case, and [CCHS] for the general case) using i j ⇒ i+j the spectral sequence Hq(ExtS(N,ωS)) Hq (N,ωS). 2. The duality theorem

If F• and G• are two complexes of R-modules, HomgrR(F•,G•) is the coho- i mological complex with modules C = p−q=i HomR(Fp,Gq). If either F• or G• is finite, then HomgrR(F•,G•) is the totalisation of the double complex with p,−q C =HomR(Fp,Gq). We first consider the case where R := k[X1,...,Xm,Y1,...,Yn]isapolynomial ring over a field k, p := (X1,...,Xm)andq := (Y1,...,Yn). From this point on, we will assume that R is Γ-graded, for an abelian group Γ, − and that the grading of Ris p-sharp (equivalently q-sharp). One has ωR = R[ σ], where σ := deg(X )+ deg(Y ). i i j j • Let C (M)betheCechˇ complex 0→M→ M →···→M ··· →0and p i Xi X1 Xm C• ˇ → → →···→ → •{ } q (M)betheCech complex 0 M i MYi MY1···Yn 0. Denote by C i • ∗ ∗ the complex C shifted in cohomological degree by i and set — := HomR(—,k). Recall that —∗ is exact on the category of graded R-modules. By [CCHS, 1.4 (b)] one has

Lemma 2.1. If F• is a graded complex of finite free R-modules, then i j (i) Hp(F•)=0for i = m, Hq(F•)=0for j = n, C• { }→C• { }∗ (ii) there is a natural graded map of complexes d : p (R) m q (ωR) n that C• { }→C• { }∗ induces a map of double complexes δ : p (F•) m q (HomR(F•,ωR)) n which gives rise to a functorial isomorphism m n ∗ Hp (F•) Hq (HomR(F•,ωR)) . Proof. We remark that the map defined in [CCHS, 1.4 (b)] is induced by the map Cm Cn ∗ −s−1 p t −q−1 from p (R)to q (ωR) defined by d(X Y )(X Y )=1ifs = t,andp = q

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and 0 otherwise (here p, q, r and s are tuples of integers, with p ≥ 0andt ≥ 0). C• { } C• { }∗ Notice that p (R) m is zero in positive cohomological degrees and q (ωR) n is zero in negative cohomological degrees. Hence this map extends uniquely to a map of complexes. Lemma 2.2. Let M, N and N be ﬁnitely generated graded R-modules and f : N→N be a graded homomorphism. For any integer i, there exist functorial iso- i i morphisms ψN and ψN that give rise to the following commutative diagram where the vertical maps are the natural ones:

M N N Proof. Let F• be a minimal graded free R-resolution of M and f• : F• →F• be a lifting of f to minimal graded free R-resolutions of N and N . Recall that there is a spectral sequence i j M N ⇒ i+j H (Hp(HomgrR(F• ,F• )) Hp (M,N), § i by [CD, 2.12] and [B, 6, Th´eor`eme 1, b]. As Hp(F•)=0fori = m if F• is a complex of free R-modules, there is a natural commutative diagram

i / i−m m M N Hp(M,N) H (Hp (HomgrR(F• ,F• )))

can can i / i−m m M N Hp(M,N ) H (Hp (HomgrR(F• ,F• ))).

M As F• is bounded, Lemma 2.1 provides the functorial isomorphisms m M N n M N ∗ Hp (HomgrR(F• ,F• )) Hq (HomR(HomgrR(F• ,F• ),ωR)) n N M ⊗ ∗ Hq (HomgrR(F• ,F• R ωR)) and equivalent ones for N . This provides a commutative diagram i / i−m n N M ⊗ ∗ Hp(M,N) H (Hq (HomgrR(F• ,F• R ωR)) )

can can i / i−m n N M ⊗ ∗ Hp(M,N ) H (Hq (HomgrR(F• ,F• R ωR)) ). As k is a ﬁeld, we have a natural isomorphism i−m n N M ⊗ ∗ m−i n N M ⊗ ∗ H (Hq (HomgrR(F• ,F• R ωR)) ) H (Hq (HomgrR(F• ,F• R ωR))

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M for N and a similar one for N .AsF• ⊗R ωR is acyclic, it provides a free R- resolution of M ⊗R ωR, and we get a commutative diagram m−i n N M ⊗ / m+n−i ⊗ H (Hq (HomgrRO(F• ,F• R ωR))) Hq (N,MO R ωR)

can can

m−i n N M ⊗ / m+n−i ⊗ H (Hq (HomgrR(F• ,F• R ωR))) Hq (N ,M R ωR) that follows from the same argument as in the first part of this proof, applied to q,N,M ⊗R ωR and q,N ,M ⊗R ωR in place of p,M,N and p,M,N . This finally gives the claimed diagram of natural maps. Remark 2.3. Lemma 2.2 holds more generally if we replace k by a Gorenstein ring of dimension 0, and assume that either M or N has finite projective dimension. Corollary 2.4. Assume S is a Cohen-Macaulay ring and a finitely generated graded R-module. Let F• be a graded complex of finite free S-modules, and consider a graded free R-resolution F•• of F•. Then the map given by Lemma 2.1, C• { }→C• { }∗ δ : p (F••) m q (HomR(F••,ωR)) n , induces a graded functorial map of double complexes C• { }→C• ∗ p (F•) dim S q (HomS (F•,ωS )) which gives rise to isomorphisms i ∗ dim S−i Hp(F•) HomS (Hq (HomS(F•,ωS)),k). Proof. First notice that S is a Cohen-Macaulay R-module and a graded k-algebra. By Lemma 2.2, the map δ induces commutative diagrams

i / m+n−i ∗ Hp(R, Fj−1) Hq (Fj−1,ωR)

i / m+n−i ∗ Hp(R, Fj) Hq (Fj,ωR)

i / m+n−i ∗ Hp(R, Fj+1) Hq (Fj+1,ωR) i i This gives the claim together with the isomorphisms Hp(R, F•) Hp(F•)and m+n−i dim S−i Hq (F•,ωR) Hq (HomS(F•,ωS)), where the second one follows from i j the Cohen-Macaulayness of S via the collapsing spectral sequence Hq(ExtR(F•,ωR)) i+j ⇒ Hq (F•,ωR). Lemma 2.5. Assume S is a Cohen-Macaulay ring and a finitely generated graded R-module. Let M and N be finitely generated graded S-modules such that N has S → finite projective dimension and Tori (M,ωS)=0for i>0. Then the map f ⊗ f 1ωS induces natural isomorphisms i i ⊗ ⊗ ExtS (M,N) ExtS(M S ωS,N S ωS). i i ⊗ ⊗ It follows that for any S-ideal I, HI (M,N) HI (M S ωS ,N S ωS ) for all i.

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Proof. ThecasewhereM = N = S is a classical property of ωS.IfM = S the result follows by induction on the projective dimension of N, using the “ﬁve lemma” and the exact sequences of Ext modules derived from the exact sequences E : → S → → → ⊗ ⊗ S 0 Syz1 (N) F N 0andE SωS.NoticeE SωS is exact since Tori (P, ωS)=0 for i>0ifP has ﬁnite projective dimension (see [S, 3.17, 4.5]). Hence: ∗ ⊗ i ⊗ ∀ ( )HomS (ωS,N S ωS ) N and ExtS(ωS,N S ωS )=0, i>0.

M For the general case, let F• be a free S-resolution of M.By(∗), the map → ⊗ M ⊗ ⊗ M f f 1ωS induces an isomorphism HomS(F• S ωS,N S ωS ) HomS(F• ,N). i M ⊗ ⊗ ∀ M ⊗ Furthermore, as ExtS(F• S ωS ,N S ωS )=0, i>0andF• S ωS is acyclic, i M ⊗ ⊗ i ⊗ ⊗ it follows that H (HomS (F• S ωS,N S ωS)) ExtS(M S ωS ,N S ωS ). This shows the ﬁrst isomorphism. It also follows that the map from C• M C• M ⊗ ⊗ I (HomgrS(F• ,N)) to I (HomgrS(F• S ωS ,N S ωS )),

induced by tensoring by 1ωS , gives a morphism of spectral sequences which is an Cp q isomorphism between the ﬁrst terms, which are respectively I (ExtS (M,N)) and Cp q ⊗ ⊗ I (ExtS(M S ωS ,N S ωS )). Hence the total homology of these double complexes is isomorphic. This proves our claim by [CD, 2.12].

We can now prove the main result of this article.

i Proof of Theorem 1.1. Since the grading of S is p-sharp, it follows that Hp(M,N) i ∗∗ Hp(M,N) , for all i. Hence by Lemma 2.5 it suﬃces to treat the case where M has M N ﬁnite projective dimension. Let F• and F• be a minimal graded free S-resolutions M N of M and N, respectively. Set F• := HomgrS (F• ,F• ). By Corollary 2.4, if F•• is a graded free R-resolution of F•,themap C• { }→C• { }∗ δ : p (F••) m q (HomR(F••,ωR)) n given by Lemma 2.1 induces a graded morphism of double complexes C• M N { }→C• N M ⊗ ∗ p (HomgrS(F• ,F• )) dim S q (HomgrS (F• ,F• S ωS ))

that is an isomorphism on the E1 terms: i M N dim S−i N M ⊗ ∗ Hp(HomgrS (F• ,F• )) Hq (HomgrS(F• ,F• S ωS )) . It follows that these two spectral sequences have isomorphic terms at any level, in i particular the same abutment. The first one abuts to Hp(M,N) by [CD, 2.12]. The dim S−i ⊗ ∗ § second one abuts to Hq (N,M S ωS ) by [B, 6, Théorème 1, b] and [CD, S 2.12], as Tori (M,ωS)=0fori>0 (because M has finite projective dimension).

Acknowledgments We thank the referee for indicating a simpliﬁcation in the proof of Lemma 2.5. This research took place when the second named author was on sabbatical leave. He would like to express his deep thanks to the Institut de Math´ematiques de Jussieu for its kind hospitality.

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References

[B] N. Bourbaki, Algèbre, Chapitre X, Herman, Paris, 1980. [CCHS] M. Chardin, S. D. Cutkosky, J. Herzog and H. Srinivasan, Duality and Tameness, arXiv:0704.0102v1 (to appear in Michigan Math. J.). [CD] M. Chardin and K. Divaani-Aazar, Generalized local cohomology and regularity of Ext modules, arXiv:math/0701509 (to appear in J. Algebra). [H] J. Herzog, Komplex Auflösungen und Dualität in der lokalen algebra, preprint, Univer- sitüt Regensburg, 1974. [HR] J. Herzog and A. Rahimi, Local Duality for Bigraded Modules, math.AC/0604587. [S] N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto. Univ. 18 (1) (1978), 71-85. MR490835 (80i:13004)

C.N.R.S., Institut de Mathematiques´ de Jussieu, 175 rue du Chevaleret, F-75013, Paris, France E-mail address: [email protected] Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran E-mail address: [email protected]

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