Local cohomology with supports in the non-free locus

Algebra Colloquium c °2008 AMSS CAS & SUZHOU UNIV Algebra Colloquium 1? : ? (200?) ??–?? Local Cohomology with Supports in the Non-free Locus∗ Agust´ın Marcelo Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria Tafira Baja, Edificio de Inform´atica y Matem´aticas 35017-Las Palmas de Gran Canaria, Spain E-mail: [email protected] J. Mu˜noz Masqu´e Instituto de F´ısica Aplicada, CSIC, C/Serrano 144, 28006-Madrid, Spain E-mail: [email protected] C. Rodr´ıguez Mielgo Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria Tafira Baja, Edificio de Inform´atica y Matem´aticas 35017-Las Palmas de Gran Canaria, Spain E-mail: [email protected] Received 9 May 2007 Revised 17 July 2007 Communicated by Zhongming Tang Abstract. The groups of local cohomology with supports in the non-free locus of a module are used in order to obtain three classifications and one characterization of four classes of modules. 2000 Mathematics Subject Classification: primary 13C05; secondary 13E05, 13E15 Keywords: ideal module, local cohomology, non-free locus, projective dimension, reflexive module 1 Introduction and Preliminaries Let Rbe a Noetherian ring and let Mbe a finitely generated R-module. As is well known (see [5, (11.1.1)]), the set of points p∈Spec Rsuch that Mpis a free Rp-module is an open subset in the Zariski topology. Hence, its complement Cis a closed subset, called the non-free locus of M, whose corresponding radical ideal (cf. [1, II, §4.3, Proposition 11(iii)]) is denoted by a=a(M) = I(C) throughout the paper. Proposition 1.1. Let Rbe a Noetherian ring and let F1 ϕ −→ F0→M→0be a finite free presentation of the R-module M. If rank ϕ=rand rank F0=n, then ∗Supported by Ministerio de Ciencia y Tecnolog´ıa of Spain, under grants BFM2001-2718 and MTM2005-00173. 2A. Marcelo, J. Mu˜noz Masqu´e, C. Rodr´ıguez Mielgo the ideal a(M)coincides with the radical of (n−r)-th Fitting invariant of M; i.e., a(M) = rad Ir(ϕ). Proof. As ais a radical ideal, we only need to prove that V(a) = V(Ir(ϕ)), or conversely, that a6⊆ pif and only if Ir(ϕ)6⊆ pfor every p∈Spec R. If pdoes not contain a, then Mpis a free Rp-module and hence Ir(ϕ)p=Rp(for example, see [2, Proposition 1.4.9]); therefore Ir(ϕ)6⊆ p. The converse also follows from the previous reference. 2 Below, we use the ideal ain order to obtain classifications of two classes of modules and a characterization of k-th syzygies. More precisely, in Section 2, a reflexive finitely generated module Mover a Noetherian local domain whose dual module is of projective dimension one is shown to be completely determined by H2 a(M). Similarly, in Section 3, we obtain a classification of the ideal modules (in the sense of [7, Proposition 5.1]) over a regular local ring by means of H1 a(M). As a consequence of such a result (see Corollary 3.2), from a decomposition H1 a(M) = Lr i=1 R/aifor certain ideals a1, . . . , ar, we deduce that Mis stably equivalent to Lr i=1 ai. By applying these results and from the existence of a dualizing functor, in Section 4, we also obtain a classification of the torsion-free finitely generated and non-free modules of projective dimension one over a regular local ring. Finally, in Section 5, we obtain a characterization of k-th syzygies within the class of reflexive finitely generated R-modules over a regular local ring by the vanishing of the groups Hi a(M) for i= 0, . . . , k −1. To a certain extent, this result can be considered as a generalization of the characterization of the vector bundles on the punctured spectrum, which are k-th syzygies (see [4, Lemma 6.5]). 2 Reflexive Modules with Dual of Projective Dimension One Lemma 2.1. If Ris a Cohen–Macaulay ring, then H1 q(R)=0for every ideal q with height q≥2. Proof. By virtue of the hypothesis, we know that depth Rp= dim Rpfor every p∈Spec R(see [2, Theorem 2.1.3]). Therefore, depthqR= minp∈V(q)depth Rp= minp∈V(q)dim Rp≥2, and we can conclude by simply applying the cohomological interpretation of depth. 2 Theorem 2.2. Let (R, m)be a regular local ring of dimension d≥3. Then every reflexive finitely generated R-module Mwith proj dim M∨= 1 is completely determined by H2 a(M). Proof. Let 0→F1 ϕ −→ F0→M∨→0 (1) be a minimal free resolution of M∨. As Mis reflexive, dualizing (1), we obtain the exact sequence 0→M→F∨ 0 ϕ∨ −→ F∨ 1→Ext1(M∨, R)→0,(2) Local Cohomology with Supports in the Non-free Locus 3 which breaks into two short exact sequences 0→Im ϕ∨→F∨ 1→Ext1(M∨, R)→0,(3) 0→M→F∨ 0→Im ϕ∨→0.(4) First, we prove that Ext1(M∨, R) is isomorphic to H2 a(M). Owing to reflexivity, Mand M∨have the same non-free locus; hence support Ext1(M∨, R)⊆V(a), and accordingly, H0 a(Ext1(M∨, R)) = Ext1(M∨, R). Moreover, as Im ϕ∨and F∨ 1are torsion-free modules, both H0 a(Im ϕ∨) and H0 a(F∨ 1) vanish, and taking cohomology with supports in V(a) in (3), by virtue of Lemma 2.1, we obtain Ext1(M∨, R)∼ = H1 a(Im ϕ∨). Similarly, by taking cohomology with supports in V(a) in (4), we obtain an isomorphism H1 a(Im ϕ∨)∼ =H2 a(M). Hence, Ext1(M∨, R)∼ =H2 a(M), and consequently, Mis a second syzygy module of H2 a(M). In order to conclude, we only need to prove that the minimality of the resolution (1) implies that of (2). In fact, if a linear form ω0∈F∨ 0exists such that ϕ∨(ω0)/∈mF∨ 1(cf. [2, Proposition 1.3.1]), then by Nakayama’s lemma, ϕ∨(ω0) belongs to a basis of F∨ 1and hence there exists an element x1∈F1such that ϕ∨(ω0)(x1) = ω0(ϕ(x1)) = 1. Hence, Im ϕ6⊂ mF0and (1) is not minimal. 2 3 Ideal Modules Theorem 3.1. Let Mbe an ideal module over a regular Noetherian local ring (R, m)of dimension d≥2, and let r(M)be the greatest rank of a free direct summand in M. Then Mis completely determined by r(M)and H1 a(M). Proof. The module Membeds into its bidual and the quotient T=M∨∨/M is a torsion module. First of all, we prove that H1 a(M) = T. For every prime ideal pof Rwith height p≤1, Mpis a free Rp-module as Ris regular and Mis torsionless. Therefore, Mp=M∨∨ pand accordingly p/∈V(a); hence height a≥2. Since M∨∨ is free, by virtue of Lemma 2.1, we conclude that H1 a(M∨∨) = 0. This fact proves our claim by simply taking cohomology with supports in ain the sequence 0 →M→ M∨∨ →T→0, and recalling that H0 a(M) = H0 a(M∨∨) = 0 as Mand M∨∨ are torsionless, and H0 a(T) = Tas the support of Tis contained in V(a). If Mand M0are two isomorphic ideal modules, then Tand T0=M0∨∨/M0are also isomorphic. Moreover, from the very definition of r(M), it follows that there exist two submodules F, ¯ M⊆Msuch that Fis free of rank r(M) and M=F⊕¯ M. Hence, we only need to prove that if Mand M0are two ideal modules such that r(M) = r(M0) and T∼ =T0, then ¯ M∼ =¯ M0. Let π:¯ M∨∨ →T=¯ M∨∨/¯ M(resp., π0:¯ M0∨∨ →T0=¯ M0∨∨/¯ M0) be the quotient map. As ¯ M∨∨ and ¯ M0∨∨ are free Rmodules, every isomorphism φ:T→T0induces a homomorphism Φ: ¯ M∨∨ →¯ M0∨∨ making the following diagram commutative: ¯ M∨∨ π −→ T→0 Φ↓ ↓ φ ¯ M0∨∨ π0 −→ T0→0 We claim ¯ M⊂m¯ M∨∨ (resp., ¯ M0⊂m¯ M0∨∨), otherwise, every element ¯xin ¯ Mnot belonging to m¯ M∨∨ generates a submodule R¯x, which is a direct summand in ¯ M 4A. Marcelo, J. Mu˜noz Masqu´e, C. Rodr´ıguez Mielgo by Nakayama’s lemma, thus contradicting the definition of r(M). Hence, rank ¯ M∨∨ = dimR/m(T/mT), and similarly for the rank of ¯ M0∨∨. Accordingly, Φ is an isomorphism that induces an isomorphism from ¯ M= ker πonto ¯ M0= ker π0.2 Corollary 3.2. With the same hypotheses as in Theorem 3.1, assume a decomposition H1 a(M) = Lr i=1 R/aiholds for certain ideals a1, . . . , arin R. Then there exists a free module Fsuch that M∼ =F⊕¡Lr i=1 ai¢. Proof. Let M=F⊕¯ Mbe as in the proof of the previous theorem. As Ris a unique factorization domain by virtue of our assumption, and each a∨ iis a reflexive R-module of rank 1 (cf. [6, Corollary 1.2, Proposition 1.9]), we conclude a∨ i∼ =R; hence ¡Lr i=1 ai¢∨∨ ∼ =Rr. Moreover, from the first part of the proof of Theorem 3.1, we know that T=Lr i=1 R/ai, and proceeding as in the second part of that proof, we obtain ¯ M∼ =Lr i=1 ai.2 Example 3.3. Let s, t be two variables over the field k, and let A=k[s4, s3t, st3, t4] be the k-algebra, which is a classical example of a non-Cohen–Macaulay ring. We also set R=k[s4, t4] and m= (s4, t4)·R. The inclusion of Rinto Aconverts Ainto an R-module generated by 1, s3t, st3, s2t2, and we have A=s2t2m⊕F, where F is the R-module generated by 1, s3t, st3. As m∨∨ = HomR(HomR(m, R)) ∼ =R, we readily conclude that Ais an ideal R-module and, in this case, we have r(A)=3 and H1 a(A) = k. 4 Torsion-free Modules of Projective Dimension One Let Rbe a Noetherian local domain and let Fπ −→ Mbe a minimal epimorphism of a torsion-free finitely generated and non-free R-module M, where Fis a free R-module, which is completely determined by Mup to an isomorphism. Dualizing the short exact sequence 0→Nϕ −→ Fπ −→ M→0,(5) we obtain 0 →M∨π∨ −→ F∨ϕ∨ −→ N∨. The ‘codual module’ of Mis defined to be the R-module cd M= Im ϕ∨, which is also a torsion-free finitely generated and non-free R-module, as follows from the following exact sequence by virtue of the assumptions on M: 0→M∨π∨ −→ F∨ϕ∨ −→ cd M→0.(6) Proposition 4.1. Let Rbe a Noetherian local domain and let Mbe a torsion-free finitely generated and non-free R-module. We have: (i) cd (cd M) = M. (ii) a(M) = a(cd M). Proof. (i) Dualizing (6), we obtain 0 →(cd M)∨ϕ∨∨ −→ Fπ∨∨ −→ M∨∨. From this sequence and (5), taking the natural inclusion M⊆M∨∨ into account, we obtain an injection N⊆(cd M)∨. As Nand (cd M)∨have the same rank (equal to Local Cohomology with Supports in the Non-free Locus 5 rank F−rank M), there exists r∈Rsuch that r·(cd M)∨⊆N, and rmust be invertible because Mis torsion-free. (ii) If (cd M)pis a free Rp-module, then (M∨)p(and hence Mp) is also free, as follows from (6). Hence, a(M)⊆a(cd M), and we can conclude by virtue of (i). 2 Remark 4.2. More formally, we can state that ‘taking the codual module’ is a dualizing functor in the sense of [3, §21.1]. Proposition 4.3. Let Rbe a regular Noetherian local domain of dimension d≥2. Then every torsion-free finitely generated non-free R-module Mof projective dimension one is completely determined by r(cd M)and H1 a(cd M). Proof. By virtue of Theorem 3.1, we only need to prove that ‘taking the codual module’ is an equivalence between the category of torsion-free finitely generated non-free R-modules of projective dimension one and that of ideal torsion-free finitely generated non-free R-modules. Let 0 →F1 ϕ −→ F0 π −→ M→0 be a minimal free resolution of M. From [7, Proposition 5.1(e)], we know that cd M= Im ϕ∨is an ideal module. Conversely, if Mis an ideal torsion-free finitely generated non-free R-module, then again from the reference above, it follows that cd Mis of projective dimension one. 2 5 Reflexive Modules over a Regular Ring Which Are k-th Syzygies Theorem 5.1. Let Rbe a regular Noetherian local ring. A reflexive finitely generated R-module Mis a k-th syzygy if and only if Hi a(M) = 0 for every i= 0, . . . , k−1. Proof. Let Mbe a reflexive finitely generated R-module. Because of reflexivity (e.g., see [2, 1.4.19(c)]), we can assume k≥2. Let 0→Fj ϕj−1 −→ Fj−1 ϕj−2 −→ · · · ϕ1 −→ F1 ϕ0 −→ F0→M∨→0 (7) be a minimal free resolution of M∨. If Mis a k-th syzygy, then by dualizing (7), since Mis reflexive, we obtain an exact sequence (see the proof of [4, Lemma 5.1]) 0→M→F∨ 0 ϕ∨ 0 −→ F∨ 1 ϕ∨ 1 −→ · · · ϕ∨ k−3 −→ F∨ k−2 ϕ∨ k−2 −→ F∨ k−1→N→0 (8) with N= coker ϕ∨ k−2, which breaks into the following kshort exact sequences: S0: 0 →M→F∨ 0→Im ϕ∨ 0→0, Si: 0 →Im ϕ∨ i−1→F∨ i→Im ϕ∨ i→0 (i= 1, . . . , k −2), Sk−1: 0 →Im ϕ∨ k−2→F∨ k−1→N→0. Let p∈Spec Rbe a prime ideal with height p≤k. As Mis a k-th syzygy, it satisfies Serre’s Skcondition (e.g., see [4, Theorem 3.8(b)]), and accordingly, we have depth M≥height p. From the Auslander–Buchsbaum formula [2, Theorem 1.3.3], we thus conclude that proj dim Mp= 0, or in other words, Mpis a free Rp-module. Therefore, p∈V(a) implies height p≥k+ 1, so that height a≥k+ 1 and hence d= dim R≥k+ 1. Recalling the cohomological interpretation of depth, 6A. Marcelo, J. Mu˜noz Masqu´e, C. Rodr´ıguez Mielgo from the sequence S0above, we obtain Hp−1 a(Im ϕ∨ 0)∼ =Hp a(M) for p= 1, . . . , d−1, and similarly, from Sifor i= 1, . . . , k −2, we have Hp−1 a(Im ϕ∨ i)∼ =Hp a(Im ϕ∨ i−1) for p= 1, . . . , d −1. In particular, for every p= 1, . . . , k −1, we have Hp a(M)∼ =Hp−1 a(Im ϕ∨ 0)∼ =Hp−2 a(Im ϕ∨ 1)∼ =· · · ∼ =Hp−j a(Im ϕ∨ j−1)∼ =· · · ∼ =H0 a(Im ϕ∨ p−1). Moreover, we have H0 a(M) = H0 a(Im ϕ∨ 0) = · · · =H0 a(ϕ∨ k−2) = 0 as all these modules are torsion-free. Hence, Hp a(M) = 0 for p= 0, . . . , k −1, and we can conclude the first part of the proof. Conversely, assume Hi a(M) = 0 for i= 0, . . . , k −1. Again by virtue of [4, Theorem 3.8], in order to prove that Mis a k-th syzygy, we only need to state that Msatisfies Serre’s Skcondition. Given p∈Spec R, we are led to distinguish two cases: If Mpis a free Rp-module, then depth Mp= depth Rpand the Sk condition certainly holds. If Mpis not free as an Rp-module, then p∈V(a) and we have depth Mp≥depthaM= minp∈V(a)depth Mp≥k, so that Msatisfies the Sk condition again. The theorem is thus established. 2 Corollary 5.2. With the same assumptions as in Theorem 5.1, the module M satisfies the Skcondition if and only if depthaM≥k. Remark 5.3. If Mis a k-th syzygy strictly, i.e., if Mis a k-th syzygy but not a (k+ 1)-th syzygy, then the torsion submodule Tof the module Nin (8) does not vanish, and localizing (8) at any p∈Spec Rwith height p≤k, we obtain an exact sequence of free Rp-modules; hence Tp= 0, and accordingly, support T⊆V(a). Taking cohomology with supports in ain the sequence Sk−1, we have Hk a(M)∼ = H0 a(N) = T. Acknowledgement. We would like to thank Professor Peter Schenzel for his valuable comments and suggestions in preparing the manuscript. References [1] N. Bourbaki, Elements of Mathematics, Commutative Algebra, Hermann, Paris, 1972. [2] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993. [3] D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995. [4] Gr. Evans, Ph. Griffiths, Syzygies, Cambridge University Press, Cambridge, 1985. [5] A. Grothendieck, J. Dieudonn´e, EGA IV. ´ Etude locale des sch´emas et des morphismes de sch´emas, Publ. Math. IHES, 20, 24, 28, 32, IHES, Paris, 1964, 1965, 1966, 1967. [6] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980) 121–176. [7] A. Simis, B. Ulrich, W.V. Vasconcelos, Rees algebras of modules, Proc. London Math. Soc. 87 (2003) 610–646.