cofiniteness of local cohomology modules over regular local rings

COFINITENESS OF LOCAL COHOMOLOGY MODULES OVER REGULAR LOCAL RINGS GABRIEL CHIRIACESCU

1. Introduction It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules HIi (M) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then Hmi (M) are Artinian modules, which is equivalent to the following two properties: (i) SuppR (Hmi (M)) ⊆ {m}; (ii) the vector space HomR (k, Hmi (M)) has finite dimension over k, where k = R/m. Taking these facts into account, Grothendieck [9] made the following conjecture. Conjecture 1.1. If I is an ideal of a Noetherian local ring R, and M is a finitely generated R-module, then HomR (R/I, HIi (M)) is finitely generated. Later, Hartshorne [11] refined this conjecture and asked the following more general question. Conjecture 1.2. If I is an ideal of a Noetherian local ring R, and M is a finitely generated R-module, does it follow that ExtiR (R/I, HIj (M)) are finitely generated for all i, j > 0 ? In the same paper, Hartshorne gave the following general definition. Definition 1.3. If I is an ideal of a Noetherian ring R, then a module N will be called I-cofinite if it satisfies the following conditions: (i) SuppR (N) ⊆ V (I); (ii) ExtiR (R/I, N) is finitely generated for all i > 0. Hartshorne’s definition is motivated by the fact that if we have a short exact sequence of modules of which two are I-cofinite, then the third module has the same property. Hartshorne’s definition is fulfilled, for instance, in the case when the ideal I coincides with the maximal ideal of a local ring, since in this case not only is the socle HomR (k, Hmi (M)) finitely generated, but all ExtjR (k, Hmi (M)) are finitely generated for all i and j, as we can see by using Matlis duality. Hartshorne also gave a counterexample to Conjecture 1.1, which is essentially as follows. Let k be a field, and let R = k[[X, Y , Z, U]] and I = (X, U)R. If we take M = R/(XY − ZU), then HI2 (M) is not I-cofinite. In fact, he proved much more Received 5 January 1998; revised 24 September 1998. 1991 Mathematics Subject Classification 13H05, 13H10, 13D45. Bull. London Math. Soc. 32 (2000) 1–7

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than this, namely that HomR (k, HI2 (M)) is not finitely generated. Nonetheless, by using the derived category theory, he proved that if R is a complete regular local ring, then HIi (M) is I-cofinite in two cases: (i) I is a non-zero principal ideal; (ii) I is a prime ideal with dim(R/I) = 1. In [12] Huneke and Koh generalized the above result to the case of a Gorenstein complete domain and an arbitrary one-dimensional ideal. In [6] Delfino extended the above result to a complete ring, under some restrictive conditions, and finally, Delfino and Marley [7] proved the general case, as follows. Theorem 1.4. Let (R, m) be a Noetherian local ring, let I be a one-dimensional ideal of R, and let M be a finitely generated R-module. Then HIi (M) is I-cofinite for all i. The purpose of this paper is to give a new proof of this result in the case when R is a regular local ring. By [7], the general case is immediately reduced to this case. We are trying to avoid, as far as possible, the use of spectral sequences, a technique adopted both by Hartshorne and by Huneke and Koh. The hard part of the proof is to show that HId−1 (M) is I-cofinite, where I is a one-dimensional ideal of a Gorenstein ring of Krull dimension d such that HId (R) = 0 and M is a finitely generated module. In fact, we are able to show much more than this: namely, we shall prove that TorRj (M, HId−1 (R)) and ExtjR (M, HId−1 (R)) are I-cofinite for all j > 0 (see Proposition 3.2). Our terminology follows [3] and [14]. 2. Preliminaries We shall use the following remarkable result, which is known as the Hartshorne– Lichtenbaum vanishing theorem (HLVT). For a proof, see [4], [10] or [15]. Theorem 2.1. Let (R, m) be a Noetherian local ring of Krull dimension d, and let b R b + P ) > 0 for all prime I be an ideal of R. Then HId (R) = 0 if and only if dim(R/I b such that dim(R/P b ) = d. ideals P of the m-adic completion R Remark. (i) For a complete domain, the ‘only if ’ condition of HLVT is equivalent to the fact that I is not m-primary. (ii) If R is a Noetherian local ring of Krull dimension d, and I is an ideal of R such that HId (R) = 0, then HIm (M) = 0 for all R-modules M and all m > d (see [10]). As a consequence, we infer that HId−1 (−) is a right exact functor. Therefore, by [16, Theorem 3.33], we have HId−1 (M) ' M ⊗R HId−1 (R) for all R-modules M. This remark will be of essential use in this paper. To prove the finiteness of some Ext, we shall often check, by using Matlis duality, that the Matlis dual of it is an Artinian module, which is similar to checking that some Tor is an Artinian module. In this sense, the following proposition is fundamental (for a proof, see [5, VI, 5.1] or [16]).

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Proposition 2.2. Let R be a commutative ring, let M and N be two R-modules, and let I be an injective R-module. Then we have the isomorphism HomR (TorRi (M, N), I) ' ExtiR (M, HomR (N, I)). In addition, if R is Noetherian and M is finitely generated, then HomR (ExtiR (M, N), I) ' TorRi (M, HomR (N, I)). Lemma 2.3. Let (R, m, k) be a Noetherian local ring, let S be a multiplicative closed set such that m ∩ S 6= ?, and let N be an RS -module. Then HomR (k, N) = 0. Proof.

We have the following canonical isomorphisms: HomR (k, N) ' HomR (k, HomRS (RS , N)) ' HomRS (k⊗R RS , N).

But the last module is zero since, by hypothesis, we have mRS = RS . ϕ

Lemma 2.4. Let A be a Noetherian ring, and let A −→ B be a faithfully flat A-algebra. Then for any finitely generated A-module N, and each A-module M such that M ⊗A Coker(ϕ) = 0, we have TorAi (M, ExtjA (N, Coker(ϕ))) = 0

and

TorAi (M, TorAj (N, Coker(ϕ))) = 0

for all i, j > 0. (For the second equality, it is not necessary to assume that N is finitely generated.) Proof. Since B is a faithfully flat A-algebra, we infer that A is a pure submodule of B, so Coker(ϕ) is a flat A-module. Now we have the following canonical isomorphisms (see [1, Propositions 7 and 8, pp. 108–109]): TorAi (M, ExtjA (N, Coker(ϕ))) ' TorAi (M, ExtjA (N, A) ⊗A Coker(ϕ))

' TorAi (M ⊗A Coker(ϕ), ExtjA (N, A)).

But, by hypothesis, the last module is zero, and the proof for the first equality is finished. A similar proof works for the second equality. α

β

δ

Lemma 2.5. (i) Let R be a Noetherian ring, and let M −→ N −→ L −→ P be an exact sequence of R-modules, where M and P are finitely generated. If T is a finitely generated R-module, then HomR (T , N) is finitely generated if and only if HomR (T , L) has the same property. (ii) In addition, if X and Y are two finitely generated R-modules, then the following statements are equivalent. (a) HomR (X, TorRi (Y , ExtjR (T , N))) is finitely generated. (b) HomR (X, TorRi (Y , ExtjR (T , L))) has the same property. A similar conclusion can be drawn if we replace Extj by Torj . Proof.

(i) Considering Im(α) and Im(δ), we may assume that our exact sequence

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gabriel chiriacescu β

has the form 0 −→ M −→ N −→ L −→ P −→ 0. We break up this sequence into two short exact sequences: 0 −→ M −→ N −→ Im(β) −→ 0, 0 −→ Im(β) −→ L −→ P −→ 0.

(2.5.1) (2.5.2)

Applying HomR (T , −) to the first of these, we obtain 0 −→ HomR (T , M) −→ HomR (T , N) −→ HomR (T , Im(β)) −→ Ext1R (T , M) −→ · · · .

(2.5.3)

Ext1R (T , M)

are finitely generated, we infer that HomR (T , N) Since HomR (T , M) and is finitely generated if and only if HomR (T , Im(β)) has the same property. Applying the same procedure to the exact sequence (2.5.2), we deduce that HomR (T , Im(β)) is finitely generated if and only if HomR (T , L) has this property. (ii) Let us denote TorRi (Y , −) by Ti (−) and ExtjR (T , −) by E j (−). If we rewrite the long exact sequence (2.5.3), then we obtain uj

vj

wj

· · · −→ E j (M) −→ E j (N) −→ E j (Im(β)) −→ E j+1 (M) −→ · · · . Then, for every j, we have the following exact sequences: 0 −→ Im(uj ) −→ E j (N) −→ Im(vj ) −→ 0, 0 −→ Im(vj ) −→ E j (Im(β)) −→ Im(wj ) −→ 0.

(2.5.4) (2.5.5)

Tensoring (2.5.4) with Y , we obtain the long exact sequence of Tors · · · −→ Ti (Im(uj )) −→ Ti (E j (N)) −→ Ti (Im(vj )) −→ Ti−1 (Im(uj )) −→ · · · . Since the two outside terms are finitely generated, we infer that HomR (X, Ti (E j (N))) is finitely generated for all i, j > 0 if and only if HomR (X, Ti (Im(vj ))) has this property (by (i)). Similar reasoning applied to (2.5.5) gives that HomR (X, Ti (Im(vj ))) is finitely generated if and only if HomR (X, Ti (E j (Im(β)))) is finitely generated. By a similar argument applied to (2.5.2), we obtain that the finite generation of HomR (X, Ti (E j (Im(β)))) and of HomR (X, Ti (E j (L))) are equivalent. 3. The proof of Theorem 1.4 in the regular case We need the following lemma. Lemma 3.1. Let (R, m, k) be a complete Gorenstein ring of Krull dimension d, and let I be an ideal of R such that dim(R/I) = 1. If {P1 , P2 , . . . , Pr } is the set of minimal S prime ideals over I and S = R − ri=1 Pi , then there exists an exact sequence ΦS cS −→ H ∨ −→ 0, 0 −→ HId (R)∨ −→ R −→ R

(3.1.1)

which gives, by localization, ΨS

cS −→ (H ∨ )S −→ 0, 0 −→ RS −→ R

(3.1.2)

where H ∨ means the Matlis dual of HId−1 (R), and ΨS is the canonical homomorphism of completion. S Proof. For any element f ∈ m − ri=1 Pi and any natural number n, we have ˇ the following exact sequence, which arises from the Cech complex applied to the

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R-module R/I n and to the principal ideal (f): Φnf

0 1 (R/I n ) −→ R/I n −→ (R/I n )f −→ H(f) (R/I n ) −→ 0. 0 −→ H(f)

By the base ring independence of local cohomology, applied to the canonical homomorphism R → R/I n and to the ideal (f), we deduce the exact sequence Φnf

0 −→ H(I0 n ,f) (R/I n ) −→ R/I n −→ (R/I n )f −→ H(I1 n ,f) (R/I n ) −→ 0. S Since f ∈ m − ri=1 Pi and dim(R/I n ) = 1, we deduce that the radical of (I n , f) is m and that (R/I n )f is an Artinian ring, with the set of maximal ideals {P1f , P2f , . . . , Prf }. Hence we have the isomorphisms (R/I n )f '

r M (R/I n )Pif ' (R/I n )S ' RS /ISn . i=1

With these remarks, the above exact sequence becomes ΦnS

0 −→ Hm0 (R/I n ) −→ R/I n −→ (R/I n )S −→ Hm1 (R/I n ) −→ 0. All modules of the above sequence define inverse systems which satisfy the MittagLeffler condition so, taking inverse limits, we obtain ΦS cS −→ lim Hm1 (R/I n ) −→ 0. 0 −→ lim Hm0 (R/I n ) −→ R −→ R ←− ←− n

n

By using the following isomorphisms, which arise from the local duality theorem, we obtain the exact sequence (3.1.1): n lim H i (R/I n ) ' lim HomR (Extd−i R (R/I , R), E) ←− ←− m n

n

n ' HomR (lim Extd−i R (R/I , R), E) −→ n

' HomR (HId−i (R), E), where 0 6 i 6 d and E denotes the injective hull of k. It is clear that the homomorphism ΦS is obtained as the composition of the ΨS cS , where the first homomorphism continuous ring homomorphisms R −→ RS −→ R is the localization with respect to S, and ΨS is the completion homomorphism of the semilocal ring RS with respect to the topology given by its Jacobson radical. Since ΨS cS is injective, being faithfully flat, we obtain (3.1.2). the homomorphism RS −→ R Remark. As one can see, the exact sequence (3.1.1) is nothing but the exact sequence of [15, III, 2.2]. We preferred to give, in our case, the above elementary proof which avoids the use of formal function theory. In fact, if we denote U = cS . b O ) is isomorphic to R Spec(R) − {m}, then V (I) ∩ U is a finite set, therefore Γ(U, b U We are ready to prove the main result of this section. Proposition 3.2. Let (R, m, k) be a Gorenstein ring of Krull dimension d, and let I be an ideal of R such that dim(R/I) = 1. If M is a finitely generated R-module, then TorRj (M, H) and ExtjR (M, H) are I-cofinite for all j > 0, where H := HId−1 (R). In particular, if HId (R) = 0, then HId−1 (M) is I-cofinite.

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Proof. We shall prove only the cofiniteness of TorRj (M, H). A similar proof works for ExtjR (M, H). b it is clear that an R-module By the faithful flatness of the m-adic completion R, b is b I-cofinite, and since T is I-cofinite if and only if T ⊗R R R c d−1 b b b b ' Exti (R/ I, Torb ExtiR (R/I, TorRj (M, H)) ⊗R R j (M, Hb (R))), b R

I

we may assume that R is complete. We have to show that ExtiR (R/I, TorRj (M, H)) is finitely generated for all i > 0, or, by using Matlis duality, that ExtiR (R/I, TorRj (M, H))∨ ' TorRi (R/I, TorRj (M, H)∨ ) ' TorRi (R/I, ExtRj (M, H ∨ )) is an Artinian module. By using a standard characterization of Artinian modules, this is the same as proving the following two facts: (i) HomR (k, TorRi (R/I, ExtRj (M, H ∨ ))) is a finitely generated k-vector space; (ii) SuppR (TorRi (R/I, ExtRj (M, H ∨ ))) ⊆ {m}. We begin by proving (i). From the exact sequence (3.1.1), we obtain the exact sequence cS −→ H ∨ −→ 0. 0 −→ Im(ΦS ) −→ R cS ))) = 0, and since By Lemma 2.3 we know that HomR (k, TorRi (R/I, ExtRj (M, R

Im(ΦS ) is finitely generated, Lemma 2.5 shows that HomR (k, TorRi (R/I, ExtRj (M, H ∨ ))) is finitely generated. To prove (ii), we first note that SuppR (TorRi (R/I, ExtRj (M, H ∨ ))) ⊆ V (I). So it will be sufficient to show that TorRi S (RS /IS , ExtRj S (MS , (H ∨ )S )) = 0. ΨS cS is faithfully flat, and taking into account that ΨS ⊗RS RS /IS is Since RS −→ R cS /IbS ), from (3.1.2) we infer that an isomorphism (because RS /IS ' R RS /IS ⊗RS (H ∨ )S ' RS /IS ⊗RS Coker(ΨS ) = 0. Hence we may apply Lemma 2.4, and (ii) is proved. In the case when HId (R) = 0, to see that HId−1 (M) is I-cofinite, we must observe that, by HLVT, we have HId−1 (M) ' M ⊗R H = TorR0 (M, H) (see the remark after Theorem 2.1). Remark. It is easy to see that we have the following convergence of spectral sequences: E2p,q := ExtpR (M, HIq (R))

⇒

n lim Extp+q R (M/I M, R). −→ n

If R and I satisfy the conditions of Proposition 3.2, and HId (R) = 0, then HIq (R) = 0 for all q 6= d−1, hence the above spectral sequence collapses. Since R is a Gorenstein ring of dimension d, it follows that ExtiR (−, R) = 0 for all i > d, so we are interested in only the isomorphisms n HomR (M, H) ' lim Extd−1 R (M/I M, R) −→

and

n

Ext1R (M, H)

' lim ExtdR (M/I n M, R). −→ n

These modules are, in the language of Hartshorne [11], nothing but H d−1 (DI (M)) and H d (DI (M)), respectively.

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We are ready to give the proof of Theorem 1.4 in the case when R is regular. Proof of Theorem 1.4. As in the proof of Proposition 3.2, we may assume that R is complete. We shall proceed by induction on pd(M). If M is free, then HIi (M) = 0 for all i 6= d − 1 (since grade(I, R) = d − 1 and, by HLVT, we have HId (R) = 0). By Proposition 3.2, HId−1 (R) is I-cofinite, so in this case we are done. Now take a finite presentation of M: 0 → N → F → M → 0, where F is a free module of finite rank and pd(N) < pd(M). If we apply ΓI (−) to this exact sequence, taking into account that HIi (F) = 0 for all i 6= d − 1, then we obtain an exact sequence µ

0 −→ HId−2 (M) −→ HId−1 (N) −→ HId−1 (F) −→ HId−1 (M) −→ 0, and the isomorphisms HIi (M) ' HIi+1 (N) for all i < d − 2. By induction, we infer that HIi (M) is I-cofinite for all i < d − 2. We break up the above exact sequence into two short exact sequences: and

0 −→ HId−2 (M) −→ HId−1 (N) −→ Im(µ) −→ 0 0 −→ Im(µ) −→ HId−1 (F) −→ HId−1 (M) −→ 0.

Since, by Proposition 3.2, HId−1 (M) is I-cofinite, it follows that Im(µ) is I-cofinite and, finally, we deduce that HId−2 (M) is I-cofinite. References 1. N. Bourbaki, Alg`ebre, alg`ebre homologique (Mason, Paris, 1980). 2. M. Brodmann, Einige ergebnisse aus der Lokalen Kohomologietheorie und Ihre Anwendung, Osnabr¨ uck. Schrift. Math. 5 (University of Osnabr¨ uck, 1983). 3. W. Bruns and J. Herzog, Cohen–Macaulay rings (Cambridge University Press, 1993). 4. F. W. Call and R. Y. Sharp, ‘A short proof of the local Lichtenbaum–Hartshorne Theorem on the vanishing of local cohomology’, Bull. London Math. Soc. 18 (1986) 261–264. 5. H. Cartan and S. Eilenberg, Homological algebra (Princeton University Press, 1956). 6. D. Delfino, ‘On the cofiniteness of local cohomology modules’, Math. Proc. Cambridge Philos. Soc. 115 (1994). 7. D. Delfino and T. Marley, ‘Cofinite modules and local cohomology’, J. Pure Appl. Algebra (1) 121 (1997) 45–52. 8. A. Grothendieck, Local cohomology, notes by R. Hartshorne, Lecture Notes in Math. 862 (Springer, New York, 1966). 9. A. Grothendieck, Cohomologie locale des faisceaux et th´eor`emes de Lefshetz locaux et globaux (North-Holland, Amsterdam, 1969). 10. R. Hartshorne, ‘Cohomological dimension of algebraic varieties’, Ann. of Math. 88 (1968) 403–450. 11. R. Hartshorne, ‘Affine duality and cofiniteness’, Invent. Math. 9 (1970) 145–164. 12. C. Huneke and J. Koh, ‘Cofiniteness and vanishing of local cohomology modules’, Math. Proc. Cambridge Philos. Soc. 110 (1991). 13. E. Matlis, ‘Injective modules over noetherian rings’, Pacific J. Math. 8 (1958) 511–528. 14. H. Matsumura, Commutative ring theory (Cambridge University Press, 1986). ´ 15. C. Peskine and L. Szpiro, ‘Dimension projective finie et cohomologie locale’, Inst. Hautes Etudes Sci. Publ. Math. 42 (1973) 323–395. 16. J. Rotman, An introduction to homological algebra (Academic Press, New York, 1979).

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