A New Outlook on Cofiniteness
Kamran Divaani-Aazar, Hossein Faridian, Massoud Tousi

TL;DR
This paper investigates conditions under which local cohomology modules are cofinite, establishing abelian properties of certain subcategories and extending results to rings with small dimension or cohomological dimension.
Contribution
It proves the abelian nature of the subcategory of cofinite modules when cohomological dimension is at most 1 and extends cofinite properties of local cohomology modules to rings with small dimension.
Findings
Subcategory of cofinite modules is abelian when $ ext{cd}(rak{a}, R) extless= 1.
Local cohomology modules are cofinite under certain dimensional constraints.
Answers to key questions about cofinite modules in specific cases.
Abstract
Let be an ideal of a commutative noetherian (not necessarily local) ring . In the case , we show that the subcategory of -cofinite -modules is abelian. Using this and the technique of way-out functors, we show that if , or , or , then the local cohomology module is -cofinite for every -complex with finitely generated homology modules and every . We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.
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A New Outlook on Cofiniteness
Kamran Divaani-Aazar, Hossein Faridian and Massoud Tousi
K. Divaani-Aazar, Department of Mathematics, Alzahra University, Vanak, Post Code 19834, Tehran, Iran-and-School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.
H. Faridian, Department of Mathematics, Shahid Beheshti University, G.C., Evin, Tehran, Iran, Zip Code 1983963113.
M. Tousi, Department of Mathematics, Shahid Beheshti University, G.C., Evin, Tehran, Iran, P.O. Box 19395-5746.
Abstract.
Let be an ideal of a commutative noetherian (not necessarily local) ring . In the case , we show that the subcategory of -cofinite -modules is abelian. Using this and the technique of way-out functors, we show that if , or , or , then the local cohomology module is -cofinite for every -complex with finitely generated homology modules and every . We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.
Key words and phrases:
Abelian category; cofinite module; cohomological dimension; derived category; dualizing complex; local cohomology module; local homology module.
The research of the first author is supported by a grant from IPM (No. 95130212).
2010 Mathematics Subject Classification:
13D45; 13D07; 13D09.
1. Introduction
Throughout this paper, denotes a commutative noetherian ring with identity and flags the category of -modules.
In 1969, Hartshorne introduced the notion of cofiniteness for modules and complexes; see [Ha1]. He defined an -module to be -cofinite if and is finitely generated for every . Moreover, in the case where is an -adically complete regular ring of finite Krull dimension, he defined an -complex to be -cofinite if for some -complex with finitely generated homology modules. He then proceeded to pose three questions in this direction which we paraphrase as follows.
Question 1.1**.**
Is the local cohomology module , -cofinite for every finitely generated -module and every ?
Question 1.2**.**
Is the category consisting of -cofinite -modules an abelian subcategory of ?
Question 1.3**.**
Is it true that an -complex is -cofinite if and only if the homology module is -cofinite for every ?
By providing a counterexample, Hartshorne showed that the answers to these questions are negative in general; see [Ha1, Section 3]. However, he established affirmative answers to these questions in the case where is a principal ideal generated by a nonzerodivisor and is an -adically complete regular ring of finite Krull dimension, and also in the case where is a prime ideal with and is a complete regular local ring; see [Ha1, Propositions 6.1 and 6.2, Corollary 6.3, Theorem 7.5, Proposition 7.6 and Corollary 7.7]. Since then many papers are devoted to study his first two questions; see for example [HK], [DM], [Ka1], [Ka2], [Me1], [Me2] and [Y]. These results were extended in several stages to take the following form:
Theorem 1.4**.**
Let be an ideal of such that either , or , or . Then is -cofinite for every finitely generated -module and every , and is an abelian subcategory of .
For the case , refer to [Ka2, Theorem 1] and [Ka1, Theorem 2.1]. For the case , see [Me1, Theorem 2.6 and Corollary 2.12], [BNS, Corollary 2.8], and [BN, Corollary 2.7]. For the case , observe [Me2, Theorem 7.10] and [Me2, Theorem 7.4].
The significance of cofiniteness of the local cohomology modules mainly stems from the fact that if an -module is -cofinite, then its set of associated primes is finite as well as all its Bass numbers and Betti numbers with respect to every prime ideal of . It is worth mentioning that the investigation of such finiteness properties is a long-sought problem in commutative and homological algebra; see e.g. [HS] and [Ly].
In this paper, we deal with the above three questions. Theorems 2.2, 3.3, 3.5 and 4.3 are our main results.
In [PAB, Question 1], the authors asked: Is an abelian subcategory of for every ideal of with ? We answer this question affirmatively by deploying the theory of local homology; see Theorem 2.2. Note that there exists an inequality that can be strict; see Example 2.3.
It turns out that to establish the cofiniteness of for any -complex with finitely generated homology modules, all we need to know is the cofiniteness of for any finitely generated -module and the abelianness of ; see Theorem 3.3. The crucial step to achieve this is to recruit the technique of way-out functors.
To be consistent in both module and complex cases, we define an -complex to be -cofinite if and has finitely generated homology modules. Corollary 4.2 indicates that, for homologically bounded -complexes, this definition coincides with that of Hartshorne.
Questions 1.1 and 1.2 have been high-profile among researchers, whereas not much attention has been brought to Question 1.3. The most striking result on this question is [EK, Theorem 1] which confines itself to complete Gorenstein local domains and the case . We answer Hartshorne’s third question in the cases , , and with no extra assumptions on ; see Corollary 3.6 (ii). Having the results thus far obtained at our disposal, we show that the answers to Questions 1.1 and 1.2 are affirmative if and only if the answer to Question 1.3 is affirmative for all homologically bounded -complexes; see Theorem 4.3.
2. Question 1.2
We need to work in the framework of the derived category . For more information, refer to [AF], [Ha2], [Fo], [Li], and [Sp].
We let (res. ) denote the full subcategory of consisting of -complexes with for (res. ), and let . We further let denote the full subcategory of consisting of -complexes with finitely generated homology modules. We also feel free to use any combination of the subscripts and the superscript as in , with the obvious meaning of the intersection of the two subcategories involved.
Lemma 2.1**.**
Let be an ideal of and . Then the following conditions are equivalent:
- (i)
. 2. (ii)
.
Proof.
See [WW, Propositions 7.4]. ∎
In this section, we show that given an ideal of with , the subcategory of is abelian. This fact is proved in [PAB, Theorem 2.4], under the extra assumption that is local. Here we relax this assumption. The tool here is the local homology functors.
Recall that the local homology functors are the left derived functors of the completion functor. More precisely, for every , where for any -module . Further, we remind the cohomological dimension of with respect to as
[TABLE]
Theorem 2.2**.**
Let be an ideal of . Then the following assertions hold:
- (i)
An -module with is -cofinite if and only if is a finitely generated -module for every . 2. (ii)
If , then is an abelian subcategory of .
Proof.
(i): By [GM, Theorem 2.5 and Corollary 3.2], for every . Therefore, the assertion follows from Lemma 2.1.
(ii): Let and be two -cofinite -modules and an -homomorphism. The short exact sequence
[TABLE]
gives the exact sequence
[TABLE]
which in turn implies that is finitely generated -module since is so. The short exact sequence
[TABLE]
gives the exact sequence
[TABLE]
As , , and are finitely generated -modules, the exact sequence (2.2.3) shows that and are finitely generated -modules, and thus is -cofinite by (i). From the short exact sequence (2.2.2), we conclude that is -cofinite, and from the short exact sequence (2.2.1), we infer that is -cofinite. It follows that is an abelian subcategory of . ∎
It is well-known that . On the other hand, the following example shows that an ideal of with need not have . Hence Theorem 2.2 (ii) genuinely generalizes Theorem 1.4.
Example 2.3**.**
Let be a field and . Consider the elements , , and of . Let , and . Then is a noetherian local ring of dimension , , and . See [HeSt, Remark 2.1 (ii)].
3. Question 1.3
In this section, we exploit the technique of way-out functors as the main tool to depart from modules to complexes.
Definition 3.1**.**
Let and be two rings, and a covariant functor. We say that
- (i)
is way-out left if for every , there is an , such that for any -complex with , we have . 2. (ii)
is way-out right if for every , there is an , such that for any -complex with , we have . 3. (iii)
is way-out if it is both way-out left and way-out right.
The Way-out Lemma appears in [Ha2, Ch. I, Proposition 7.3]. However, we need a refined version which is tailored to our needs. Since the proof of the original result in [Ha2, Ch. I, Proposition 7.3] is left to the reader, we deem it appropriate to include a proof of our refined version for the convenience of the reader as well as bookkeeping.
Lemma 3.2**.**
Let and be two rings, and a triangulated covariant functor. Let be an additive subcategory of , and an abelian subcategory of which is closed under extensions. Suppose that for every and every . Then the following assertions hold:
- (i)
If with for every , then for every . 2. (ii)
If is way-out left and with for every , then for every . 3. (iii)
If is way-out right and with for every , then for every . 4. (iv)
If is way-out and with for every , then for every .
Proof.
(i): Let . Since , we argue by induction on . If , then . Therefore,
[TABLE]
as . Now, let and assume that the result holds for amplitude less than . Since , there is a distinguished triangle
[TABLE]
It is clear that the two -complexes and have all their homology modules in and their amplitudes are less than . Therefore, the induction hypothesis implies that and for every . Applying the functor to the distinguished triangle (3.2.1), we get the distinguished triangle
[TABLE]
which in turn yields the long exact homology sequence
[TABLE]
[TABLE]
We break the displayed part of the above exact sequence into the following exact sequences
[TABLE]
[TABLE]
[TABLE]
Since the subcategory is abelian, we conclude from the first and the third exact sequences above that . Since is closed under extensions, the second exact sequence above implies that for every .
(ii): Let . Since is way-out left, we can choose an integer corresponding to . Apply the functor to the distinguished triangle
[TABLE]
to get the distinguished triangle
[TABLE]
From the associated long exact homology sequence, we get
[TABLE]
where the vanishing is due to the choice of . Since with for every , it follows from (i) that for every , and as a consequence, for every .
(iii): Given , choose the integer corresponding to . The rest of the proof is similar to (ii) using the distinguished triangle
[TABLE]
(iv): Apply the functor to the distinguished triangle
[TABLE]
to get the distinguished triangle
[TABLE]
Since and with for every , we deduce from (ii) and (iii) that for every . Using the associated long exact homology sequence, an argument similar to (i) yields that for every . ∎
The next result provides us with a suitable transition device from modules to complexes when dealing with cofiniteness.
Theorem 3.3**.**
If is an ideal of , then the functor is triangulated and way-out. As a consequence, if is -cofinite for every finitely generated -module and every , and is an abelian category, then is -cofinite for every and every .
Proof.
By [Li, Corollary 3.1.4], the functor is triangulated and way-out. Now, let be the subcategory of finitely generated -modules, and let . It can be easily seen that is closed under extensions. It now follows from Lemma 3.2 that for every and every . ∎
Lemma 3.4**.**
Suppose that admits a dualizing complex , and is an ideal of . Further, suppose that is -cofinite for every and every . Let , and . Then is -cofinite for every .
Proof.
Set . Then clearly, . Let denote the Čech complex on a sequence of elements that generates . For any -complex , [Li, Proposition 3.1.2] yields that . Now, by applying the Tensor Evaluation Isomorphism, we get the following display:
[TABLE]
Hence for every , and so and the conclusion follows. ∎
The next result answers Hartshorne’s third question.
Theorem 3.5**.**
Let be an ideal of and . Then the following assertions hold:
- (i)
If is -cofinite for every , then is -cofinite. 2. (ii)
Assume that admits a dualizing complex , is contained in the Jacobson radical of , and is -cofinite for every and every . If is -cofinite in the sense of Hartshorne, then is -cofinite for all .
Proof.
(i) Suppose that is -cofinite for all . The spectral sequence
[TABLE]
from the proof of [Ha1, Proposition 6.2], together with the assumption that is finitely generated for every , conspire to imply that is finitely generated. On the other hand, one has
[TABLE]
Thus is -cofinite.
(ii) Suppose that is -cofinite in the sense of Hartshorne. Then by definition, there is such that . Now, the Affine Duality Theorem [Li, Theorem 4.3.1] implies that
[TABLE]
Since and , we conclude that . As the functor is faithfully flat, it turns out that . Now, the claim follows by Lemma 3.4. ∎
Corollary 3.6**.**
Let be an ideal of such that either , or , or . Then the following assertions hold:
- (i)
* is -cofinite for every and every .* 2. (ii)
Assume that admits a dualizing complex and is contained in the Jacobson radical of . If is -cofinite in the sense of Hartshorne, then is -cofinite for all .
Proof.
(i) Follows from Theorem 1.4, [Me2, Corollary 3.14], Theorem 2.2 (ii) and Theorem 3.3.
(ii) Follows by (i) and Theorem 3.5 (ii). ∎
4. Correlation between Questions 1.1, 1.2 and 1.3
In this section, we probe the connection between Hartshorne’s questions as highlighted in the Introduction.
Some special cases of the following result is more or less proved in [PSY, Theorem 3.10 and Proposition 3.13]. However, we include it here with a different and shorter proof due to its pivotal role in the theory of cofiniteness.
Lemma 4.1**.**
Let be an ideal of and . Then the following assertions are equivalent:
- (i)
. 2. (ii)
* for some .* 3. (iii)
* for some , provided that enjoys a dualizing complex .*
Proof.
(i) (ii): By Lemma 2.1, . Then by [AJL, Corollary after (0.3)∗], we have
[TABLE]
(ii) (iii): Set . If , then there is a semi-injective resolution of such that for every or . In particular, is bounded. On the other hand, , so there is a bounded -complex such that . Therefore,
[TABLE]
But it is obvious that is bounded, so .
Now, let denote the Čech complex on a sequence of elements that generates . We have
[TABLE]
The second isomorphism is due to the fact that is a dualizing -module, and the fifth isomorphism follows from the application of the Tensor Evaluation Isomorphism. The other isomorphisms are straightforward.
(iii) (i): Similar to the argument of the implication (ii) (iii), we conclude that . We further have
[TABLE]
The first and the fourth isomorphisms use [AJL, Corollary after (0.3)∗], the third isomorphism follows from the application of the Tensor Evaluation Isomorphism just as in the previous paragraph, and the fifth isomorphism follows from [PSY, Theorem 1.21], noting that as , its homology modules are -adically complete -modules. Now, the results follows from Lemma 2.1. ∎
Corollary 4.2**.**
Let be an ideal of for which is -adically complete and . Then the following assertions are equivalent:
- (i)
* is -cofinite.* 2. (ii)
* for some .* 3. (iii)
* for some , provided that enjoys a dualizing complex .*
Proof.
For any two -complexes and , one may easily see that
[TABLE]
Also, for any , [Li, Corollary 3.2.1] yields that if and only if . Hence the assertions follow from Lemma 4.1. ∎
The next result reveals the correlation between Hartshorne’s questions.
Theorem 4.3**.**
Let be an ideal of . Consider the following assertions:
- (i)
* is -cofinite for every finitely generated -module and every , and is an abelian subcategory of .* 2. (ii)
* is -cofinite for every and every .* 3. (iii)
An -complex is -cofinite if and only if is -cofinite for every .
Then the implications and hold. Furthermore, if is -adically complete, then all three assertions are equivalent.
Proof.
(i) (ii): Follows from Theorem 3.3.
(iii) (i): Let be a finitely generated -module. Since for every or , we have . However, [Li, Proposition 3.2.2] implies that
[TABLE]
showing that is -cofinite. The hypothesis now implies that is -cofinite for every .
Now, let and be two -cofinite -modules and an -homomorphism. Let be the morphism in represented by the roof diagram . From the long exact homology sequence associated to the distinguished triangle
[TABLE]
we deduce that . In addition, applying the functor to (4.3.1), gives the distinguished triangle
[TABLE]
whose associated long exact homology sequence shows that
[TABLE]
Hence, the -complex is -cofinite. However, we have
[TABLE]
so . Thus the hypothesis implies that is -cofinite for every . It follows that and are -cofinite, and as a consequence is an abelian subcategory of .
Now, suppose that is -adically complete.
(ii) (iii): Let . Suppose that is -cofinite for every . Then Theorem 3.5 (i) yields that is -cofinite.
Conversely, assume that is -cofinite. Then by Corollary 4.2, for some . Thus the hypothesis implies that
[TABLE]
is -cofinite for every . ∎
In view of Corollary 4.2, the next result answers Hartshorne’s third question for homologically bounded -complexes.
Corollary 4.4**.**
Let be an ideal of for which is -adically complete. Suppose that either , or , or . Then an -complex is -cofinite if and only if is -cofinite for every .
Proof.
Obvious in light of Corollary 3.6 and Theorem 4.3. ∎
**Acknowledgement **.
The authors are deeply grateful to Professor Robin Hartshorne for his invaluable comments on an earlier draft of this paper.
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