Abstract
Let denote a commutative Noetherian ring, be an ideal of , be an arbitrary dense subcategory of -modules, and be the full subcategory of finite -modules. In this paper, we study --cofinite modules with respect to the extension subcategory when . We also study -cofiniteness with respect to a new dimension.
1. Introduction
Throughout this paper, let denote a commutative Noetherian ring, be an ideal of , and be a non-negative integer. A full subcategory of -modules is called dense (Serre) provided that for every exact sequence of modules, if and only if and are in . Given a dense subcategory of -modules, an -module is called --cofinite, abbreviated to --cof, if and for all integers . Let be the class of finite -modules.
We indicate by , the extension subcategory of and , consisting of all modules fitting into an exact sequence such that and . It has been proved by Quy [1] that is dense. A well-known example for these subcategories is , the class of minimax modules [2] where is the class of Artinian modules. Another example is , the class of FSF modules introduced in [3], where consists of all modules of finite support. When , an --cof module is called -cof which was introduced for the first time by Hartshorne [4], giving a negative answer to a question of [5, Expos XIII, Conjecture 1.1]. Several authors [6–12] have investigated -cofiniteness in various aspects.
The main aim of this paper is to develop the fundamental results about -cof modules at small dimensions to --cof modules. We recall that a class admits the condition if for every -module , we have the following implication.
For every , we indicate by , the smallest dense subcategory of -modules containing . For every -module , means the dimension of which is the length of the longest chain of prime ideals in . In Section 2, we prove the following result.
Theorem 1. Let be a local ring such that admits for every and let for every where . Then, for all and any finite -module with and .
Suppose that is an -module with and admits for all . Melkersson [10, Theorem 2.3] showed that if , then is -cof if and only if and are finite. As a conclusion of the above theorem, we deduce that if is a local ring with , then is --cof if and only if and . Furthermore, if is a local ring of dimension 2 such that admits for every prime ideal of with , then we show that is --cof if and only if and .
Bahmanpour et al. [8, Theorem 3.5] proved that if is a local ring such that , then is -cof if and only if are finite for . As the other conclusion of the above theorem, we prove that if , then is --cof if and only if for . Furthermore, if is a local ring of dimension 3 such that admits for every prime ideal with , then we show that is --cof if and only if for . We prove the following result about local cohomology which generalizes [12, Theorem 3.7]. For the basic properties of local cohomology, the reader can study the textbook by Brodmann and Sharp [13].
Theorem 2. Let be a local ring, let be of dimension two, and let admit for any . for all ; then, the following statements are equivalent.(i) is --cof for all .(ii) for all .
In Section 3, we define a new dimension of modules which is an upper bound of the dimension mentioned in Section 2. For every -module , we define . We indicate by , the class of all -modules admitting . It is proved that is dense for every . We show that the class of -cof modules in is dense and the full subcategory of -cof modules in is abelian. Finally, we show that the kernel and cokernel of a homomorphism of -cof modules in are -cof if and only if is finite. Throughout this paper, the “exact sequence” is abbreviated to e.s.
2. Cofiniteness with respect to Extension Subcategories
Assume that is an abelian category and is a full subcategory of . The smallest full subcategory of containing and closed under taking subobjects is indicated by . This subcategory can be identified as follows:
The full subcategory of is defined dually as follows:
Given another full subcategory of , the extension subcategory of and is indicated by , that is:
For every , put and set . The subcategory is said to be closed under taking extensions whenever . The smallest subcategory of containing and closed under taking extensions is indicated by , that is, ,
The subcategory of is said to be dense (or Serre) provided that for every e.s of modules, if and only if and are in . The smallest dense subcategory of containing is indicated by .
Lemma 3. Let be a subcategory of . Then, we have the following conditions.(i).(ii).
Proof. (i) See [14, Proposition 2.4]. (ii) Given , there is a non-negative integer with . We prove by induction on that there is a filtration of subobjects of of length with for every . If , there is nothing to prove. For , since , there is an e.s with and . By the inductive hypothesis, there is a finite filtration with for every . Consequently, is the desired filtration.
We indicate by -Mod, the category of -modules. In the rest of this paper, is an ideal of , is a non-zero dense subcategory of -Mod, and is the full subcategory of finite modules in its corresponding module category (such as in the category of -modules or in the category of -modules for some prime ideal of ).
Lemma 4. The extension subcategory is dense for every dense subcategory of -Mod. Moreover, .
Proof. (see [1, Corollary 3.3 and Theorem 3.2]).
An -module is said to be --cof if and for all integers . An --cof module is called -cof. We recall that admits provided that for every -module , the following implication holds.
Lemma 5. The dense subcategory of -Mod admits if and only if it admits .
Proof. (see [15, Proposition 2.4]).
Lemma 6. Let be a finite -module and let be an arbitrary -module such that for a non-negative integer , we have for every . Then, for any finite -module with and every .
In this section, for every -module , is the largest non-negative integer for which there is a chain of prime ideals in .
Proposition 7. Assume that for all where and (e.g., if is a local ring). Then,(i) is --cof for all and all ideals containing with and admits . Moreover, .(ii) for all and all finite -modules with with and admits the condition .
Proof. (i) Assume that is an ideal of containing with . Then, for all by Lemma 6. If , there is an e.sin which has finite length and . Since , we deduce that so that and since admits , we have , and hence is --cof (we observe that for all ). Assume that and we proceed by induction on . If , for all . Using a similar argument as mentioned above, for all ; and hence by [6, Theorem 2.9], we have for all , and consequently, the assertion is clear in this case. If , by [16, Theorem 3.5], the module is --cof for all and (by the vanishing theorem of local cohomology, for every ). Since , using a similar argument as mentioned in the case , we deduce that and since admits , we deduce that . (ii) By Lemma 6, we have for all if and only if for all , and hence we may assume that . Since , we have , and hence (i) implies that is --cof for all . Then, using [16, Theorem 3.5], the module for all . Consequently, Lemmas 4 and 6 imply that for all .
For every dense subcategory of -Mod and every , we indicate by , the smallest dense subcategory of -Mod containing . We have the following lemma.
Lemma 8. Let be an ideal of . Then, is closed under taking subobjects and quotients. In particular, .
Proof. Given and a submodule of , there is a submodule of with . Since , we have and which yields . The second assertion follows from Lemma 3.
By virtue of Lemma 8, if is a local ring, then we have .
Lemma 9. Let be a prime ideal of and . Then, there is an -module such that is an essential -submodule of and .
Proof. Since , there is with . If is the canonical homomorphism, then is the desired module.
Theorem 10. Let be a local ring and let admit for every . If with for every , then for every and every finite -module with and .
Proof. If , then is -primary. By the assumption and Lemma 5, the class admits , and hence Proposition 7 implies that for every . Now, assume that , and for every and every . If we set , then for every and we have . It thus follows from Lemma 6 that for every , and hence we may assume that for some with . It is trivial that for every , and hence by Proposition 7, we have for every . Assume that and . We show that . In view of the previous argument, there is a finite submodule of with . Consider the canonical morphism with where is a submodule of containing . Clearly, , and since and , we have . Considering , there is an e.swhich yields the following e.s:We observe that is -primary and according to the assumption and Lemma 5, admits , and hence Proposition 7 implies that . Thus, . Since , using Lemma 3, there is a natural number such that . Without loss of generality, we may assume that , and the other cases are similar. Then, there is an e.s of -modules.with . Using Lemma 9, we may assume that and are -submodules of and , respectively. Since is an essential -submodule of , the -submodule is non-zero and and replacing by , we may assume that is a submodule of . Suppose that for some submodule of containing , and we have . Applying the functor to the e.s and the fact that , we deduce that . Then, there is an e.sin which is finite and . Since , the module has finite length and hence as . This implies that and hence . Since admits , we deduce that so that and consequently . Consider the canonical homomorphism with . Thus, so that . From the induced essential monomorphism and using a similar argument as mentioned above, we may assume that is a submodule of and so there is a submodule of containing with and . Therefore, we have . The e.s , and Lemma 4 and the fact that imply that , and so using a similar argument as mentioned above, we deduce that so that , and hence . Now applying to the e.s and using again a similar argument as mentioned before, we deduce that .
The following corollary develops a result due to Melkersson [11, Theorem 2.3].
Corollary 11. Let be a local ring with and let admit for every . If is an -module with and , then is --cof.
Proof. The assertion follows immediately from Theorem 10.
Theorem 12. Let be a local ring and let admit for every . If and for every , then for every and any finite -module with and .
Proof. In view of Theorem 10, we may assume that and similar to the proof of Theorem 10, we may assume that for some with . According to the assumption, for every . Assume that and and we show that . There is finite submodule of with . Consider the canonical morphism with where is a submodule of . Clearly, and since and , every with is a minimal prime ideal of . Then, the setis finite. Assume that . Then, , and so . Then, there is an e.swhich yields the following e.s:Since , by Theorem 10, we have , and hence using Lemma 4. Now assume that where . Then, we have . Continuing this way, assume that where for , and we deduce that for every , , and we have a chain of submodulessuch that the induced morphism is an essential monomorphism for every . We observe that for every . Since , we have . Now applying the functor to the e.s , we deduce that and so there is an e.s of -modulesin which is finite and . Using the same argument as mentioned in the proof of Theorem 10, we deduce that so that . Since , , so that and a similar proof as mentioned above gives . To be convenient, set . In view of Lemma 8, we may assume that , and the other cases are similar. Then, there is an e.sUsing Lemma 9, we may assume that and are -submodules of and , respectively. Since is an essential submodule of , the submodule is non-zero and , and hence replacing by , we may assume that is a submodule of . Assume that for some submodule of and so . Thus, . Consider the canonical e.s which forces . Since and , using a similar argument as above, we deduce that . The essential monomorphism and a similar argument as above imply that is a submodule of and so for some submodule of . The fact that implies that . Since , we deduce that . A similar argument as above implies that , and hence . Continuing this manner, we deduce that . Therefore, we may assume that is an essential monomorphism. According to Lemma 8, there is a natural number with . Set . Without loss of generality, we may assume that and so by a similar argument as mentioned above, there is an e.s of -modules such is a submodule of and is an -submodule of and so , and hence . Since and , every with is a minimal prime ideal of , and hence the setis finite. Assume that and set . We notice that . Using Proposition 7, for every . Then, for every , there is an e.swhere is a finite -module and . In view of the preceding arguments, we may assume that so that there is with . Furthermore, we may assume that is a submodule of and . Then, there is a submodule of such that and so this implies that for every . Now, set and so clearly and . We notice that there is a submodule of with and so . On the other hand, there is a submodule of with which implies that and . Moreover, since is a submodule of , there is a submodule of containing with and hence . We observe by Lemma 4 that . Considering the following e.s:we have and using a similar proof as mentioned before, we deduce that . Now the fact that forces .
The following corollary develops [8, Theorem 3.5].
Corollary 13. Let be a local ring with and let admit for every . If and for all , then is --cof.
Proof. The assertion follows immediately from Theorem 12.
The class of all -modules of finite support is indicated by . It is clear that is dense and it admits for every ideal of .
Corollary 14. Let be a local ring with and let . If for all , then is --cof.
Proof. We claim that admits for every prime ideal of . If for some , there is nothing to prove. If , there is a non-zero -module such that . Since and , we deduce that . If , then is maximal, and the result is clear. If , we claim that -Mod , and hence admits clearly. The inclusion follows by Lemma 3. For the other inclusion, if is a non-zero -module with , then for any non-zero element , there is some natural number with so that . This forces that so that is a finite set and consequently . Therefore, . Now, the assertion follows immediately from Theorem 12.
The following result about local cohomology generalizes [12, Theorem 3.7].
Theorem 15. Let be a local ring with and let admit for any . If for all ; then, the following conditions are equivalent.(i) is --cof for all .(ii) for all .
Proof. We proceed by induction on . The case is clear. For , the modules fit into an e.s in which is injective with . This e.s implies that and for every ; furthermore, . Moreover, for every , we have an e.s of -modules.If , the e.s and Corollary 13 imply that (i) and (ii) are equivalent. Let . Since , in view of and the previous isomorphisms and force for all ; consequently (i) and (ii) are equivalent for and the non-negative integer . Now, using again the previous isomorphisms, the conditions (i) and (ii) are equivalent for and non-negative integer .
Corollary 16. Let be a local ring with and let for all . Then, the following conditions are equivalent.(i) is --cof for all .(ii) for all .
Proof. Since admits for every prime ideal of , the result is obtained by Theorem 15.
Given an arbitrary class of -modules, admits provided that the following implication holds for any -module :
Theorem 17. Let be a ring of dimension admitting for all ideals of dimension (for example, if ). Then, admits for every ideal of .
Proof. Suppose that is an ideal of and is a module with and for all . If there is some natural number with , then . For every , there exist a natural number and the e.s of modules and . Since is dense, applying to e.s and using an easy induction on , we deduce that . Since , the module is -cof. Now, assume that is not nilpotent. Since is Noetherian, for some natural number . Putting and , it is trivial that is a -module and since , contains a -regular element so that . We observe that and by Lemma 6, we have for all . Thus, the assumption implies that is --cof. We now show that for every . Consider the Grothendieck spectral sequenceFor , we have . For , we have for all . Since , by Lemma 6, for all and all . The e.s and the inductive hypothesis imply that if . Continuing this manner, we deduce that if . But there is the following filtration.such that is a submodule of , and hence it is in . Therefore, for all . Now consider the Grothendieck spectral sequenceUsing again Lemma 6, we deduce for all . For any , the -module is a subquotient of and so an easy induction yields that for all so that for all . For any , there is a finite filtration.such that where . Since for all and , we deduce that for all , and hence is --cof. On the other hand, since , we conclude that is --cof.
Corollary 18. Let be a local ring of dimension 2 such that admits for every ideal of with . Then, admits for every ideal of .
Proof. By virtue of [16, Theorem 3.2] and Corollary 11, the ring admits for all ideals with . Now, the assertion is obtained using Theorem 17.
Corollary 19. Let be a local ring of dimension 2 such that admits for every prime ideal with . Then, admits for every ideal of .
Proof. By virtue of Corollary 11, the ring admits for all ideals with . Now, the results follow from Theorem 17.
Corollary 20. Let be a local ring of dimension 3 such that admits for every prime ideal with . Then, admits for every ideal of .
Proof. According to Corollary 13, the ring admits for all ideals with . Now, the claim is obtained using Theorem 17.
Corollary 21. Let be a local ring of dimension 3. Then, admits .
Proof. According to Corollary 14, the ring admits for all ideals with . Now, the result follows from Theorem 17.
3. Cofiniteness with respect to a New Dimension
For every -module , it is trivial that and for the case where is finite, they are equal. We define the upper dimension of and we indicate it by which is . Clearly, . We notice that these two dimensions are not equal in general when is not finite. To be more precise, consider a local ring of dimension and an ideal of of positive height. Then, . If we consider , then , and hence , while .
We first recall some lemmas which are needed in this section.
Lemma 22. Let be a finite -algebra and let be an -module. Then, is -cof if and only if is -cof (as an -module).
Proof. (see [17, Proposition 2]).
Lemma 23. Let be a finite -algebra and let be an -module. Then, admits if and only if admits .
Proof. (see [18, Proposition 2.15]).
Lemma 24. Assume that . Then, is Artinian and -cof if and only if has finite length.
Proof. (see [10, Proposition 4.1]).
For every non-negative integer , we indicate by , the class of all -modules with . We also indicate by , the class of all -modules where is a finite set.
Lemma 25. If is an e.s of -modules, then .
Proof. Since , we conclude that . Now assume that and . Then, there is and so for every , we have which implies that . Consequently, .
Corollary 26. The following conditions hold.(i)The class is dense.(ii)The class is dense.
Proof. The proof is straightforward by Lemma 25.
Proposition 27. Assume that . Then, is -cof if and only if and are finite.
Proof. Let . Using [18, Proposition 2.15], we may assume that . Now, the result follows from [12, Corollary 2.4].
Proposition 28. Assume that . Then, is -cof if and only if is finite for .
Proof. Let . Using [18, Proposition 2.15], we may assume that . Now, the result follows from [12, Corollary 2.5].
Proposition 29. The class of -cof modules in is dense.
Proof. If , then and so the module has finite length. Thus, by Lemma 24, the module is -cof. Now, assume that and so where . It is trivial that is finite and . By [10, Proposition 4.5], the module is -cof. Finally, using Lemma 22, is -cof. The second assertion is straightforward.
Corollary 30. Let with . Then, is -cof if and only if is finite.
Proof. Since , there is an e.s of -modules in which is finite and has finite support. We notice that is finite, and it suffices to show that is -cof and so we may assume that is a finite set so that . By [10, Proposition 4.5], the module is -cof where and . Now Lemma 22 implies that is -cof.
Proposition 31. The class of -cof modules in is abelian.
Proof. Assume that is a morphism of -cof modules and assume that and . The assumption implies that is finite -module where . If , the module has finite length and so is Artinian. Now, Lemma 24 implies that is -cof. If , by Proposition 29, is -cof as is -cof, and hence is -cof using Lemma 22. If , using [12, Corollary 2.6], the module is -cof and so is -cof by using Lemma 22. Now, using the e.s of -modulesit is straightforward to show that and are -cof modules.
Proposition 32. The kernel and cokernel of a homomorphism of -cof modules in are -cof if and only if is finite.
Proof. By the assumption, we have and also . If we put and , we have and further and are -module. Clearly, is a finite -module. It now follows from [12, Theorem 10] that and are -cof and so using Lemma 22, they are -cof.
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Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.