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Vahidi A, Hassani F, Hoseinzade E. Extension Functors of Generalized Local Cohomology Modules. mmr 2021; 7 (1) :151-164
URL: http://mmr.khu.ac.ir/article-1-2857-en.html
URL: http://mmr.khu.ac.ir/article-1-2857-en.html
1- Payame Noor University , vahidi.ar@pnu.ac.ir
2- Payame Noor University
2- Payame Noor University
Abstract: (2273 Views)
Introduction
Throughout this paper,
is a commutative Noetherian ring with non-zero identity, 
is an ideal of , 
is a finitely generated -module, and 
is an arbitrary -module which is not necessarily finitely generated.
Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that
is finitely generated for all 
. In [12], Hartshorne gave a counter-example and raised the question whether 
is finitely generated for all 
and . The th generalized local cohomology module of and with respect to ,
was introduced by Herzog in [14]. It is clear that
is just the ordinary local cohomology module 
of with respect to . As a generalization of Hartshorne's question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]).
Question. When is
finitely generated for all and ?
In this paper, we study
in general for a finitely generated -module 
and an arbitrary -module .
Material and methods
The main tool used in the proofs of the main results of this paper is the spectral sequences.
Results and discussion
We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers
, 
, 
, and 
with 
, 
and the -modules 
and 
are in a Serre subcategory of the category of -modules (i.e. the class of -modules which is closed under taking submodules, quotients, and extensions).
Conclusion
We apply the main results of this paper to some Serre subcategories (e.g. the class of zero-modules and the class of finitely generated -modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an -module is said to be -cofinite (resp. minimax) if 
and 
is finitely generated for all [12] (resp. there is a finitely generated submodule 
of such that 
is Artinian [26]) where
. We show that if is finitely generated for all 
and 
is minimax for all 
, then is -cofinite for all and 
is finitely generated (Corollary 3.5). We prove that if is finitely generated for all 
, where 
is the arithmetic rank of , and is -cofinite for all 
, then 
is also an -cofinite -module (Corollary 3.6). We show that if is local, 
, and is finitely generated for all 
, then is -cofinite for all if and only if 
is finitely generated for all (Corollary 3.7). We also prove that if is local, , is finitely generated for all , and 
(or 
) is -cofinite for all , then is -cofinite for all (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules../files/site1/files/71/15.pdf
Throughout this paper,
is a commutative Noetherian ring with non-zero identity, is an ideal of , is a finitely generated -module, and is an arbitrary -module which is not necessarily finitely generated.Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that
is finitely generated for all . In [12], Hartshorne gave a counter-example and raised the question whether is finitely generated for all and . The th generalized local cohomology module of and with respect to ,was introduced by Herzog in [14]. It is clear that
is just the ordinary local cohomology module of with respect to . As a generalization of Hartshorne's question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]).Question. When is
finitely generated for all and ?In this paper, we study
in general for a finitely generated -module and an arbitrary -module .Material and methods
The main tool used in the proofs of the main results of this paper is the spectral sequences.
Results and discussion
We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers
, , , and with , and the -modules and are in a Serre subcategory of the category of -modules (i.e. the class of -modules which is closed under taking submodules, quotients, and extensions).Conclusion
We apply the main results of this paper to some Serre subcategories (e.g. the class of zero
-modules and the class of finitely generated -modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an -module is said to be -cofinite (resp. minimax) if and is finitely generated for all [12] (resp. there is a finitely generated submodule of such that is Artinian [26]) where. We show that if is finitely generated for all and is minimax for all , then is -cofinite for all and is finitely generated (Corollary 3.5). We prove that if is finitely generated for all , where is the arithmetic rank of , and is -cofinite for all , then is also an -cofinite -module (Corollary 3.6). We show that if is local, , and is finitely generated for all , then is -cofinite for all if and only if is finitely generated for all (Corollary 3.7). We also prove that if is local, , is finitely generated for all , and (or ) is -cofinite for all , then is -cofinite for all (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules../files/site1/files/71/15.pdf
Keywords: Associated prime ideals, Bass numbers, Cofinite modules, Extension functors, Generalized local cohomology modules, Injective dimensions, Minimax modules.
Type of Study: Original Manuscript |
Subject:
alg
Received: 2018/10/17 | Revised: 2021/05/29 | Accepted: 2019/08/5 | Published: 2021/05/31 | ePublished: 2021/05/31
Received: 2018/10/17 | Revised: 2021/05/29 | Accepted: 2019/08/5 | Published: 2021/05/31 | ePublished: 2021/05/31
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