Extension Functors of Generalized Local Cohomology Modules

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 Hartshornechr('39')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 .

The main tool used in the proofs of the main results of this paper is the spectral sequences.

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).

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