پژوهشهای ریاضی (May 2021)
Extension Functors of Generalized Local Cohomology Modules
Abstract
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
