# UPPER BOUNDS FOR FINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Document Type: Original Manuscript

Author

Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran.

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity and $a$ an ideal of $R$. Let $M$ be a finite $R$--module
of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $H^{i}_{a}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the category of $R$--modules and $n > pd M$ be an integer such that $H^{i}_{a}(M,N)$ belongs to $\mathcal S$ for all $i> n$. Then, for any ideal $b\supseteq a$, it is also shown that the module $H^{n}_{a}(M,N)/{b}H^{n}_{a}(M,N)$ belongs to $\mathcal S$.

Keywords