Upper bounds for finiteness of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $\lc^{i}_{\fa}(M,N)$ 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 $n \geqslant \pd M$ be an integer such that $\lc^{i}_{\fa}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If $\fb$ is an ideal of $R$ such that $\lc^{n}_{\fa}(M,N/{\fb}N)$ belongs to $\mathcal S$, It is also shown that the module $\lc^{n}_{\fa}(M,N)/{\fb}\lc^{n}_{\fa}(M,N)$ belongs to $\mathcal S$.