Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules

Abstract

Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ and $N$ two finitely generated $R$-modules, and $X$ an arbitrary $R$-module. In this paper, we study cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules. In some cases, we show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $(\operatorname{FD}_{<n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,X)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. If $R$ is semi-local, we observe that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))$ is finite for all $i$ when $\dim_R(M)\leq{3}$ or $\dim_R(N)\leq{3}$. Also, in some situations, we prove that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $\mathfrak{a}$-cofinite $R$-module for all $i$.