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 $H_{\mathfrak{a}}^i(M,X)$ is an $({\mathrm{FD}}_{<n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in {\mathrm{Ass}}_R(H_{\mathfrak{a}}^i(M,X)):\dim (R/\mathfrak{p})\ge n\}$ is a finite set for all $i$. If $R$ is semi-local, we observe that ${\mathrm{Ass}}_R(H_{\mathfrak{a}}^i(M,N))$ is finite for all $i$ when ${\dim }_R(M)\le 3$ or ${\dim }_R(N)\le 3$. Also, in some situations, we prove that $H_{\mathfrak{a}}^i(M,X)$ is an $\mathfrak{a}$-cofinite $R$-module for all $i$.