Finiteness dimensions and cofiniteness of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ and ideal of $R$, $M$ a finite $R$--module, and $n$ a non-negative integer. In this paper, for an arbitrary $R$--module $X$ which is not necessarily finite, we study the finiteness dimension $f_{\mathfrak{a}}(M,X)$ and the $n$-th finiteness dimension $f_{\mathfrak{a}}^n(M,X)$ of $M$ and $X$ with respect to $\mathfrak{a}$. Assume that ${\mathrm{Ext}}_R^i(R/\mathfrak{a},X)$ is finite for all $i\le f_{\mathfrak{a}}^2(M,X)$ (resp. $i<f_{\mathfrak{a}}^1(M,X)$). We show that $H_{\mathfrak{a}}^i(M,X)$ is $\mathfrak{a}$--cofinite for all $i<f_{\mathfrak{a}}^2(M,X)$ (resp. $i<f_{\mathfrak{a}}^1(M,X)$) and ${\mathrm{Ass}}_R(H_{\mathfrak{a}}^{f_{\mathfrak{a}}^2(M,X)}(M,X))$ (resp. if ${\mathrm{Ext}}_R^{f_{\mathfrak{a}}^1(M,X)}(R/\mathfrak{a},X)$ is finite, then ${\mathrm{Ass}}_R(H_{\mathfrak{a}}^{f_{\mathfrak{a}}^1(M,X)}(M,X))$) is finite.

arXiv commentThe proof of Theorem 4.6 holds a flaw