A New Outlook on Cofiniteness

Abstract

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$, or $\dim (R/\mathfrak{a})\le 1$, or $\dim (R)\le 2$, then the local cohomology module $H_{\mathfrak{a}}^i(X)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i\in \mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.

arXiv commentIt will appear in Kyoto Journal of Mathematics