A new outlook on cofiniteness

Abstract

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