Hartshorne's question on cofinite complexes

TL;DR

This paper addresses Hartshorne's question on the cofiniteness of complexes over Noetherian rings, providing complete answers in specific dimension cases and establishing criteria for cofiniteness related to homology modules and Ext groups.

Contribution

It offers a comprehensive resolution to Hartshorne's question on cofinite complexes in certain dimension scenarios and develops new criteria involving Ext groups and spectral sequences.

Findings

Complete characterization of $rak{a}$-cofiniteness for complexes when $ ext{dim} R=d$ or related conditions.

Equivalence of cofiniteness of complexes and their homology modules for $d extless 3$.

Finiteness conditions for Ext groups determining cofiniteness when $d extgreater 2$.

Abstract

Let $\mathfrak{a}$ be a proper ideal of a commutative noetherian ring $R$ and $d$ a positive integer. We answer Hartshorne's question on cofinite complexes completely in the cases $\mathrm{dim}R=d$ or $\mathrm{dim}R/\mathfrak{a}=d-1$ or $\mathrm{ara}(\mathfrak{a})=d-1$, show that if $d\leq2$ then an $R$-complex $X\in\mathrm{D}_\sqsubset(R)$ is $\mathfrak{a}$-cofinite if and only if each homology module $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite; if $\mathfrak{a}$ is a perfect ideal and $R$ is regular local with $d\leq2$ then an $R$-complex $X\in\mathrm{D}(R)$ is $\mathfrak{a}$-cofinite if and only if $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite for every $i\in\mathbb{Z}$; if $d\geq3$ then for an $R$-complex $X$ of $\mathfrak{a}$-cofinite $R$-modules, each $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite if and only if $\mathrm{Ext}^j_R(R/\mathfrak{a},\mathrm{coker}d_i)$ are finitely generated for $j\leq d-2$. We also study cofiniteness of local cohomology $\mathrm{H}^i_\mathfrak{a}(X)$ for an $R$-complex $X\in\mathrm{D}_\sqsubset(R)$ in the above cases. The crucial step to achieve these is to recruit the technique of spectral sequences.