Annihilation of cohomology, generation of modules and finiteness of derived dimension

TL;DR

This paper establishes a connection between the annihilation of cohomology, module generation, and the finiteness of derived categories in commutative noetherian local rings, providing new characterizations of isolated singularities.

Contribution

It introduces criteria linking cohomology annihilators to module construction from syzygies and extends these results to derived and singularity categories, especially for Gorenstein rings.

Findings

Cohomology annihilator is m-primary iff certain syzygies are constructed from the residue field.

Rings with these properties are isolated singularities with finite derived and singularity category dimensions.

Modules free on the punctured spectrum are generated from finite length modules via extensions.

Abstract

Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and $d$ extensions. This result is exploited to investigate several ascent and descent problems between $R$ and its completion $\widehat R$.