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

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 contains a power of m if and only if, for some n≥ 0, the nth 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 Dsg(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 R. © 2016 Published by Oxford University Press. All rights reserved.