Annihilation of cohomology, generation of modules and finiteness of derived dimension
Let \((R,\mathfrak{m},k)\) be a commutative noetherian local ring and \(mod~R\) the category of all finitely generated \(R\)-modules. Let \(\mathrm{ca}(R)\) be the cohomology annihilator of \(R\). The authors study properties of \(\mathrm{ca}(R)\) and the connections to singularity theory. They prove that there is some integer \(n\geq0\) such that \(\mathfrak{m}^n\subseteq \mathrm{ca}\;(R)\) if and only if the \(n\)th syzygies in \(mod~R\) are constructed from syzygies of the residue field o \(R\) in a finite number of extensions, direct sums and direct summands. The authors also show that, if \(d\) is the Krull dimension of \(R\) and \(M\) is an \(R\)-module that is locally free on the punctured spectrum of \(R\), then \(M\) is build out of syzygies of finite length by taking \(d\) extensions in \(mod~R\) up to finite direct sums and direct summands.
scientific article; zbMATH DE number 6654443