Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

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 |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 |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|⁠.