  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 $\widehat
R$.
