ANNIHILATION OF COHOMOLOGY, GENERATION OF MODULES AND FINITENESS OF DERIVED DIMENSION

Let
$(R,\mathfrak {m},k)$
be a commutative noetherian local ring of Krull dimension
$d$
. We prove that the cohomology annihilator
$\mathsf {ca}\,(R)$
of
$R$
contains a power of
$\mathfrak {m}$
if and only if, for some
$n\ge 0,$
the
$n$
th syzygies in
$\mathsf {mod}\,R$
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
$\mathsf {D^b}\,(R)$
and the singularity category
$\mathsf {D_{sg}}\,(R)$
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
$\hat R$
.