Let
be a commutative noetherian ring. Denote by
the derived category of
cochain complexes
of
finitely generated
-modules
with
for .
Then
has the structure of a tensor triangulated category with tensor product
and unit object
. In this paper, we study
thick tensor ideals of
,
i.e., thick subcategories closed under the tensor action by each object in
, and investigate the
Balmer spectrum
of
, i.e., the set of prime
thick tensor ideals of
.
First, we give a complete classification of the thick tensor ideals of
generated
by bounded complexes, establishing a generalized version of the Hopkins–Neeman smash
nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum
and the Zariski
spectrum
,
and study their topological properties. After that, we compare several classes of thick tensor
ideals of
,
relating them to specialization-closed subsets of
and Thomason
subsets of
,
and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor
ideals of
in the
case where
is a discrete valuation ring.
Keywords
thick tensor ideal, Balmer spectrum, derived category,
specialization-closed subset, support