Thick tensor ideals of right bounded derived categories

TL;DR

This paper classifies thick tensor ideals in the derived category of a noetherian ring, explores their spectral properties, and relates them to the ring's prime spectrum, extending classical theorems and providing counterexamples.

Contribution

It provides a complete classification of thick tensor ideals generated by bounded complexes and investigates the structure of the Balmer spectrum for $D^-(R)$.

Findings

Classified thick tensor ideals generated by bounded complexes.

Established a generalized Hopkins-Neeman theorem.

Constructed a counterexample to Balmer's conjecture.

Abstract

Let $R$ be a commutative noetherian ring. Denote by $D^-(R)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with $H^i(X)=0$ for $i\gg0$. Then $D^-(R)$ has the structure of a tensor triangulated category with tensor product $-\otimes_R^L-$ and unit object $R$. In this paper, we study thick tensor ideals of $D^-(R)$, i.e., thick subcategories closed under the tensor action by each object in $D^-(R)$, and investigate the Balmer spectrum $Spc\,D^-(R)$ of $D^-(R)$, i.e., the set of prime thick tensor ideals of $D^-(R)$. First, we give a complete classification of the thick tensor ideals of $D^-(R)$ 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 $Spc\,D^-(R)$ and the Zariski spectrum $Spec\,R$, and study their topological properties. After that, we compare several classes of thick tensor ideals of $D^-(R)$, relating them to specialization-closed subsets of $Spec\,R$ and Thomason subsets of $Spc\,D^-(R)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of $D^-(R)$ in the case where $R$ is a discrete valuation ring.