Thick tensor ideals of right bounded derived categories

Abstract

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