Thick tensor ideals of right bounded derived categories

Matsui, Hiroki; Takahashi, Ryo

doi:10.2140/ant.2017.11.1677

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Thick tensor ideals of right bounded derived categories

Hiroki Matsui and Ryo Takahashi

Vol. 11 (2017), No. 7, 1677–1738

DOI: 10.2140/ant.2017.11.1677

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 $i ≫ 0$ . Then $D^{-} (R)$ 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 $D^{-} (R)$ , 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.

Keywords

thick tensor ideal, Balmer spectrum, derived category, specialization-closed subset, support

Mathematical Subject Classification 2010

Primary: 13D09

Secondary: 18D10, 18E30, 19D23

Milestones

Received: 15 April 2017

Revised: 9 June 2017

Accepted: 16 July 2017

Published: 7 September 2017

Authors

Hiroki Matsui
Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya Japan

Ryo Takahashi
Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya Japan

Vol. 11, No. 7, 2017