Vol. 11, No. 7, 2017

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Thick tensor ideals of right bounded derived categories

Hiroki Matsui and Ryo Takahashi

Vol. 11 (2017), No. 7, 1677–1738

Let R be a commutative noetherian ring. Denote by D(R) the derived category of cochain complexes X of finitely generated R-modules with Hi(X) = 0 for i 0. Then D(R) has the structure of a tensor triangulated category with tensor product RL 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 SpcD(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 SpcD(R) and the Zariski spectrum SpecR, 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 SpecR and Thomason subsets of SpcD(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.

thick tensor ideal, Balmer spectrum, derived category, specialization-closed subset, support
Mathematical Subject Classification 2010
Primary: 13D09
Secondary: 18D10, 18E30, 19D23
Received: 15 April 2017
Revised: 9 June 2017
Accepted: 16 July 2017
Published: 7 September 2017
Hiroki Matsui
Graduate School of Mathematics
Nagoya University
Ryo Takahashi
Graduate School of Mathematics
Nagoya University