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2017 Thick tensor ideals of right bounded derived categories
Hiroki Matsui, Ryo Takahashi
Algebra Number Theory 11(7): 1677-1738 (2017). 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 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 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 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.

Citation

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Hiroki Matsui. Ryo Takahashi. "Thick tensor ideals of right bounded derived categories." Algebra Number Theory 11 (7) 1677 - 1738, 2017. https://doi.org/10.2140/ant.2017.11.1677

Information

Received: 15 April 2017; Revised: 9 June 2017; Accepted: 16 July 2017; Published: 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06775557
MathSciNet: MR3697152
Digital Object Identifier: 10.2140/ant.2017.11.1677

Subjects:
Primary: 13D09
Secondary: 18D10, 18E30, 19D23

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.11 • No. 7 • 2017
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