Algebra & Number Theory

Thick tensor ideals of right bounded derived categories

Hiroki Matsui and Ryo Takahashi

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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.

Article information

Source
Algebra Number Theory, Volume 11, Number 7 (2017), 1677-1738.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513096743

Digital Object Identifier
doi:10.2140/ant.2017.11.1677

Mathematical Reviews number (MathSciNet)
MR3697152

Zentralblatt MATH identifier
06775557

Subjects
Primary: 13D09: Derived categories
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18E30: Derived categories, triangulated categories 19D23: Symmetric monoidal categories [See also 18D10]

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

Citation

Matsui, Hiroki; Takahashi, Ryo. Thick tensor ideals of right bounded derived categories. Algebra Number Theory 11 (2017), no. 7, 1677--1738. doi:10.2140/ant.2017.11.1677. https://projecteuclid.org/euclid.ant/1513096743


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