Thick tensor ideals of right bounded derived categories

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 \(i\gg0\). Then \(D^-(R)\) has the structure of a tensor triangulated category with tensor product \(-\otimes_R^L-\) and unit obj...

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Bibliographic Details
Published in:arXiv.org
Main Authors: Matsui, Hiroki, Takahashi, Ryo
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 17.07.2017
Subjects:
Mathematical analysis
Tensors
ISSN:2331-8422
Online Access:Get full text
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Summary: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 \(i\gg0\). Then \(D^-(R)\) has the structure of a tensor triangulated category with tensor product \(-\otimes_R^L-\) 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.
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ISSN:2331-8422
DOI:10.48550/arxiv.1611.02826