Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 7 (2017), 1677-1738.
Thick tensor ideals of right bounded derived categories
Let be a commutative noetherian ring. Denote by the derived category of cochain complexes of finitely generated -modules with for . Then has the structure of a tensor triangulated category with tensor product and unit object . In this paper, we study thick tensor ideals of , i.e., thick subcategories closed under the tensor action by each object in , and investigate the Balmer spectrum of , i.e., the set of prime thick tensor ideals of . First, we give a complete classification of the thick tensor ideals of 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 and the Zariski spectrum , and study their topological properties. After that, we compare several classes of thick tensor ideals of , relating them to specialization-closed subsets of and Thomason subsets of , and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of in the case where is a discrete valuation ring.
Algebra Number Theory, Volume 11, Number 7 (2017), 1677-1738.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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