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Group actions on monoidal triangulated categories and Balmer spectra

Hongdi Huang, Kent B. Vashaw

Abstract

Let $G$ be a group acting on a left or right rigid monoidal triangulated category ${\mathbf K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of ${\mathbf K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of ${\mathbf K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of ${\mathbf K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of ${\mathbf K}$ by $G$ is also homeomorphic to the space of $G$-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory.

Group actions on monoidal triangulated categories and Balmer spectra

Abstract

Let be a group acting on a left or right rigid monoidal triangulated category which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of by is homeomorphic to the space of -prime ideals of , give a concrete description of this space, and classify the -invariant thick ideals of . Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of by is also homeomorphic to the space of -prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory.
Paper Structure (7 sections, 17 theorems, 78 equations)

This paper contains 7 sections, 17 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Monoidal triangular preliminaries
  3. Group action preliminaries
  4. $G$-ideals and $G$-prime ideals
  5. Ideals and primes in the crossed product category
  6. Ideals and primes in the equivariantization
  7. The core map and a description of $G\text{-}\mathop{\mathrm{Spc}}\nolimits$

Key Result

Theorem 1.1

(See Proposition pup-down-ideals, Proposition pequivariantization, Proposition ptopol-gspc, Proposition pideal-class-gspc). Let $G$ be a group acting on a monoidal triangulated category $\mathbf K$. Then

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Example 2.3
  • Example 2.4
  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • ...and 41 more