Group actions on monoidal triangulated categories and Balmer spectra

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.

arXiv commentAssumption that the category has a thick generator has been removed throughout, section on the equivariantization has been added, examples have been added, comments from experts in the field have been addressed. 24 pages. Comments welcome!