Group actions on monoidal triangulated categories and Balmer spectra

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.