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.