Costratification and actions of tensor-triangulated categories

Abstract

We develop the theory of costratification in the setting of relative tensor-triangular geometry, in the sense of Stevenson, providing a unified approach to classification results of Neeman and Benson–Iyengar–Krause. In addition, we introduce and study prime localizing submodules and prime colocalizing $\mathrm{hom}$-submodules, in the first case, generalizing objectwise-prime localizing tensor-ideals. We apply our results to show that the derived category of quasi-coherent sheaves over a noetherian separated scheme is costratified. An application of the results proved in this paper, which appears in separate work, is that the singularity category of a locally hypersurface ring (more generally a noetherian separated scheme with hypersurface singularities) is costratified.