Costratification and actions of tensor-triangulated categories

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, while laying the foundations for future applications. In addition, we introduce and study prime localizing submodules and prime colocalizing $\mathrm{h}\mathrm{o}\mathrm{m}$-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.