Costratification and actions of tensor-triangulated categories

TL;DR

This paper develops the theory of costratification in tensor-triangulated categories, unifying existing classification results and establishing new foundational concepts with applications to derived categories of schemes.

Contribution

It introduces the concept of costratification in relative tensor-triangular geometry and generalizes prime localizing submodules, providing a unified framework for classification.

Findings

Derived category of quasi-coherent sheaves over a noetherian separated scheme is costratified.

Unified approach to classification results of Neeman and Benson--Iyengar--Krause.

Introduced prime localizing and colocalizing hom-submodules.

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, while laying the foundations for future applications. 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.