Separable extensions in tensor-triangular geometry and generalized Quillen stratification

TL;DR

This paper investigates how separable extensions affect spectra in tensor-triangular geometry, generalizing Quillen's Stratification Theorem to equivariant derived categories through a novel descent approach.

Contribution

It introduces a new analysis of spectra maps induced by separable extensions and extends Quillen's Stratification Theorem in a broader categorical context.

Findings

Determined the image of the spectra map induced by separable extensions.

Related the fiber cardinality to the extension degree.

Established a weak descent principle 'up-to-nilpotence'.

Abstract

We study the continuous map induced on spectra by a separable extension of tensor-triangulated categories. We determine the image of this map and relate the cardinality of its fibers to the degree of the extension. We then prove a weak form of descent, "up-to-nilpotence", which allows us to generalize Quillen's Stratification Theorem to equivariant derived categories.