Convexity in tensor triangular geometry

TL;DR

This paper classifies dualizable localizing ideals in certain tensor-triangulated categories, linking them to convex subsets of the Balmer spectrum, thereby extending previous results to broader algebraic and topological contexts.

Contribution

It provides a new classification of dualizable localizing ideals in cohomologically stratified tensor-triangulated categories, generalizing Efimov's theorem to more categories.

Findings

Dualizable localizing ideals correspond to convex subsets of the Balmer spectrum.

The classification applies to categories with noetherian Balmer spectrum.

Results include derived categories of noetherian schemes.

Abstract

We classify the dualizable localizing ideals of rigidly-compactly generated tt-$\infty$-categories that are cohomologically stratified. By definition, these are the localizing ideals that are dualizable with respect to the Lurie tensor product. We prove that these ideals correspond to the convex subsets of the Balmer spectrum. More generally, we establish this classification for categories which are locally cohomologically stratified and whose Balmer spectrum is noetherian. The classification thus applies to many categories arising in algebra and topology, including derived categories of noetherian schemes. Our result generalizes, and is motivated by, a recent theorem of Efimov which establishes this classification for derived categories of commutative noetherian rings.