Weakly approximable triangulated categories and enhancements: a survey

TL;DR

This survey explores recent advances in the intrinsic properties and enhancement uniqueness of weakly approximable triangulated categories, highlighting their implications for derived invariance in algebraic geometry.

Contribution

It provides a comprehensive review of the intrinsicness and enhancement uniqueness in weakly approximable triangulated categories, extending Rickard's results.

Findings

Intrinsic subcategories of weakly approximable categories are well-understood.

Enhancement uniqueness results are generalized to broader classes of categories.

Derived invariance results are extended to schemes and rings.

Abstract

This paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the aim of showing the fruitful interplay. In particular, we show how this leads to a vast generalization of a result by Rickard about derived invariance for schemes and rings.