Approximable Triangulated Categories and Reflexive DG-categories

TL;DR

This paper explores the conditions under which proper DG-categories are reflexive, using approximable triangulated categories, and applies these results to various algebraic structures including schemes and DG-algebras.

Contribution

It introduces a new criterion for reflexivity of proper DG-categories based on approximable triangulated categories and describes their completions under properness assumptions.

Findings

Provided a new description of the completion of an approximable triangulated category.

Established conditions for reflexivity of proper DG-categories.

Applied results to proper schemes, DG-algebras, and Azumaya algebras.

Abstract

We use the theory of approximable triangulated categories to give a condition for a proper DG-category to be reflexive in the sense of Kuznetsov and Shinder. To do this we provide another description of the completion of an approximable triangulated category under a properness assumption. We apply our results to proper schemes, proper connective DG-algebras and Azumaya algebras over proper schemes. We include an appendix by Raedschelders and Stevenson showing that proper connective DG-algebras admit finite dimensional models over any field.