Gluing Approximable Triangulated Categories

TL;DR

This paper explores the concept of approximability in triangulated categories, demonstrating how it can be preserved under certain constructions and applying it to categories in noncommutative algebraic geometry, thus broadening its usefulness.

Contribution

It proves that the recollement of two approximable triangulated categories remains approximable under weak conditions, extending the applicability of approximability in noncommutative geometry.

Findings

Recollement preserves approximability under weak hypotheses

Many categories in noncommutative algebraic geometry are approximable

Develops tools with broad applications in triangulated category theory

Abstract

Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which has interesting applications in existing and forthcoming work by the authors.