Approximable triangulated categories

TL;DR

This survey introduces the concept of approximable triangulated categories, demonstrating their naturalness and utility in deriving new results and insights about derived categories in algebraic geometry.

Contribution

It presents the theory of approximable triangulated categories and applies it to improve classical theorems in algebraic geometry.

Findings

$D_{qc}(X)$ is approximable for quasicompact separated schemes

New proofs and improvements of classical theorems by Bondal, Rickard, Rouquier, and Van den Bergh

Enhanced understanding of $D^{perf}(X)$ and $D^b_{coh}(X)$ categories

Abstract

In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects. In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is approximable whenever $X$ is a quasicompact separated scheme. As corollaries of this (seemingly technical) statement one can prove striking improvements on old theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much smaller) categories $D^{\text{perf}}(X)$ and $D^b_{\text{coh}}(X)$.