Geometricity for derived categories of algebraic stacks

TL;DR

This paper proves that the dg category of perfect complexes on certain algebraic stacks is geometric, implying it can be embedded into the derived category of a smooth projective variety, thus linking stacks to classical geometry.

Contribution

It establishes the geometricity of the dg category of perfect complexes for smooth, proper Deligne-Mumford stacks and tame Artin stacks, extending Orlov's framework.

Findings

The dg category of perfect complexes is geometric for smooth, proper Deligne-Mumford stacks.

The derived category embeds as an admissible subcategory into a smooth, projective variety.

The results hold over fields of characteristic zero and arbitrary fields for tame Artin stacks.

Abstract

We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.