Equivalences of derived categories of sheaves on tame stacks

TL;DR

This paper extends Orlov's representability theorem to smooth, proper, and tame algebraic stacks, establishing equivalences of derived categories of sheaves and generalizing prior results for Deligne--Mumford stacks.

Contribution

It proves Orlov's theorem for a broader class of algebraic stacks, including tame stacks, expanding the understanding of derived category equivalences.

Findings

Proves Orlov's representability theorem for tame stacks.

Establishes Fourier--Mukai transform equivalences for these stacks.

Generalizes previous results for Deligne--Mumford stacks.

Abstract

Building on Olander's work on algebraic spaces, we prove Orlov's representability theorem relating fully faithful functors and Fourier--Mukai transforms between the bounded derived category of coherent sheaves to the case of smooth, proper, and tame algebraic stacks. This extends previous results of Kawamata for Deligne--Mumford stacks with generically trivial stabilizers and projective coarse moduli spaces.