An Explicit Derived Equivalence of Azumaya Algebras on K3 Surfaces via Koszul Duality

TL;DR

This paper constructs explicit derived equivalences between Azumaya algebras on K3 surfaces using Koszul duality, revealing new relations between their second Chern classes and moduli space properties.

Contribution

It provides a novel explicit derived equivalence construction for Azumaya algebras on K3 surfaces via Koszul duality, and establishes divisibility conditions for their second Chern classes.

Findings

Derived equivalence corresponds to twisted sheaves equivalence

Divisibility condition: 2r divides c_2(A) - c_2(A')

Moduli stack of Azumaya algebras with fixed gerbe is proper under certain conditions

Abstract

We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.