The Resolution Property via Azumaya Algebras

TL;DR

This paper proves that separated, normal tame Artin surfaces possess the resolution property by reducing the problem to the existence of Azumaya algebras through a series of reductions involving tame Artin stacks and gerbes.

Contribution

It introduces a novel approach using Azumaya algebras and formal-local methods to establish the resolution property for a broad class of algebraic surfaces.

Findings

Normal tame Artin surfaces have the resolution property.

Normal tame Artin stacks can be rigidified.

Existence of Azumaya algebras on these surfaces is established.

Abstract

Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.