On stacky surfaces and noncommutative surfaces

TL;DR

This paper establishes a geometric description of tame orders of global dimension 2 over surfaces as categories of twisted sheaves on certain algebraic stacks, extending classical results to positive characteristic and global settings.

Contribution

It constructs explicit algebraic stacks associated with tame orders on surfaces, generalizing Reiten and Van den Bergh's results to finite characteristic and global cases.

Findings

Equivalence between modules over tame orders and twisted sheaves on stacks

Explicit construction of stacks via root stacks, canonical stacks, and gerbes

Applications to noncommutative geometry and Hochschild cohomology

Abstract

Let $\mathbf{k}$ be an algebraically closed field of characteristic $\geq 7$ or zero. Let $\mathcal{A}$ be a tame order of global dimension $2$ over a normal surface $X$ over $\mathbf{k}$ such that $\operatorname{Z}(\mathcal{A})=\mathcal{O}_{X}$ is locally a direct summand of $\mathcal{A}$. We prove that there is a $\mu_N$-gerbe $\mathcal{X}$ over a smooth tame algebraic stack whose generic stabilizer is trivial, with coarse space $X$ such that the category of 1-twisted coherent sheaves on $\mathcal{X}$ is equivalent to the category of coherent sheaves of modules on $\mathcal{A}$. Moreover, the stack $\mathcal{X}$ is constructed explicitly through a sequence of root stacks, canonical stacks, and gerbes. This extends results of Reiten and Van den Bergh to finite characteristic and the global situation. As applications, in characteristic $0$ we prove that such orders are geometric noncommutative schemes in the sense of Orlov, and we study relations with Hochschild cohomology and Connes' convolution algebra.