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On stacky surfaces and noncommutative surfaces

Authors :
Faber, Eleonore
Ingalls, Colin
Okawa, Shinnosuke
Satriano, Matthew
Publication Year :
2022

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.<br />Comment: v2:many minor revisions. Section 6 of v1 is removed. 34 pages

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2206.13359
Document Type :
Working Paper