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Non-commutative coordinate rings and stacks

Published online by Cambridge University Press:  13 January 2004

Daniel Chan
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia. E-mail: [email protected]
Colin Ingalls
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada. E-mail: [email protected]
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Abstract

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne–Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.

Type
Research Article
Copyright
2004 London Mathematical Society

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