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hereditary orders

Proceedings of the London Mathematical Society (1)

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Non-commutative coordinate rings and stacks
Daniel Chan, Colin Ingalls
Journal:

Proceedings of the London Mathematical Society / Volume 88 / Issue 1 / January 2004

Published online by Cambridge University Press:

13 January 2004, pp. 63-88

Print publication:

January 2004
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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.