Daniel Chan and Colin Ingalls

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.

2000 Mathematical Subject Classification: 14A22, 16S38, 14A20.

Keywords: algebraic stacks, hereditary orders, groupoids.

E-mail:
danielch@maths.unsw.edu.au
colin@math.unb.ca