Non-commutative coordinate rings and stacks

• Autores: Daniel Chan, Colin Ingalls
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 88, Nº 1, 2004, págs. 63-88
• Idioma: inglés
• DOI: 10.1112/s0024611503014278
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• Resumen

  • 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.