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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Serre duality for non-commutative ${\mathbb{P}}^{1}$-bundles


Author: Adam Nyman
Journal: Trans. Amer. Math. Soc. 357 (2005), 1349-1416
MSC (2000): Primary 14A22; Secondary 16S99
Published electronically: July 16, 2004
MathSciNet review: 2115370
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Abstract: Let $X$ be a smooth scheme of finite type over a field $K$, let $\mathcal{E}$ be a locally free $\mathcal{O}_{X}$-bimodule of rank $n$, and let $\mathcal{A}$ be the non-commutative symmetric algebra generated by $\mathcal{E}$. We construct an internal $\operatorname{Hom}$ functor, ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (-,-)$, on the category of graded right $\mathcal{A}$-modules. When $\mathcal{E}$ has rank 2, we prove that $\mathcal{A}$ is Gorenstein by computing the right derived functors of ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (\mathcal{O}_{X},-)$. When $X$ is a smooth projective variety, we prove a version of Serre Duality for ${\mathsf{Proj}} \mathcal{A}$ using the right derived functors of $\underset{n \to \infty}{\lim} \underline{\mathcal{H}\textit{om}}_{\mathsf{Gr} \mathcal{A}} (\mathcal{A}/\mathcal{A}_{\geq n}, -)$.


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Additional Information

Adam Nyman
Affiliation: Department of Mathematical Sciences, Mathematics Building, University of Montana, Missoula, Montana 59812-0864
Email: nymana@mso.umt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03523-8
PII: S 0002-9947(04)03523-8
Keywords: Non-commutative geometry, Serre duality, non-commutative projective bundle
Received by editor(s): September 20, 2002
Received by editor(s) in revised form: September 16, 2003
Published electronically: July 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society