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- W2953173653 abstract "Suppose $X$ is a smooth projective scheme of finite type over a field $K$, $mathcal{E}$ is a locally free ${mathcal{O}}_{X}$-bimodule of rank 2, $mathcal{A}$ is the non-commutative symmetric algebra generated by $mathcal{E}$ and ${sf Proj}A$ is the corresponding non-commutative $mathbb{P}^{1}$-bundle. We use the properties of the internal $operatorname{Hom}$ functor $HU(-,-)$ to prove versions of Serre finiteness and Serre vanishing for ${sf Proj}A$. As a corollary to Serre finiteness, we prove that ${sf Proj}A$ is Ext-finite. This fact is used in cite{izu} to prove that if $X$ is a smooth curve over $operatorname{Spec}K$, ${sf Proj }A$ has a Riemann-Roch theorem and an adjunction formula." @default.
- W2953173653 created "2019-06-27" @default.
- W2953173653 creator A5037888064 @default.
- W2953173653 date "2002-10-05" @default.
- W2953173653 modified "2023-09-27" @default.
- W2953173653 title "Serre finiteness and Serre vanishing for non-commutative P^1-bundles" @default.
- W2953173653 cites W2019487252 @default.
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- W2953173653 doi "https://doi.org/10.48550/arxiv.math/0210080" @default.
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