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A note on the geometric criteria for the factoriality of an affine ring


Author: James Hornell
Journal: Proc. Amer. Math. Soc. 53 (1975), 45-50
MSC: Primary 13F15; Secondary 14C20
DOI: https://doi.org/10.1090/S0002-9939-1975-0376657-6
MathSciNet review: 0376657
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Abstract: Let $ R$ be a normal integral domain finitely generated over a field $ k$, let $ U$ be an affine algebraic variety with coordinate ring $ R$, and let $ V$ be a projective completion of $ U$ over $ k$. $ R$ is factorial if and only if the class group of $ V$ over $ k$ is generated by the $ k$-rational cycles at infinity with respect to $ U$. If $ k$ is algebraically closed, $ R$ is shown to be factorial if and only if the Picard group of $ V$ is zero and the Néron-Severi group of $ V$ is generated by the $ k$-rational cycles at infinity. If $ k$ is finitely generated over its prime field, some well-known arithmetic results are applied to show the existence of affine localizations of $ V$ which have a factorial coordinate ring over $ k$. The relationship between the existence of an affine localization of $ V$ with a factorial coordinate ring, and the birationality of $ V$ is also discussed.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0376657-6
Article copyright: © Copyright 1975 American Mathematical Society

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