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

Hornell, James

doi:10.1090/S0002-9939-1975-0376657-6

A note on the geometric criteria for the factoriality of an affine ring
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Proc. Amer. Math. Soc. 53 (1975), 45-50 Request permission

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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