Simple reference for valuative criterion of integrality? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:47:35Z http://mathoverflow.net/feeds/question/78702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78702/simple-reference-for-valuative-criterion-of-integrality Simple reference for valuative criterion of integrality? Allen Knutson 2011-10-20T20:15:23Z 2011-10-20T20:25:44Z <p>I'd like to see a complete proof of the simplest version of the following rough statement: "If \$f/g\$ is a rational function on a reduced scheme (\$g\$ not a zero divisor), and \$f/g\$ doesn't have poles in codimension \$1\$, then \$f/g\$ is a well-defined function on the normalization."</p> <p>I figure this should be called the valuative criterion of integrality (and if not, then maybe that's my problem). I can't find it in [Eisenbud], [Vakil], [Hartshorne], or [Vasconcelos]. There is a version of it in [Huneke-Swanson], but they're treating the more general case of integral closure of ideals than merely of rings, which obviously I should be able to downgrade from but would rather not if I don't have to. Where should I read about it?</p> http://mathoverflow.net/questions/78702/simple-reference-for-valuative-criterion-of-integrality/78703#78703 Answer by ACL for Simple reference for valuative criterion of integrality? ACL 2011-10-20T20:25:44Z 2011-10-20T20:25:44Z <p>This is corollary 11.4 in Eisenbud's book, namely <em>a normal domain is the intersection of its localizations at primes of codimension 1.</em></p>