In algebra , an analytically normal ring is a local ring whose completion is a normal ring , in other words a domain that is integrally closed in its quotient field .
Zariski (1950) proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem . Nagata (1958 , 1962 , Appendix A1, example 7) gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.
References
Nagata, Masayoshi (1958), "An example of a normal local ring which is analytically reducible" , Mem. Coll. Sci. Univ. Kyoto. Ser. A Math. , 31 : 83–85, MR 0097395 Nagata, Masayoshi (1962), Local rings , Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers, ISBN 978-0470628652 Zariski, Oscar (1948), "Analytical irreducibility of normal varieties", Annals of Mathematics , Second Series, 49 (2): 352–361, doi :10.2307/1969284 , JSTOR 1969284 , MR 0024158 Zariski, Oscar (1950), "Sur la normalité analytique des variétés normales" , Annales de l'Institut Fourier , 2 : 161–164, doi :10.5802/aif.27 MR 0045413 Zariski, Oscar ; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II , Berlin, New York: Springer-Verlag , ISBN 978-0-387-90171-8 MR 0389876
