Analytically normal ring

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, 13, New York-London: Interscience Publishers, ISBN 978-0470628652 
• Zariski, Oscar (1948), "Analytical irreducibility of normal varieties", Annals of Mathematics, Second Series, 49: 352–361, MR 0024158, doi:10.2307/1969284 
• Zariski, Oscar (1950), "Sur la normalité analytique des variétés normales", Annales de l'Institut Fourier, 2: 161–164, 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