In commutative algebra, an analytically normal ring is a local ring whose completion is a normal ring; in other words, an integral domain that is integrally closed in its quotient field.
Zariski[1] 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[2][3] gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.
Notes
- ^ Zariski (1950).
- ^ Nagata (1958).
- ^ Nagata (1962), appendix A1, example 7.
References
- Nagata, Masayoshi (1958). "An example of a normal local ring which is analytically reducible". Memoirs of the College of Science, University of Kyoto Series A, Mathematics. 31: 83–85. doi:10.1215/kjm/1250776950. MR 0097395.
- Nagata, Masayoshi (1962). Local Rings. Interscience Tracts in Pure and Applied Mathematics. Vol. 13. New York: Interscience. Reprinted, 1974, ISBN 978-0-470-62865-2.
- Zariski, Oscar (1948). "Analytical irreducibility of normal varieties". Annals of Mathematics. 49 (2): 352–361. doi:10.2307/1969284. MR 0024158.
- Zariski, Oscar (1950). "Sur la normalité analytique des variétés normales". Annales de l'Institut Fourier (in French). 2: 161–164. doi:10.5802/aif.27. MR 0045413.
- Zariski, Oscar; Samuel, Pierre (1960). Commutative Algebra, Volume II. Graduate Texts in Mathematics. Vol. 29. Berlin, Heidelberg: Springer. doi:10.1007/978-3-662-29244-0.
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