Analytically normal ring
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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.
- Nagata, Masayoshi (1962). Local Rings. Interscience Tracts in Pure and Applied Mathematics. 13 (1974 reprint ed.). New York: Interscience. 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.
- Zariski, Oscar (1950). "Sur la normalité analytique des variétés normales" (in fr). Annales de l'Institut Fourier 2: 161–164. doi:10.5802/aif.27. http://www.numdam.org/item?id=AIF_1950__2__161_0.
- Zariski, Oscar; Samuel, Pierre (1960). Commutative Algebra, Volume II. Graduate Texts in Mathematics. 29. Berlin, Heidelberg: Springer. ISBN 978-3-662-27753-9.
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