Analytically unramified ring
In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent.)
The following rings are analytically unramified:
- pseudo-geometric reduced ring.
- excellent reduced ring.
There are two classical theorems of David Rees that characterize analytically unramified rings. The first says that a noetherian local ring (R, m) is analytically unramified if and only if there are a m-primary ideal J and a sequence such that , where the bar means the integral closure of an ideal. The second says that a noetherian local domain is analytically unramified if and only if, for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first.
Chevalley (1945) showed that any local ring of an algebraic variety is analytically unramified. It is not easy to find reduced local rings that are not analytically unramified: Nagata (1955) gave an example of one that is normal, answering a question of Zariski (1948).
References
- Chevalley, Claude (1945), "Intersections of algebraic and algebroid varieties", Trans. Amer. Math. Soc., 57: 1–85, JSTOR 1990167, MR 0012458
- Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432
- Nagata, Masayoshi (1955), "An example of normal local ring which is analytically ramified", Nagoya Math. J., 9: 111–113, MR 0073572
- Rees, D. (1961), "A note on analytically unramified local rings", J. London Math. Soc., 36: 24–28, MR 0126465
- Zariski, Oscar (1948), "Analytical irreducibility of normal varieties", Ann. of Math. (2), 49: 352–361, doi:10.2307/1969284, MR 0024158
- Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR0389876