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 $n_{j}\to \infty$ such that ${\overline {J^{j}}}\subset J^{n_{j}}$, 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.

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, 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
• 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