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.

Chevalley (1945) showed that every local ring of an algebraic variety is analytically unramified. Schmidt (1936) gave an example of an analytically ramified reduced local ring. Krull (1930) showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more precisely he showed that a 1-dimensional normal Noetherian local domain is analytically unramified if and only if its integral closure is a finite module. This prompted Zariski (1948) to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However Nagata (1955) gave an example of a 2-dimensional normal analytically ramified Noetherian local ring. Nagata also showed that a slightly stronger version of Zariski's question is correct: if the normalization of every finite extension of a given Noetherian local ring R is a finite module, then R is analytically unramified.

There are two classical theorems of David Rees (1961) 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.

Nagata's example

Suppose that K is a field of characteristic p>0 containing infinitely many algebraically independent elements b_n, c_n such that K has infinite degree over K^p. For example, K could be the purely transcendental extension of the finite field F_p generated by elements b_n, c_n. Let T be the subring of the formal power series ring K[[x,y]] generated by K and the power series with coefficients in K^p and the element Σb_nxⁿ+ c_nyⁿ. Then the ring T is a normal local Noetherian domain, but its completion has nilpotent elements, so T is analytically ramified.

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
• Schmidt, Friedrich Karl (1936), "Über die Erhaltung der Kettensätze der Idealtheorie bei beliebigen endlichen Körpererweiterungen", Mathematische Zeitschrift, 41 (1): 443–450, doi:10.1007/BF01180433
• 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