Analytically unramified ring

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In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent).

The following rings are analytically unramified:

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 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.[citation needed] 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 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.

Nagata's example

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Let K0 be a perfect field of characteristic 2, such as F2. Let K be K0({un, vn : n ≥ 0}), where the un and vn are indeterminates. Let T be the subring of the formal power series ring K [[x,y]] generated by K and K2 [[x,y]] and the element Σ(unxn+ vnyn). Nagata proves that T is a normal local noetherian domain whose completion has nonzero nilpotent elements, so T is analytically ramified.

References

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  • Chevalley, Claude (1945), "Intersections of algebraic and algebroid varieties", Trans. Amer. Math. Soc., 57: 1–85, doi:10.1090/s0002-9947-1945-0012458-1, 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, archived from the original on 2019-11-15, retrieved 2013-07-13
  • 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, MR 0389876