Analytically unramified ring: Difference between revisions

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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:

pseudo-geometric reduced ring.
excellent reduced ring.

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).

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.

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

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 *[[Pseudo-geometric ring|pseudo-geometric]] reduced ring.
 *[[Excellent 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 <math>n_j \to \infty</math> such that <math>\overline{J^j} \subset J^{n_j}</math>, 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.
 {{harvtxt|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: {{harvtxt|Nagata|1955}} gave an example of one that is normal, answering a question of {{harvtxt|Zariski|1948}}.
+There are two classical theorems of {{harvs|txt|authorlink=David Rees|first=David|last=Rees|year=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 <math>n_j \to \infty</math> such that <math>\overline{J^j} \subset J^{n_j}</math>, 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.
 == References ==