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aralytically
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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 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 * * * → * ∞ * * * {\displaystyle n_{j}\to \infty } * such that * * * * * * * J * * j * * * ¯ * * * ⊂ * * J * * * n * * j * * * * * * * {\displaystyle {\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. |