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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. showed that every local ring of an algebraic variety is analytically unramified. 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. This prompted to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However 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 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. (en)
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- 4789 (xsd:nonNegativeInteger)
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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. (en)
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- Analytically unramified ring (en)
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