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- Analytically_unramified_ring abstract "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 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.".
- Analytically_unramified_ring wikiPageExternalLink index.html.
- Analytically_unramified_ring wikiPageExternalLink 1118799688.
- Analytically_unramified_ring wikiPageID "39952976".
- Analytically_unramified_ring wikiPageRevisionID "577944235".
- Analytically_unramified_ring authorlink "David Rees".
- Analytically_unramified_ring first "David".
- Analytically_unramified_ring last "Rees".
- Analytically_unramified_ring year "1961".
- Analytically_unramified_ring subject Category:Commutative_algebra.
- Analytically_unramified_ring comment "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.".
- Analytically_unramified_ring label "Analytically unramified ring".
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- Analytically_unramified_ring sameAs Q17097738.
- Analytically_unramified_ring sameAs Q17097738.
- Analytically_unramified_ring wasDerivedFrom Analytically_unramified_ring?oldid=577944235.
- Analytically_unramified_ring isPrimaryTopicOf Analytically_unramified_ring.