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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Analytically unramified ring ： ウィキペディア英語版 Analytically unramified ring 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. 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 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 $n\_j\; \backslash to\; \backslash infty$ such that $\backslash overline\; \backslash subset\; J^$, 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. ==Nagata's example== Let ''K''0 be a perfect field of characteristic 2, such as F2. Let ''K'' be ''K''0(), where the ''u''''n'' and ''v''''n'' are indeterminates. Let ''T'' be the subring of the formal power series ring ''K''  generated by ''K'' and ''K''''2''  and the element ∑(''u''''n''''x''''n''+ ''v''''n''''y''''n''). Nagata proves that ''T'' is a normal local noetherian domain whose completion has nonzero nilpotent elements, so ''T'' is analytically ramified. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Analytically unramified ring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.