翻訳と辞書 Words near each other ・ Cotillion ball ・ Cotillion Ballroom ・ Cotillion Handicap ・ Cotillion Records ・ Cotinga ・ Cotinga (genus) ・ Cotinga (journal) ・ Cotingo River ・ Cotinguiba Esporte Clube ・ Cotinguiba River ・ Cotinine ・ Cotinis ・ Cotinis aliena ・ Cotinis antonii ・ Cotinis barthelemyi ・ Cotangent sheaf ・ Cotangent space ・ Cotap ・ Cotaparaco District ・ Cotapata National Park and Integrated Management Natural Area ・ Cotara ・ Cotard delusion ・ Cotarsina ・ Cotaruse District ・ Cotati ・ Cotati (comics) ・ Cotati Jazz Festival ・ Cotati Naval Outer Landing Field ・ Cotati, California ・ Cotati-Rohnert Park Unified School District Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cotangent sheaf ： ウィキペディア英語版 Cotangent sheaf In algebraic geometry, given a morphism ''f'': ''X'' → ''S'' of schemes, the cotangent sheaf on ''X'' is the sheaf of $\backslash mathcal\_X$-modules that represents (or classifies) ''S''-derivations 〔http://stacks.math.columbia.edu/tag/08RL〕 in the sense: for any $\backslash mathcal\_X$-modules ''F'', there is an isomorphism :$\backslash operatorname\_(\backslash Omega\_,\; F)\; =\; \backslash operatorname\_S(\backslash mathcal\_X,\; F)$ that depends naturally on ''F''. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d:\; \backslash mathcal\_X\; \backslash to\; \backslash Omega\_$ such that any ''S''-derivation $D:\; \backslash mathcal\_X\; \backslash to\; F$ factors as $D\; =\; \backslash alpha\; \backslash circ\; d$ with some $\backslash alpha:\; \backslash Omega\_\; \backslash to\; F$. In the case ''X'' and ''S'' are affine schemes, the above definition means that $\backslash Omega\_$ is the module of Kähler differentials. The standard way to construct a cotangent sheaf (e.g., Hartshorne, Ch II. § 8) is through a diagonal morphism (which amounts to gluing modules of Kähler differentials on affine charts to get the globally-defined cotangent sheaf.) The dual module of the cotangent sheaf on a scheme ''X'' is called the tangent sheaf on ''X'' and is sometimes denoted by $\backslash Theta\_X$.〔In concise terms, this means: :$\backslash Theta\_X\; \backslash oversetom\_(\backslash Omega\_X,\; \backslash mathcal\_X)\; =\; \backslash mathcaler(\backslash mathcal\_X).$ 〕 There are two important exact sequences: #If ''S'' →''T'' is a morphism of schemes, then #:$f^$ * \Omega_ \to \Omega_ \to \Omega_ \to 0. #If ''Z'' is a closed subscheme of ''X'' with ideal sheaf ''I'', then #:$I/I^2\; \backslash to\; \backslash Omega\_\; \backslash otimes\; \backslash mathcal\_Z\; \backslash to\; \backslash Omega\_\; \backslash to\; 0.$〔http://mathoverflow.net/questions/79956/jacobian-criterion-for-smoothness-of-schemes as well as 〕 The cotangent sheaf is closely related to smoothness of a variety or scheme. For example, an algebraic variety is smooth of dimension ''n'' if and only if Ω''X'' is a locally free sheaf of rank ''n''. == Construction through a diagonal morphism == Let $f:\; X\; \backslash to\; S$ be a morphism of schemes as in the introduction and Δ: ''X'' → ''X'' ×''S'' ''X'' the diagonal morphism. Then the image of Δ is locally closed; i.e., closed in some open subset ''W'' of ''X'' ×''S'' ''X'' (the image is closed if and only if ''f'' is separated). Let ''I'' be the ideal sheaf of Δ(''X'') in ''W''. One then puts: :$\backslash Omega\_\; =\; \backslash Delta^$ * (I/I^2) and checks this sheaf of modules satisfies the required universal property of a cotangent sheaf (Hartshorne, Ch II. Remark 8.9.2). The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if ''S'' is Noetherian and ''f'' is of finite type. 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「Cotangent sheaf」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.