翻訳と辞書 Words near each other ・ Torslanda Church ・ Torslanda IK ・ Torslandavallen ・ Torslandaverken ・ Torsnes ・ Torsnäs ・ Torso ・ Torso (1973 film) ・ Torso (comics) ・ Torso (disambiguation) ・ Torso (Image Comics) ・ Torso Fragment ・ Torso of Venus and a Landscape ・ Torsobear ・ Torson, Count of Toulouse ・ Torsor (algebraic geometry) ・ Torstad ・ Torstad Chapel ・ Torstar ・ Torstar Syndication Services ・ Torsted ・ Torstein Aagaard-Nilsen ・ Torstein Andersen Aase ・ Torstein Blixfjord ・ Torstein Børte ・ Torstein Dahle ・ Torstein Dale Sjøtveit ・ Torstein Eckhoff ・ Torstein Ellingsen ・ Torstein Flakne 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 Torsor (algebraic geometry) ： ウィキペディア英語版 Torsor (algebraic geometry) In algebraic geometry, given a smooth algebraic group ''G'', a ''G''-torsor or a principal ''G''-bundle ''P'' over a scheme ''X'' is a scheme (or even algebraic space) with the action of ''G'' that is locally trivial in the given Grothendieck topology in the sense that the base change $Y\; \backslash times\_X\; P$ along "some" covering map $Y\; \backslash to\; X$ is the trivial torsor $Y\; \backslash times\; G\; \backslash to\; Y$ (''G'' acts only on the second factor). Equivalently, a ''G''-torsor ''P'' on ''X'' is a principal homogeneous space for the group scheme $G\_X\; =\; X\; \backslash times\; G$ (i.e., $G\_X$ acts simply transitively on $P$.) The definition may be formulated in the sheaf-theoretic language: a sheaf ''P'' on the category of ''X''-schemes with some Grothendieck topology is a ''G''-torsor if there is a covering $\backslash$ in the topology, called the local trivialization, such that the restriction of ''P'' to each $U\_i$ is a trivial $G\_$-torsor. A line bundle is nothing but a $\backslash mathbb\_m$-bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably (by permitting ''P'' to be a stack like an algebraic space if necessary). == Examples and basic properties == Examples *A $GL\_n$-torsor on ''X'' is a vector bundle of rank ''n'' (i.e., a locally free sheaf) on ''X''. *If $L/K$ is a finite Galois extension, then $\backslash operatorname\; L\; \backslash to\; \backslash operatorname\; K$ is a $\backslash operatorname(L/K)$-torsor (roughly because the Galois group acts simply transitively on the roots.) This fact is a basis for Galois descent. See integral extension for a generalization. Remark: A ''G''-torsor ''P'' over ''X'' is isomorphic to a trivial torsor if and only if $P(X)\; =\; \backslash operatorname(X,\; P)$ is nonempty. (Proof: if there is an $s:\; X\; \backslash to\; P$, then $X\; \backslash times\; G\; \backslash to\; P,\; (x,\; g)\; \backslash mapsto\; s(x)g$ is an isomorphism.) Let ''P'' be a ''G''-torsor with a local trivialization $\backslash$ in étale topology. A trivial torsor admits a section: thus, there are elements $s\_i\; \backslash in\; P(U\_i)$. Fixing such sections $s\_i$, we can write uniquely $s\_i\; g\_\; =\; s\_j$ on $U\_$ with $g\_\; \backslash in\; G(U\_)$. Different choices of $s\_i$ amount to 1-coboundaries in cohomology; that is, the $g\_$ define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf coefficient) group $H^1(X,\; G)$. A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in $H^1(X,\; G)$ defines a ''G''-torsor on ''X'', unique up to an isomorphism. If ''G'' is a connected algebraic group over a finite field $\backslash mathbf\_q$, then any ''G''-bundle over $\backslash operatorname\; \backslash mathbf\_q$ is trivialy. (Lang's theorem.) 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「Torsor (algebraic geometry)」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.