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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 Tangent space to a functor ： ウィキペディア英語版 Tangent space to a functor In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let ''X'' be a scheme over a field ''k''. :To give a $k()/(\backslash epsilon)^2$-point of ''X'' is the same thing as to give a ''k''-rational point ''p'' of ''X'' (i.e., the residue field of ''p'' is ''k'') together with an element of $(\backslash mathfrak\_/\backslash mathfrak\_^2)^$ *; i.e., a tangent vector at ''p''. (To see this, use the fact that any local homomorphism $\backslash mathcal\_p\; \backslash to\; k()/(\backslash epsilon)^2$ must be of the form :$\backslash delta\_p^v:\; u\; \backslash mapsto\; u(p)\; +\; \backslash epsilon\; v(u),\; \backslash quad\; v\; \backslash in\; \backslash mathcal\_p^$ *.) Let ''F'' be a functor from the category of ''k''-algebras to the category of sets. Then, for any ''k''-point $p\; \backslash in\; F(k)$, the fiber of $\backslash pi:\; F(k()/(\backslash epsilon)^2)\; \backslash to\; F(k)$ over ''p'' is called the tangent space to ''F'' at ''p''. The tangent space may be given the structure of a vector space over ''k''. If ''F'' is a scheme ''X'' over ''k'' (i.e., $F\; =\; \backslash operatorname\_(\backslash operatorname-,\; X)$), then each ''v'' as above may be identified with a derivation at ''p'' and this gives the identification of $\backslash pi^(p)$ with the space of derivations at ''p'' and we recover the usual construction. The construction may be thought of as defining an analog of the tangent bundle in the following way. Let $T\_X\; =\; X(k()/(\backslash epsilon)^2)$. Then, for any morphism $f:\; X\; \backslash to\; Y$ of schemes over ''k'', one sees $f^(\backslash delta\_p^v)\; =\; \backslash delta\_^$; this shows that the map $T\_X\; \backslash to\; T\_Y$ that ''f'' induces is precisely the differential of ''f'' under the above identification. == References == *A. Borel, ''Linear algebraic groups'' * * 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「Tangent space to a functor」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.