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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 Morphism of schemes ： ウィキペディア英語版 Morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. More generally, morphisms ''p'':''X'' →''S'' with various schemes ''X'' but fixed scheme ''S'' form the category of schemes over ''S'' (the slice category of the category of schemes with the base object ''S''.) An object in the category is called an ''S''-scheme and a morphism in the category an ''S''-morphism; explicitly, an ''S''-morphism from ''p'':''X'' →''S'' to ''q'':''Y'' →''S'' is a morphism ƒ:''X'' →''Y'' of schemes such that ''p'' = ''q'' ∘ ƒ. == Definition == By definition, a morphism of schemes is just a morphism of locally ringed spaces. A scheme, by definition, has an open affine chart and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:''X''→''Y'' be a morphism of schemes. If ''x'' is a point of ''X'', since ƒ is continuous, there are open affine subsets ''U'' = Spec ''A'' of ''X'' containing ''x'' and ''V'' = Spec ''B'' of ''Y'' such that ƒ(''U'') ⊂ ''V''. Then ƒ: ''U'' → ''V'' is a morphism of affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:''X''→''Y'' is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts. *Note: It would not be desirable to define a morphism of schemes as a morphism of ringed spaces. One trivial reason is that there is an example of a ringed-space morphism between affine schemes that is not induced by a ring homomorphism (for example, a morphism of ringed spaces: *:$\backslash operatorname\; k(x)\; \backslash to\; \backslash operatorname\; k()\_\; =\; \backslash$ :that sends the unique point to ''s'' and that comes with $k()\_\; \backslash to\; k(x),\; \backslash ,\; y\; \backslash mapsto\; x$.) More conceptually, the definition of a morphism of schemes needs to capture "Zariski-local nature" or localization of rings; this point of view (i.e., a local-ringed space) is essential for a generalization (topos). Let ƒ:''X''→''Y'' be a morphism of schemes with $\backslash phi:\; \backslash mathcal\_Y\; \backslash to\; f\_$ *\mathcal_X. Then, for each point ''x'' of ''X'', the homomorphisms on the stalks: :$\backslash phi:\; \backslash mathcal\_\; \backslash to\; \backslash mathcal\_$ is a local ring homomorphism: i.e., $\backslash phi(\backslash mathfrak\_)\; \backslash subset\; \backslash mathfrak\_x$ and so induces an injective homomorphism of residue fields :$\backslash phi:\; k(f(x))\; \backslash hookrightarrow\; k(x)$. (In fact, φ maps th ''n''-th power of a maximal ideal to the ''n''-th power of the maximal ideal and thus induces the map between the (Zariski) cotangent spaces.) For each scheme ''X'', there is a natural morphism :$\backslash theta:\; X\; \backslash to\; \backslash operatorname\; \backslash Gamma(X,\; \backslash mathcal\_X),$ which is an isomorphism if and only if ''X'' is affine; θ is obtained by gluing ''U'' → target which come from restrictions to open affine subsets ''U'' of ''X''. This fact can also be stated as follows: for any scheme ''X'' and a ring ''A'', there is a natural bijection: :$\backslash operatorname(X,\; \backslash operatorname(A))\; \backslash cong\; \backslash operatorname(A,\; \backslash Gamma(X,\; \backslash mathcal\; O\_X)).$ (Proof: The map $\backslash phi\; \backslash mapsto\; \backslash operatorname(\backslash phi)\; \backslash circ\; \backslash theta$ from the right to the left is the required bijection. In short, θ is an adjunction.) Moreover, this fact (adjoint relation) can be used to characterize an affine scheme: a scheme ''X'' is affine if and only if for each scheme ''S'', the natural map :$\backslash operatorname(S,\; X)\; \backslash to\; \backslash operatorname(\backslash Gamma(X,\; \backslash mathcal\_X),\; \backslash Gamma(S,\; \backslash mathcal\_S))$ is bijective. (Proof: if the maps are bijective, then $\backslash operatorname(-,\; X)\; \backslash simeq\; \backslash operatorname(-,\; \backslash operatorname\; \backslash Gamma(X,\; \backslash mathcal\_X))$ and ''X'' is isomorphic to $\backslash operatorname\; \backslash Gamma(X,\; \backslash mathcal\_X)$ by Yoneda's lemma; the converse is clear.) 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「Morphism of schemes」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.