翻訳と辞書 Words near each other ・ Finite element updating ・ Finite extensions of local fields ・ Finite field ・ Finite field arithmetic ・ Finite Fourier transform ・ Finite geometry ・ Finite group ・ Finite impulse response ・ Finite intersection property ・ Finite lattice representation problem ・ Finite Legendre transform ・ Finite map ・ Finite mathematics ・ Finite model property ・ Finite model theory ・ Finite morphism ・ Finite number ・ Finite part ・ Finite pointset method ・ Finite potential well ・ Finite Records ・ Finite ring ・ Finite Risk insurance ・ Finite set ・ Finite state machine with datapath ・ Finite state transducer ・ Finite strain theory ・ Finite strip method ・ Finite subdivision rule ・ Finite thickness 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 Finite morphism ： ウィキペディア英語版 Finite morphism In algebraic geometry, a branch of mathematics, a morphism $f:\; X\; \backslash rightarrow\; Y$ of schemes is a finite morphism if $Y$ has an open cover by affine schemes :$V\_i\; =\; \backslash mbox\; \backslash ;\; B\_i$ such that for each $i$, :$f^(V\_i)\; =\; U\_i$ is an open affine subscheme $\backslash mbox\; \backslash ;\; A\_i$, and the restriction of ''f'' to $U\_i$, which induces a map of rings :$B\_i\; \backslash rightarrow\; A\_i,$ makes $A\_i$ a finitely generated module over $B\_i$. == Properties of finite morphisms == In the following, ''f'' : ''X'' → ''Y'' denotes a finite morphism. * The composition of two finite maps is finite. * Any base change of a finite morphism is finite, i.e. if $g:\; Z\; \backslash rightarrow\; Y$ is another (arbitrary) morphism, then the canonical morphism $X\; \backslash times\_Y\; Z\; \backslash rightarrow\; Z$ is finite. This corresponds to the following algebraic statement: if ''A'' is a finitely generated ''B''-module, then the tensor product $A\; \backslash otimes\_B\; C$ is a finitely generated ''C''-module, where $C\; \backslash rightarrow\; B$ is any map. The generators are $a\_i\; \backslash otimes\; 1$, where $a\_i$ are the generators of ''A'' as a ''B''-module. * Closed immersions are finite, as they are locally given by $A\; \backslash rightarrow\; A\; /\; I$, where ''I'' is the ideal corresponding to the closed subscheme. * Finite morphisms are closed, hence (because of their stability under base change) proper. Indeed, replacing ''Y'' by the closure of ''f''(''X''), one can assume that ''f'' is dominant. Further, one can assume that ''Y''=''Spec B'' is affine, hence so is ''X=Spec A''. Then the morphism corresponds to an integral extension of rings ''B'' ⊂ ''A''. Then the statement is a reformulation of the going up theorem of Cohen-Seidenberg. * Finite morphisms have finite fibres (i.e. they are quasi-finite). This follows from the fact that any finite ''k''-algebra, for any field ''k'' is an Artinian ring. Slightly more generally, for a finite surjective morphism ''f'', one has ''dim X=dim Y''. * Conversely, proper, quasi-finite locally finite-presentation maps are finite. (EGA IV, 8.11.1.) * Finite morphisms are both projective and affine. 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「Finite morphism」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.