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In algebraic geometry, a branch of mathematics, a morphism of schemes is a finite morphism if has an open cover by affine schemes : such that for each , : is an open affine subscheme , and the restriction of ''f'' to , which induces a map of rings : makes a finitely generated module over . == 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 is another (arbitrary) morphism, then the canonical morphism is finite. This corresponds to the following algebraic statement: if ''A'' is a finitely generated ''B''-module, then the tensor product is a finitely generated ''C''-module, where is any map. The generators are , where are the generators of ''A'' as a ''B''-module. * Closed immersions are finite, as they are locally given by , 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
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