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In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well. == Definition == If ''X'' and ''Y'' are closed subvarieties of A''n'' and A''m'' (so they are affine varieties), then a regular map ƒ:''X''→''Y'' is the restriction of a polynomial map A''n''→A''m''. Explicitly, it has the form : where the 's are in the coordinate ring of ''X'': :, ''I'' the ideal defining ''X'', so that the image lies in ''Y''; i.e., satisfying the defining equations of ''Y''. (Two polynomials ''f'' and ''g'' define the same function on ''X'' if and only if ''f'' - ''g'' is in ''I''.) More generally, a map ƒ:''X''→''Y'' between two varieties is regular at a point ''x'' if there is a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ƒ(''x'') such that ƒ(''U'') ⊂ ''V'' and the restricted function ƒ:''U''→''V'' is regular as a function on some affine charts of ''U'' and ''V''. Then ƒ is called regular, if it is regular at all points of ''X''. *Note: It is not immediately obvious that the two definitions coincide: if ''X'' and ''Y'' are affine varieties, then a map ƒ:''X''→''Y'' is regular in the first sense if and only if it is so in the second sense.〔Here is the argument showing the definitions coincide. Clearly, we can assume ''Y'' = A1. Then the issue here is whether the "regular-ness" can be patched together; this answer is yes and that can be seen from the construction of the structure sheaf of an affine variety as described at affine variety#Structure sheaf.〕 Also, it is not immediately clear whether a regularity depends on a choice of affine charts (it doesn't.〔It is not clear how to prove this, though. If ''X'', ''Y'' are quasi-projective, then the proof can be given. The non-quasi-projective case strongly depends on one's definition of an abstract variety.〕) This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an (abstract) algebraic variety is defined to be a particular kind of a locally ringed space (see for example "projective variety" for the ringed space structure of a projective variety). When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces. The composition of regular maps is again regular; thus, algebraic varieties form the category of algebraic varieties where the morphisms are the regular maps. Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if ƒ:''X''→''Y'' is a morphism of affine varieties, then it defines the algebra homomorphism : where are the coordinate rings of ''X'' and ''Y''; it is well-defined since is a polynomial in elements of . Conversely, if is an algebra homomorphism, then it induces the morphism : given by: writing , : where are the images of 's.〔The image of lies in ''Y'' since if ''g'' is a polynomial in ''J'', then, a priori thinking is a map to the affine space, since ''g'' is in ''J''.〕 Note as well as since φ is an algebra homomorphism. Also, 〕 In particular, ''f'' is an isomorphism of affine varieties if and only if ''f''# is an isomorphism of the coordinate rings. For example, if ''X'' is a closed subvariety of an affine variety ''Y'' and ƒ is the inclusion, then ƒ# is the restriction of regular functions on ''Y'' to ''X''. See #Examples below for more examples. 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 algebraic varieties」の詳細全文を読む スポンサード リンク
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