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In algebraic geometry, a closed immersion of schemes is a regular embedding of codimension ''r'' if each point ''x'' in ''X'' has an open affine neighborhood ''U'' in ''Y'' such that the ideal of is generated by a regular sequence of length ''r''. For example, if ''X'' and ''Y'' are smooth over a scheme ''S'' and if ''i'' is an ''S''-morphism, then ''i'' is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If is regularly embedded into a regular scheme, then ''B'' is a complete intersection ring. The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when ''i'' is a regular embedding, if ''I'' is the ideal sheaf of ''X'' in ''Y'', then the normal sheaf, the dual of , is locally free (thus a vector bundle) and the natural map is an isomorphism: the normal cone coincides with the normal bundle. A flat morphism of finite type is called a (local) complete intersection morphism if each point ''x'' in ''X'' has an open affine neighborhood ''U'' so that ''f'' |''U'' factors as where ''j'' is a regular embedding and ''g'' is smooth. For example, if ''f'' is a morphism between smooth varieties, then ''f'' factors as where the first map is the graph morphism and so is a complete intersection morphism. == References == *, section B.7 *E. Sernesi: ''(Deformations of algebraic schemes )'' 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「Regular embedding」の詳細全文を読む スポンサード リンク
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