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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 Castelnuovo–Mumford regularity ： ウィキペディア英語版 Castelnuovo–Mumford regularity In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf ''F'' over projective space P''n'' is the smallest integer ''r'' such that it is r-regular, meaning that :$H^i(\backslash mathbf^n,\; F(r-i))=0\; \backslash ,$ whenever ''i'' > 0. The regularity of a subscheme is defined to be the regularity of its sheaf of ideals. The regularity controls when the Hilbert function of the sheaf becomes a polynomial; more precisely dim ''H''0(''P''''n'', ''F''(''m'')) is a polynomial in ''m'' when ''m'' is at least the regularity. The concept of ''r''-regularity was introduced by , who attributed the following results to Guido Castelnuovo: *An ''r''-regular sheaf is ''s''-regular for any ''s'' ≥ ''r''. *If a coherent sheaf is ''r''-regular then ''F''(''r'') is generated by its global sections. ==Graded modules== A related idea exists in commutative algebra. Suppose ''R'' = ''k''() is a polynomial ring over a field ''k'' and ''M'' is a finitely generated graded ''R''-module. Suppose ''M'' has a minimal graded free resolution :$\backslash cdots\backslash rightarrow\; F\_j\; \backslash rightarrow\backslash cdots\backslash rightarrow\; F\_0\backslash rightarrow\; M\backslash rightarrow\; 0$ and let ''b''''j'' be the maximum of the degrees of the generators of ''F''''j''. If ''r'' is an integer such that ''b''''j'' - ''j'' ≤ ''r'' for all ''j'', then ''M'' is said to be ''r''-regular. The regularity of ''M'' is the smallest such ''r''. These two notions of regularity coincide when ''F'' is a coherent sheaf such that Ass(''F'') contains no closed points. Then the graded module is finitely generated and has the same regularity as ''F''. 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「Castelnuovo–Mumford regularity」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.