|
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. ==Definitions== For a commutative ring ''R'' and an ''R''-module ''M'', an element ''r'' in ''R'' is called a non-zero-divisor on ''M'' if ''r m'' = 0 implies ''m'' = 0 for ''m'' in ''M''. An ''M''-regular sequence is a sequence :''r''1, ..., ''r''''d'' in ''R'' such that ''r''''i'' is a non-zero-divisor on ''M''/(''r''1, ..., ''r''''i''-1)''M'' for ''i'' = 1, ..., ''d''.〔N. Bourbaki. ''Algèbre. Chapitre 10. Algèbre Homologique.'' Springer-Verlag (2006). X.9.6.〕 Some authors also require that ''M''/(''r''1, ..., ''r''''d'')''M'' is not zero. Intuitively, to say that ''r''1, ..., ''r''''d'' is an ''M''-regular sequence means that these elements "cut ''M'' down" as much as possible, when we pass successively from ''M'' to ''M''/(''r''1)''M'', to ''M''/(''r''1, ''r''2)''M'', and so on. An ''R''-regular sequence is called simply a regular sequence. That is, ''r''1, ..., ''r''''d'' is a regular sequence if ''r''1 is a non-zero-divisor in ''R'', ''r''2 is a non-zero-divisor in the ring ''R''/(''r''1), and so on. In geometric language, if ''X'' is an affine scheme and ''r''1, ..., ''r''''d'' is a regular sequence in the ring of regular functions on ''X'', then we say that the closed subscheme ⊂ ''X'' is a complete intersection subscheme of ''X''. For example, ''x'', ''y''(1-''x''), ''z''(1-''x'') is a regular sequence in the polynomial ring C(''y'', ''z'' ), while ''y''(1-''x''), ''z''(1-''x''), ''x'' is not a regular sequence. But if ''R'' is a Noetherian local ring and the elements ''r''''i'' are in the maximal ideal, or if ''R'' is a graded ring and the ''r''''i'' are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence. Let ''R'' be a Noetherian ring, ''I'' an ideal in ''R'', and ''M'' a finitely generated ''R''-module. The depth of ''I'' on ''M'', written depth''R''(''I'', ''M'') or just depth(''I'', ''M''), is the supremum of the lengths of all ''M''-regular sequences of elements of ''I''. When ''R'' is a Noetherian local ring and ''M'' is a finitely generated ''R''-module, the depth of ''M'', written depth''R''(''M'') or just depth(''M''), means depth''R''(''m'', ''M''); that is, it is the supremum of the lengths of all ''M''-regular sequences in the maximal ideal ''m'' of ''R''. In particular, the depth of a Noetherian local ring ''R'' means the depth of ''R'' as a ''R''-module. That is, the depth of ''R'' is the maximum length of a regular sequence in the maximal ideal. For a Noetherian local ring ''R'', the depth of the zero module is ∞,〔A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.〕 whereas the depth of a nonzero finitely generated ''R''-module ''M'' is at most the Krull dimension of ''M'' (also called the dimension of the support of ''M'').〔N. Bourbaki. ''Algèbre Commutative. Chapitre 10.'' Springer-Verlag (2007). Th. X.4.2.〕 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 sequence」の詳細全文を読む スポンサード リンク
|