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			{{for\|regular Cauchy sequence\|Cauchy sequence#In constructive mathematics}}
			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 (mathematics)\|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

	Hi and ~~also~~. ~~Let me start by introducing~~ the ~~author~~, ~~her name~~ is ~~Consuelo~~ and ~~she gets comfortable~~ a ~~lot~~ of ~~use~~ the ~~full name~~. The ~~thing she adores~~ most is to ~~play with dogs~~ and ~~she~~'~~d never give it up~~. ~~Nevada~~ is ~~where her house~~ is ~~normally~~. ~~After being out~~ of ~~his project~~ for ~~years he became~~ an ~~administrative assistant but her promotion never comes~~. ~~She~~'~~s been working on her website for~~ a ~~while now~~. ~~Continue reading here~~: ~~https~~://~~www~~.~~youtube~~.~~com~~/~~watch?v=pns2W~~-~~mSPU8~~<br><br>~~Review my web page~~; [~~https~~://www.~~youtube~~.~~com~~/~~watch~~?v=~~pns2W-mSPU8 Quantum Pendants~~]		:''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> in ''R''

			such that ''r''<sub>''i''</sub> is a non-zero-divisor on ''M''/(''r''<sub>1</sub>, ..., ''r''<sub>''i''-1</sub>)''M'' for ''i'' = 1, ..., ''d''.<ref>N. Bourbaki. ''Algèbre. Chapitre 10. Algèbre Homologique.'' Springer-Verlag (2006). X.9.6.</ref> Some authors also require that ''M''/(''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>)''M'' is not zero. Intuitively, to say that
			''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> 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''<sub>1</sub>)''M'', to ''M''/(''r''<sub>1</sub>, ''r''<sub>2</sub>)''M'', and so on.

			An ''R''-regular sequence is called simply a '''regular sequence'''. That is, ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence if ''r''<sub>1</sub> is a non-zero-divisor in ''R'', ''r''<sub>2</sub> is a non-zero-divisor in the ring ''R''/(''r''<sub>1</sub>), and so on. In geometric language, if ''X'' is an [[Spectrum of a ring\|affine scheme]] and ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence in the ring of regular functions on ''X'', then we say that the closed subscheme {''r''<sub>1</sub>=0, ..., ''r''<sub>''d''</sub>=0} ⊂ ''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'''[''x'', ''y'', ''z''], while ''y''(1-''x''), ''z''(1-''x''), ''x'' is not a regular sequence. But if ''R'' is a [[Noetherian ring\|Noetherian]] [[local ring]] and the elements ''r''<sub>''i''</sub> are in the maximal ideal, or if ''R'' is a [[Graded algebra\|graded ring]] and the ''r''<sub>''i''</sub> 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 (ring theory)\|depth]]''' of ''I'' on ''M'', written depth<sub>''R''</sub>(''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<sub>''R''</sub>(''M'') or just depth(''M''), means depth<sub>''R''</sub>(''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 ∞,<ref>A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.</ref> whereas the depth of a nonzero finitely generated ''R''-module ''M'' is at most the [[Krull dimension#Krull dimension of a module\|Krull dimension]] of ''M'' (also called the dimension of the support of ''M'').<ref>N. Bourbaki. ''Algèbre Commutative. Chapitre 10.'' Springer-Verlag (2007). Th. X.4.2.</ref>

			==Examples==

			*For a prime number ''p'', the local ring '''Z'''<sub>(''p'')</sub> is the subring of the rational numbers consisting of fractions whose denominator is not a multiple of ''p''. The element ''p'' is a non-zero-divisor in '''Z'''<sub>(''p'')</sub>, and the quotient ring of '''Z'''<sub>(''p'')</sub> by the ideal generated by ''p'' is the field '''Z'''/(''p''). Therefore ''p'' cannot be extended to a longer regular sequence in the maximal ideal (''p''), and in fact the local ring '''Z'''<sub>(''p'')</sub> has depth 1.

			*For any field ''k'', the elements ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> in the polynomial ring ''A'' = ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] form a regular sequence. It follows that the [[Localization of a ring\|localization]] ''R'' of ''A'' at the maximal ideal ''m'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) has depth at least ''n''. In fact, ''R'' has depth equal to ''n''; that is, there is no regular sequence in the maximal ideal of length greater than ''n''.

			*More generally, let ''R'' be a [[regular local ring]] with maximal ideal ''m''. Then any elements ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> of ''m'' which map to a basis for ''m''/''m''<sup>2</sub> as an ''R''/''m''-vector space form a regular sequence.

			An important case is when the depth of a local ring ''R'' is equal to its [[Krull dimension]]: ''R'' is then said to be '''[[Cohen-Macaulay ring\|Cohen-Macaulay]]'''. The three examples shown are all Cohen-Macaulay rings. Similarly, a finitely generated ''R''-module ''M'' is said to be '''Cohen-Macaulay''' if its depth equals its dimension.

			==Applications==

			*If ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence in a ring ''R'', then the [[Koszul complex]] is an explicit [[Resolution (algebra)\|free resolution]] of ''R''/(''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>) as an ''R''-module, of the form:

			:<math>0\rightarrow R^{\binom{d}{d}} \rightarrow\cdots \rightarrow
			R^{\binom{d}{1}} \rightarrow R \rightarrow R/(r_1,\ldots,r_d)
			\rightarrow 0</math>

			In the special case where ''R'' is the polynomial ring ''k''[''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>], this gives a resolution of ''k'' as an ''R''-module.

			*If ''I'' is an ideal generated by a regular sequence in a ring ''R'', then the associated graded ring

			:<math>\oplus_{j\geq 0} I^j/I^{j+1}</math>

			is isomorphic to the polynomial ring (''R''/''I'')[''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>]. In geometric terms, it follows that a [[Complete intersection ring\|local complete intersection]] subscheme ''Y'' of any scheme ''X'' has a [[normal bundle]] which is a vector bundle, even though ''Y'' may be singular.

			==See also==
			*[[Complete intersection ring]]
			*[[Koszul complex]]
			*[[Depth (ring theory)]]
			*[[Cohen-Macaulay ring]]

			==Notes==
			{{reflist}}

			== References ==
			* {{Citation \| last1=Bourbaki \| first1=Nicolas \| author1-link=Nicolas Bourbaki \| title=Algèbre. Chapitre 10. Algèbre Homologique \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-3-540-34492-6 \| doi=10.1007/978-3-540-34493-3 \| mr=2327161 \| year=2006 }}
			* {{Citation \| last1=Bourbaki \| first1=Nicolas \| author1-link=Nicolas Bourbaki \| title=Algèbre Commutative. Chapitre 10 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-3-540-34394-3 \| doi=10.1007/978-3-540-34395-0 \| mr=2333539 \| year=2007 }}
			* Winfried Bruns; Jürgen Herzog, ''Cohen-Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1
			* [[David Eisenbud]], ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
			*{{Citation \| last1=Grothendieck \| first1=Alexander \| author1-link=Alexander Grothendieck \| title=Éléments de géometrie algébrique IV. Première partie \| url=http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1964__20__5_0 \| mr=0173675 \| year=1964 \| journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques \| volume=20 \| pages=1–259}}

			[[Category:Commutative algebra]]
			[[Category:Dimension]]

en>M97uzivatel: cs:Pětiúhelníkové číslo

2012-09-02T09:12:35Z

cs:Pětiúhelníkové číslo

New page

Hi and also. Let me start by introducing the author, her name is Consuelo and she gets comfortable a lot of use the full name. The thing she adores most is to play with dogs and she'd never give it up. Nevada is where her house is normally. After being out of his project for years he became an administrative assistant but her promotion never comes. She's been working on her website for a while now. Continue reading here: https://www.youtube.com/watch?v=pns2W-mSPU8<br><br>Review my web page; [https://www.youtube.com/watch?v=pns2W-mSPU8 Quantum Pendants]

Pentagonal number - Revision history

en>ClueBot NG: Reverting possible vandalism by 65.210.52.210 to version by Magioladitis. False positive? Report it. Thanks, ClueBot NG. (2055997) (Bot)

46.121.232.249 at 10:31, 24 February 2014

en>Toshio Yamaguchi: Undid revision 591507874 by Kmonsoor (talk). The Wayback Machine has a snapshot of the website. Replacing dead link with Wayback link.

en>M97uzivatel: cs:Pětiúhelníkové číslo