Editing Regular sequence

{{about||regular Cauchy sequence|Cauchy sequence#In constructive mathematics|a k-regular sequence of integers|k-regular sequence}}
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==
Given 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
''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> of elements of ''R'' such that ''r''<sub>1</sub> is a not a zero-divisor on ''M'' and ''r''<sub>''i''</sub> is a not a zero-divisor on ''M''/(''r''<sub>1</sub>, ..., ''r''<sub>''i''−1</sub>)''M'' for ''i'' = 2, ..., ''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''.

Being a regular sequence may depend on the order of the elements. 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==

*Given an integral domain <math>R</math> any nonzero <math>f \in R</math> gives a regular sequence.
*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</sup> 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.

=== Non-Examples ===
A simple non-example of a regular sequence is given by the sequence <math>(xy,x^2)</math> of elements in <math>\mathbb{C}[x,y]</math> since
:<math>
\cdot x^2 : \frac{\mathbb{C}[x,y]}{(xy)} \to \frac{\mathbb{C}[x,y]}{(xy)}
</math>
has a non-trivial kernel given by the ideal <math>(y) \subset \mathbb{C}[x,y]/(xy)</math> . Similar examples can be found by looking at minimal generators for the ideals generated from reducible schemes with multiple components and taking the subscheme of a component, but fattened.

==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]]