Regular sequence

{{Short description|Well-behaved sequence in a commutative ring}} {{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 ''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'' of elements of ''R'' such that ''r''₁ is a not a zero-divisor on ''M'' and ''r''_''i'' is a not a zero-divisor on ''M''/(''r''₁, ..., ''r''_''i''−1)''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''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 {''r''₁=0, ..., ''r''_''d''=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. Geometrically, in ''xyz''-space '''C'''³, successively intersecting the varieties V(''x''), V(''y''(1-''x'')), V(''z''(1-''x'')) gives the plane (''x'' = 0), then the line (''x = y ='' 0), and finally the point (''x = y = z ='' 0), decreasing dimension by 1 at each step. However, successively intersecting V(''y''(1-''x'')), V(''z''(1-''x'')), V(''x'') gives: the union of the planes (''y ='' 0) and (''x ='' 1); then the union of the ''x''-axis (''y = z ='' 0) and the plane (''x ='' 1); and finally the point (''x = y = z ='' 0). The second step contains a plane, failing to decrease dimension, and indeed ''z''(1-''x'') is a zero-divisor in the ring '''C'''[''x'',''y'',''z'']/(''y''(1-''x'')) since ''z''(1-''x''), ''y'' ≠ 0 but ''z''(1-''x'')''y'' = 0.

However, 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. Indeed, in the example above, the failure of regularity occurred because of an extra plane far away from the eventual intersection point (''x = y = z ='' 0): this could not happen in a local ring, whose ideals see only the neighborhood of the intersection point.

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 ∞,<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 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'''_(''p'') 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'''_(''p''), and the quotient ring of '''Z'''_(''p'') 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'''_(''p'') has depth 1. *For any field ''k'', the elements ''x''₁, ..., ''x''_''n'' in the polynomial ring ''A'' = ''k''[''x''₁, ..., ''x''_''n''] form a regular sequence. It follows that the localization ''R'' of ''A'' at the maximal ideal ''m'' = (''x''₁, ..., ''x''_''n'') 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''₁, ..., ''r''_''d'' of ''m'' which map to a basis for ''m''/''m''² 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'''. 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''₁, ..., ''r''_''d'' is a regular sequence in a ring ''R'', then the Koszul complex is an explicit free resolution of ''R''/(''r''₁, ..., ''r''_''d'') 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''₁, ..., ''r''_''d''], 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''₁, ..., ''x''_''d'']. In geometric terms, it follows that a 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 | access-date=2013-04-11 | archive-date=2012-07-13 | archive-url=https://web.archive.org/web/20120713043750/http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1964__20__5_0 | url-status=dead }}

Category:Commutative algebra Category:Dimension