Regular Sequence (algebra)

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 not a zero-divisor on ''M''/(''r''₁, ..., ''r''_''i''-1)''M'' for ''i'' = 1, ..., ''d''. 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''.

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 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 ∞, 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.

Examples

*Given an integral domain $R$ any nonzero $f \in R$ 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 $(xy,x^2)$ of elements in \mathbb ,y/math> since
:$\cdot x^2 : \frac \to \frac$
has a non-trivial kernel given by the ideal (y) \subset \mathbb ,y(xy) . 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:

:$0\rightarrow R^ \rightarrow\cdots \rightarrow R^ \rightarrow R \rightarrow R/(r_1,\ldots,r_d) \rightarrow 0$

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

:$\oplus_ I^j/I^$

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

References

*
*
* Winfried Bruns; Jürgen Herzog, ''Cohen-Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.
* David Eisenbud, ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150.
*{{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

Commutative algebra
Dimension