# 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

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

**is a sequence**

*M*-regular 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*.[1] 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. 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 ∞,[2] 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*).[3]

## Examples

- Given an integral domain any nonzero 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 of elements in since

has a non-trivial kernel given by the ideal . 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:

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

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.

## Notes

- N. Bourbaki.
*Algèbre. Chapitre 10. Algèbre Homologique.*Springer-Verlag (2006). X.9.6. - A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.
- N. Bourbaki.
*Algèbre Commutative. Chapitre 10.*Springer-Verlag (2007). Th. X.4.2.

## References

- Bourbaki, Nicolas (2006),
*Algèbre. Chapitre 10. Algèbre Homologique*, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-34493-3, ISBN 978-3-540-34492-6, MR 2327161 - Bourbaki, Nicolas (2007),
*Algèbre Commutative. Chapitre 10*, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-34395-0, ISBN 978-3-540-34394-3, MR 2333539 - 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 - Grothendieck, Alexander (1964), "Éléments de géometrie algébrique IV. Première partie",
*Publications Mathématiques de l'Institut des Hautes Études Scientifiques*,**20**: 1–259, MR 0173675