Regular sequence

Last updated September 27, 2019

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.

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers $; and p -adic integers .$

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pⁿ, there should exist n − m homogeneous polynomials

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

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module. A module taking its scalars from a ring R is called an R-module.

r₁, ..., r_d in R

such that r_i is a not a zero-divisor on M/(r₁, ..., r_i-1)M for i = 1, ..., d.^[1] Some authors also require that M/(r₁, ..., r_d)M is not zero. Intuitively, to say that r₁, ..., 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/(r₁)M, to M/(r₁, r₂)M, and so on.

An R-regular sequence is called simply a regular sequence. That is, r₁, ..., r_d is a regular sequence if r₁ is a non-zero-divisor in R, r₂ is a non-zero-divisor in the ring R/(r₁), and so on. In geometric language, if X is an affine scheme and r₁, ..., r_d is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r₁=0, ..., r_d=0} ⊂ X is a complete intersection subscheme of X.

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by $, is the set of all prime ideals of R . It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme .$

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.

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals, which means there is no infinite ascending sequence of left ideals; that is, given any chain of left ideals,

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

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.

In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary property of depth is the inequality

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 $R$ 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 x₁, ..., x_n in the polynomial ring A = k[x₁, ..., x_n] form a regular sequence. It follows that the localization R of A at the maximal ideal m = (x₁, ..., 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 r₁, ..., r_d of m which map to a basis for m/m² 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 $\mathbb {C} [x,y]$ since

\cdot x^{2}:{\frac {\mathbb {C} [x,y]}{(xy)}}\to {\frac {\mathbb {C} [x,y]}{(xy)}}

has a non-trivial kernel given by the ideal $(y)\subset \mathbb {C} [x,y]/(xy)$ . 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 r₁, ..., r_d is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r₁, ..., r_d) as an R-module, of the form:

0\rightarrow R^{\binom {d}{d}}\rightarrow \cdots \rightarrow R^{\binom {d}{1}}\rightarrow R\rightarrow R/(r_{1},\ldots ,r_{d})\rightarrow 0

In the special case where R is the polynomial ring k[r₁, ..., 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 _{j\geq 0}I^{j}/I^{j+1}

is isomorphic to the polynomial ring (R/I)[x₁, ..., 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.

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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

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[1] N. Bourbaki. Algèbre. Chapitre 10. Algèbre Homologique. Springer-Verlag (2006). X.9.6.

[2] A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.

[3] N. Bourbaki. Algèbre Commutative. Chapitre 10. Springer-Verlag (2007). Th. X.4.2.