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 M-regular sequence is a sequence

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.

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 $R$ 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.

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

[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.

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