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.
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<sub>1</sub>, ..., r<sub>d</sub> of elements of R such that r<sub>1</sub> is a not a zero-divisor on M and r<sub>i</sub> is a not a zero-divisor on M/(r<sub>1</sub>, ..., r<sub>iâÂÂ1</sub>)M for i = 2, ..., d. Some authors also require that M/(r<sub>1</sub>, ..., r<sub>d</sub>)M is not zero. Intuitively, to say that r<sub>1</sub>, ..., r<sub>d</sub> 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<sub>1</sub>)M, to M/(r<sub>1</sub>, r<sub>2</sub>)M, and so on.
An R-regular sequence is called simply a regular sequence. That is, r<sub>1</sub>, ..., r<sub>d</sub> is a regular sequence if r<sub>1</sub> is a non-zero-divisor in R, r<sub>2</sub> is a non-zero-divisor in the ring R/(r<sub>1</sub>), and so on. In geometric language, if X is an affine scheme and r<sub>1</sub>, ..., r<sub>d</sub> is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r<sub>1</sub>=0, ..., r<sub>d</sub>=0} â 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<sup>3</sup>, 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 â 0 but z(1-x)y = 0.
However, if R is a Noetherian local ring and the elements r<sub>i</sub> are in the maximal ideal, or if R is a graded ring and the r<sub>i</sub> 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<sub>R</sub>(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<sub>R</sub>(M) or just depth(M), means depth<sub>R</sub>(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 âÂÂ, 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).
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.
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.
In the special case where R is the polynomial ring k[r<sub>1</sub>, ..., r<sub>d</sub>], this gives a resolution of k as an R-module.
is isomorphic to the polynomial ring (R/I)[x<sub>1</sub>, ..., x<sub>d</sub>]. 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.