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

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
r1,..., rd of elements of R such that r1 is a not a zero-divisor on M and ri is a not a zero-divisor on M/''M for i'' = 2,..., d. Some authors also require that M/''M is not zero. Intuitively, to say that
r''1,..., rd is an M-regular sequence means that these elements "cut M down" as much as possible, when we pass successively from M to M/''M, to M''/M, and so on.
An R-regular sequence is called simply a regular sequence. That is, r1,..., rd is a regular sequence if r1 is a non-zero-divisor in R, r2 is a non-zero-divisor in the ring R/, and so on. In geometric language, if X is an affine scheme and r1,..., rd 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, z is a regular sequence in the polynomial ring C, while y, z, x is not a regular sequence. But if R is a Noetherian local ring and the elements ri are in the maximal ideal, or if R is a graded ring and the ri 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 depthR or just depth, 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 depthR or just depth, means depthR; 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.

Examples

  • Given an integral domain any nonzero gives a regular sequence.
  • For a prime number p, the local ring Z 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, and the quotient ring of Z by the ideal generated by p is the field Z/. Therefore p cannot be extended to a longer regular sequence in the maximal ideal, and in fact the local ring Z has depth 1.
  • For any field k, the elements x1,..., xn in the polynomial ring A = k form a regular sequence. It follows that the localization R of A at the maximal ideal m = 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,..., rd 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 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

In the special case where R is the polynomial ring k, this gives a resolution of k as an R-module.
is isomorphic to the polynomial ring . 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.