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, ..., rd in R

such that ri is a not a zero-divisor on M/(r1, ..., ri-1)M for i = 1, ..., d. Some authors also require that M/(r1, ..., rd)M is not zero. Intuitively, to say that r1, ..., 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/(r1)M, to M/(r1, r2)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/(r1), 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 {r1=0, ..., rd=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 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(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 depthR(M) or just depth(M), means depthR(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 ∞, 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).

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, ..., xn in the polynomial ring A = k[x1, ..., xn] form a regular sequence. It follows that the localization R of A at the maximal ideal m = (x1, ..., xn) 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 $$(xy,x^2)$$ 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, ..., rd is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r1, ..., rd) 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, ..., rd], 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, ..., xd]. 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.