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.

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 r 1 , ..., r d of elements of R such that r 1 is a not a zero-divisor on M and r i is a not a zero-divisor on M/(r 1 , ..., r i−1 )M for i = 2, ..., d. Some authors also require that M/(r 1 , ..., r d )M is not zero. Intuitively, to say that r 1 , ..., 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/(r 1 )M, to M/(r 1 , r 2 )M, and so on. An R-regular sequence is called simply a regular sequence. That is, r 1 , ..., r d is a regular sequence if r 1 is a non-zero-divisor in R, r 2 is a non-zero-divisor in the ring R/(r 1 ), and so on. In geometric language, if X is an affine scheme and r 1 , ..., r d is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r 1 =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. Geometrically, in xyz-space C 3 , 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 ≠ 0 but z(1-x)y = 0. However, 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. 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 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 ∞, 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 x 1 , ..., x n in the polynomial ring A = k[x 1 , ..., x n ] form a regular sequence. It follows that the localization R of A at the maximal ideal m = (x 1 , ..., 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 r 1 , ..., r d of m which map to a basis for m/m 2 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. ==Applications== If r 1 , ..., r d is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r 1 , ..., 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[r 1 , ..., 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)[x 1 , ..., 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. ==See also== Complete intersection ring Koszul complex Depth (ring theory) Cohen-Macaulay ring ==Notes==

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