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.
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.[1] 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 ⊂ 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 ∞,[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]
R
f\inR
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
(xy,x2)
C[x,y]
⋅ x2:
| C[x,y] | |
| (xy) |
\to
| C[x,y] | |
| (xy) |
(y)\subsetC[x,y]/(xy)
0 → R\binom{d{d}} → … → R\binom{d{1}} → R → R/(r1,\ldots,rd) → 0
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.
⊕ j\geqIj/Ij+1
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.