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.

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

such that r_i is a non-zero-divisor on M/(r₁, ..., r_i-1)M for i = 1, ..., d. Some authors also require that M/(r₁, ..., r_d)M is not zero. Intuitively, to say that r₁, ..., 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/(r₁)M, to M/(r₁, r₂)M, and so on.

An R-regular sequence is called simply a regular sequence. That is, r₁, ..., r_d 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, ..., r_d is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r₁=0, ..., r_d=0} ⊂ X is a complete intersection subscheme of X.

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 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.

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