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
such that "r""i" is a not a zero-divisor on "M"/("r"1, ..., "r""i"-1)"M" for "i" = 1, ..., "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. 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.
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.
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 formula_3 of elements in formula_4 since
has a non-trivial kernel given by the ideal formula_6 . 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"["r"1, ..., "r""d"], this gives a resolution of "k" as an "R"-module.
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.