Regular sequence (original) (raw)

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Well-behaved sequence in a commutative ring

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.

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. [1] 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 ∞,[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]

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 ( x y , x 2 ) {\displaystyle (xy,x^{2})} {\displaystyle (xy,x^{2})} of elements in C [ x , y ] {\displaystyle \mathbb {C} [x,y]} {\displaystyle \mathbb {C} [x,y]} since

⋅ x 2 : C [ x , y ] ( x y ) → C [ x , y ] ( x y ) {\displaystyle \cdot x^{2}:{\frac {\mathbb {C} [x,y]}{(xy)}}\to {\frac {\mathbb {C} [x,y]}{(xy)}}} {\displaystyle \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 ) ⊂ C [ x , y ] / ( x y ) {\displaystyle (y)\subset \mathbb {C} [x,y]/(xy)} {\displaystyle (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.

0 → R ( d d ) → ⋯ → R ( d 1 ) → R → R / ( r 1 , … , r d ) → 0 {\displaystyle 0\rightarrow R^{\binom {d}{d}}\rightarrow \cdots \rightarrow R^{\binom {d}{1}}\rightarrow R\rightarrow R/(r_{1},\ldots ,r_{d})\rightarrow 0} {\displaystyle 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.

⊕ j ≥ 0 I j / I j + 1 {\displaystyle \oplus _{j\geq 0}I^{j}/I^{j+1}} {\displaystyle \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.

  1. ^ N. Bourbaki. Algèbre. Chapitre 10. Algèbre Homologique. Springer-Verlag (2006). X.9.6.
  2. ^ A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.
  3. ^ N. Bourbaki. Algèbre Commutative. Chapitre 10. Springer-Verlag (2007). Th. X.4.2.