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 r1, ..., rd in R such that ri is a not a zero-divisor on M/(r1, ..., ri-1)M for i = 1, ..., d. 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 {r1=0, ..., rd=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 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.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (4)

Related people (1)

Related concepts (5)

Related lectures (2)

Hilbert's syzygy theorem

In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings.

Gorenstein ring

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring).

Dimension theory (algebra)

In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety).

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the

p

-adic pushforward of the Haar measure under a ...

EPFL2017

Chow-Witt groups and Grothendieck-Witt groups of regular schemes

Jean Fasel

Let A be a noetherian commutative Z[1/2]-algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck-Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d

2009

A vanishing theorem for oriented intersection multiplicities

Jean Fasel

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements ...

2008

Homogeneous Ideals and Regular Rings MATH-510: Algebraic geometry II - schemes and sheaves

Explores homogeneous ideals, regular rings, N-graded rings, and closed sets in projective geometry.

Spectral Decomposition MATH-111(c): Linear Algebra

Explores spectral and singular value decompositions of matrices.