Regular sequence

Regular sequences play a role in commutative algebra and algebraic geometry . They are needed to define the depth of a module and Cohen-Macaulay rings and to make statements about complete averages.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

Definitions

Regular sequence

If a Noetherian module is over a ring , then an element is called -regular if it always follows for a . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle a\in R}$ ${\displaystyle M}$ ${\displaystyle a\cdot x=0}$ ${\displaystyle x\in M}$ ${\displaystyle x=0}$

A sequence of elements from is called -regular sequence if the following conditions are met: ${\displaystyle (a_{1},\dotsc ,a_{n})}$ ${\displaystyle R}$ ${\displaystyle M}$

• ${\displaystyle M\neq (a_{1},\dotsc ,a_{n})M}$
• For the picture of no zero divisor in ${\displaystyle i<n}$ ${\displaystyle a_{i+1}}$ ${\displaystyle M/(a_{1},\dotsc ,a_{i})M}$

The addition “ -” is omitted if it is clear from the context which module is meant. ${\displaystyle M}$

The special case when there is a local ring and the module is itself is most important. In this case, all terms in the sequence are in the maximum ideal. ${\displaystyle R}$ ${\displaystyle R}$

Regular parameter system

If local and the maximal ideal, then a minimal generating system of is called a regular parameter system . ${\displaystyle R}$ ${\displaystyle m}$ ${\displaystyle m}$

properties

• A maximal -regular sequence is finite and all maximal -regular sequences have the same length. ${\displaystyle M}$ ${\displaystyle M}$
• If a finite module is over a Noetherian local ring and is a regular sequence, then: ${\displaystyle M\neq 0}$ ${\displaystyle (a_{1},\dots ,a_{n})}$

${\displaystyle \mathrm {dim} (M/(a_{1}M+\dots a_{n}M))=\mathrm {dim} (M)-n}$

( is the dimension of .) ${\displaystyle \mathrm {dim} (M)}$ ${\displaystyle M}$

• For a regular ring local with maximal ideal and is equivalent: ${\displaystyle R}$ ${\displaystyle m}$ ${\displaystyle \{a_{1},\dots ,a_{n}\}\subset R}$

${\displaystyle (a_{1},\dots ,a_{n})}$ is part of a regular parameter system

${\displaystyle \{{\bar {a}}_{1},\dots ,{\bar {a}}_{n}\}}$(modulo ) is a linearly independent subset of the vector space over the body . ${\displaystyle (m^{2})}$ ${\displaystyle m/m^{2}}$ ${\displaystyle R/m}$

In particular, a minimal generating system of is a regular sequence. ${\displaystyle m}$

• Conversely, if a Noetherian local ring with maximal ideal , which is generated by a regular sequence of length , then is regular and . ${\displaystyle R}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle \mathrm {dim} (R)=n}$

• General: If a Noetherian local ring and a regular sequence, then every permutation of the sequence is regular. (This does not apply to any Noetherian rings.) ${\displaystyle R}$ ${\displaystyle (a_{1},\dots ,a_{n})}$

Examples

• In the polynomial ring over a body , every sequence of the variables is a regular sequence. ${\displaystyle R=K[X_{1},\dots ,X_{n}]}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle R}$
• The local ring body corresponds geometrically to the intersection of two affine surfaces in four-dimensional space. The ring is two-dimensional, but regular sequences have the length 1, since the ring only contains zero divisors and units modulo a non-zero divisor that is not a unit. In particular, this ring is not a Cohen-Macaulay ring. ${\displaystyle K[X_{1},X_{2},X_{3},X_{4}]/((X_{1},X_{2})(X_{3},X_{4}))_{({\bar {X}}_{1},{\bar {X}}_{2},{\bar {X}}_{3},{\bar {X}}_{4})}}$ ${\displaystyle K}$

literature

• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Robin Hartshorne : Algebraic Geometry . Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9