Cohen Macaulay Ring

In the mathematical sub-area of commutative algebra , a Cohen-Macaulay ring is understood to be a Noetherian ring , which is no longer necessarily regular, but whose depth is equal to its Krull dimension . A Cohen-Macaulay singularity is a singularity whose local ring is a Cohen-Macaulay ring. The rings were named after Irvin Cohen and Francis Macaulay .

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 module is above a ring , an element is called regular if it always follows from for a . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle a\in R}$ ${\displaystyle a\cdot x=0}$ ${\displaystyle x\in M}$ ${\displaystyle x=0}$

A sequence of elements 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 ${\displaystyle i<m}$ ${\displaystyle a_{i+1}}$ ${\displaystyle M/(a_{1},\dotsc ,a_{i})M}$

Depth of a module

If a module is above a ring , then the depth of is the thickness of a maximum regular sequence of elements . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \mathrm {t} (M)}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle R}$

Dimension of a module

The dimension of a module over a ring is defined as the Krull dimension of . ( is the annihilator of M.) ${\displaystyle \mathrm {dim} (M)}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle R/\mathrm {Ann} (M)}$ ${\displaystyle \mathrm {Ann} (M)}$

If a finitely generated module is over a Noetherian ring, then: ${\displaystyle M}$

${\displaystyle \mathrm {dim} (M)=\mathrm {max} \{\mathrm {dim} (R/p)|p\in \mathrm {Ass} (M)\}=\mathrm {max} \{\mathrm {dim} (R/p)|p\in \mathrm {Supp} (M)\}}$

(For notation: denotes the set of prime ideals to be associated , the carrier of the module .) ${\displaystyle \mathrm {Ass} (M)}$ ${\displaystyle M}$ ${\displaystyle \mathrm {Supp} (M)}$

For a finitely generated module over a Noetherian local ring, the following applies: ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle \mathrm {t} (M_{m})\leq \mathrm {min} \{\mathrm {dim} (R/p)|p\in \mathrm {Ass} (M)\}\leq \mathrm {dim} (M)}$

Cohen-Macaulay

A finitely generated module over a Noetherian ring is called a Cohen-Macaulay module if for all maximal ideals of : ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle m}$ ${\displaystyle R}$

${\displaystyle \mathrm {t} (M_{m})=\mathrm {dim} (M_{m})}$

${\displaystyle R}$is called a Cohen-Macaulay ring if the module is a Cohen-Macaulay module. ${\displaystyle R}$ ${\displaystyle R}$

Cohen-Macaulay rings

• Each location of a Cohen-Macaulay ring is a Cohen-Macaulay ring.
• Every 0-dimensional Noetherian ring is a Cohen-Macaulay ring.
• Every reduced Noetherian one-dimensional ring is a Cohen-Macaulay ring.
• Every regular Noetherian ring is a Cohen-Macaulay ring.
• Every Gorenstein ring is a Cohen-Macaulay ring.
• Every Cohen-Macaulay ring is a chain ring .

Examples

• If there is a body , the variety consisting of the X-axis and the Y-axis is described by the coordinate ring . ${\displaystyle K}$ ${\displaystyle K[x,y]/(x\cdot y)}$

The intersection is through the ring

${\displaystyle R=(K[x,y]/(x\cdot y))_{(x,y)}}$

described. It is a singularity because it is one-dimensional, but the maximum ideal of can only be generated by two elements. On the other hand, there is a Cohen-Macaulay ring (even Gorenstein), since the maximal ideal doesn't just contain zero divisors. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

• A more complicated singularity is in the ring

${\displaystyle K[x,y]/(x^{2},x\cdot y)}$

The local ring associated with the singularity

${\displaystyle (K[x,y]/(x^{2},x\cdot y))_{(x,y)}}$

is not a Cohen-Macaulay ring. It is one-dimensional, but the maximum ideal consists only of zero divisors, so there is no regular sequence.

literature

• 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 0-201-00361-9 .
• Robin Hartshorne : Algebraic Geometry . Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9 .
• W. Bruns, J. Herzog: Cohen-Macaulayrings. Cambridge University Press, 1993, ISBN 0-521-56674-6 .