Gorensteinring

A Gorenstein ring is a ring that is studied in commutative algebra , a branch of mathematics. A Gorenstein ring is a Cohen-Macaulay ring with certain additional properties. A Gorenstein singularity is a singularity whose local ring is a Gorenstein ring .

The rings were named after Daniel Gorenstein , although the latter always claimed that he didn't even understand the definition.

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 .

definition

If a Noetherian local - dimensional ring with maximum ideal , then a set is called a parameter system of , if this set generates a - primary ideal . (One can show that a Noetherian local ring always has a parameter system.) ${\displaystyle R}$ ${\displaystyle d}$ ${\displaystyle m}$ ${\displaystyle \{a_{1},\dots ,a_{d}\}}$ ${\displaystyle R}$ ${\displaystyle m}$

Is

${\displaystyle R}$a local Cohen-Macaulay ring with maximum ideal , ${\displaystyle m}$

${\displaystyle \{a_{1},\dots ,a_{n}\}}$ a parameter system and

${\displaystyle q=(a_{1},\dots ,a_{n})}$the corresponding -primary ideal, ${\displaystyle m}$

so is the number

${\displaystyle r:=\mathrm {dim} _{R/m}\{{\bar {r}}\in R/q|m\cdot r=0\}}$

regardless of the selected parameter system.

This number is called the type of . ${\displaystyle r}$ ${\displaystyle R}$

A local Gorenstein ring is a Type 1 Cohen-Macaulay ring.

A Noetherian ring R is called a Gorenstein ring if all of its locations of maximum ideals are local Gorenstein rings.

(This definition follows Kunz 1980. A Gorenstein ring is often defined via the injective dimension, see below.)

properties

• If a local Cohen-Macaulay ring is then a Gorenstein ring if and only if the ideal generated by a parameter system is irreducible . ${\displaystyle R}$ ${\displaystyle R}$
• A local Noetherian ring is a Gorenstein ring if and only if its injective dimension is finite.
• Any local ring that is whole intersection is a Gorenstein ring . In particular, every regular local ring is a Gorenstein ring.

Examples

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

The intersection is through the ring

${\displaystyle R=(K[X,Y]/(XY))_{(X,Y)}}$

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 Gorenstein ring, since every regular element contained in the maximal ideal creates an irruducible sub-variety. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

• The ring is a -dimensional local ring. He is therefore Cohen-Macaulay. But he is not Gorenstein, since the null ideal is -primary, but not irreducible, since it is the intersection of the ideals and . ${\displaystyle K[X,Y]/(X^{2},Y^{2},X\cdot Y)}$ ${\displaystyle 0}$ ${\displaystyle m}$ ${\displaystyle (X)}$ ${\displaystyle (Y)}$

literature

• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, 1980, ISBN 3-528-07246-6 .
• Michael Francis Atiyah , Ian Macdonald : Introduction to Commutative Algebra. Addison-Wesley, 1969, ISBN 0-201-00361-9 .
• Rainer Brüske, Friedrich Ischebeck, Ferdinand Vogel: Commutative Algebra. Bibliographical Institute, 1989, ISBN 3-411-14041-0 .
• Hideyuki Matsumura: Commutative algebra. Cummings, 1980, ISBN 0-8053-7026-9 .

Individual evidence

1. ^ D. Eisenbud: Commutative Algebra. Springer, 2004, ISBN 0-387-94269-6 , p. 530.