nLab Cohen-Macaulay ring

Idea

A property of Noetherian local commutative rings to be Cohen-Macaulay is a weak version of a smoothness condition.

Definition

A local commutative ring is Cohen-Macaulay iff it is a Cohen-Macaulay module as a module on itself.

A variety is Cohen-Macaulay if for all points $x$ the corresponding local ring is Cohen-Macauley, that is, its Krull dimension is equal to its depth.

Relation to other classes

Every regular local ring is Cohen-Macauley and, more generally, every complete intersection ring $\mathcal{O}_x$ is Cohen-Macauley.

References

• Wikipedia, Cohen-Macaulay ring
• Winfried Bruns, Jürgen Herzog, Cohen-Macaulay rings, Cambridge University Press (1993)
• Ch. 18 of David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150

category: algebra