Throughout, let $R$ be a commutative Noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k=R/\mathfrak{m}$.

Introduction

The notion of Cohen-Macaulay ring is sufficiently general to a wealth of examples in algebraic geometry, invariance theory and combinatorics; meanwhile it is sufficiently strict to allow a rich theory. The notion of Cohen-Macaulay is a workhorse of commutative algebra. In this post, we discover an important subclass of Cohen-Macaulay ring - regular local rings (one would be thinking about $k[[x_1,\dots,x_n]]$). See also “Why Cohen-Macaulay rings have become important in commutative algebra?” on MathOverflow.

It is recommended to be familiar with basic commutative algebra tools such as Nakayama’s lemma and minimal prime ideals.

The content can be generalised to modules to a good extent, but we are not doing it for sake of quick accessibility.

Embedding Dimension, Krull Dimension and Grade

Definition 1. The Krull dimension of $R$, written as $\dim{R}$, is the supremum taking over the length of prime ideal chains

$$\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \dots \subsetneq \mathfrak{p}_d.$$

This definition was introduced to define dimension of affine varieties, in a global sense. Locally, we have the following definition.

Definition 2. The embedding dimension of $R$ is the dimension of a vector...