Every Regular Local Ring is Cohen-Macaulay

Throughout, let be a commutative Noetherian local ring with maximal ideal and residue field .

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 ). 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 , written as , is the supremum taking over the length of prime ideal chains

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 is the dimension of a vector space The right hand side is the dimension of a -vector space .

Let be the local ring of a complex variety at a point , in other words we write . Then is the Zariski tangent space of at , whose dimension equals . The embedding dimension of is the smallest integer such that some analytic neighbourhood of in embeds into . If this dimension equals the dimension of , then is "smooth" at . For this reason we define regular local ring.

Definition 3. The ring is called regular if .

The most immediate intuitive example of regular local ring has to be rings of the form where is a field. These kind of rings are regular local rings of Krull dimension . As one would imagine, this ring contains much more information than . Power series in complex analysis is much more powerful than polynomials.

But by working on regular local rings, we are not essentially restricting ourselves into the ring of power series over a field. For example, the ring is also a regular local ring, but it does not even contain a field.

Nevertheless, our primary model of regular local rings is still a ring of the form , which has a maximal ideal . To study local rings in the flavour of , we develop an analogy of elements .

Definition 4. A regular sequence of , also written as -sequence, is a sequence of elements in such that is a non-zero-divisor in , and such that given , each is a non-zero-divisor in .

The grade of , , is the longest length of regular sequences. If , then is called Cohen-Macaulay.

It is quite intuitive that, for , the longest -sequence has to be , and therefore is Cohen-Macaulay. But such an argument does not bring us to the conclusion that quick. We will show later, anyway, every regular local ring is a Cohen-Macaulay ring.

The Sequence That Forms a Basis

Amongst many sequences, we are in particular interested in the sequence that are mapped onto a basis of the -vector space . We will show later that this "regular" sequence is indeed the regular sequence.

Proposition 1. Let be elements in whose images form a basis of , then generate the maximal ideal .

Proof. Nakayama's lemma (8). Notice that as is local, the Jacobson radical is . Besides, we take .

Proposition 2. If is a regular local ring of dimension and map to a basis of , then is a regular local ring of dimension .

Proof. By proposition 1, we have . The dimension of is determined by the chain in : which has length . That is, . On the other hand, the maximal ideal in is isomorphic to , and map to a basis of , which consequently has dimension .

It looks quite promising now that the sequence of basis can get everything down to earth, and we will show that in the following section.

Regular Local Rings Are Integral Domains and Cohen-Macaulay

Proposition 3. If is regular, then is an integral domain.

Proof. We use induction on . When and is regular, has to be a field, hence an integral domain by definition. Next we assume that and the argument has been proved for .

Pick . Then this element map to a nonzero element in . There exists a basis of that contains . Therefore by proposition 2, is a regular local ring of dimension , which is an integral domain by assumption. It follows that is prime.

We claim that there exists such that has height . If not, then for all , is a minimal. It follows that there exists finitely many minimal prime ideals such that and consequently for some . It follows that , contradicting our assumption that . [Note: the prime avoidance allows at most two ideals to be non-prime. See P. 90 of Eisenbud's Commutative Algebra, with a View Toward Algebraic Geometry.]

Thus, as our claim is true, we can write with prime and . We see for all because if , then and therefore we write or equivalently . When this is the case, we have . Therefore is an integral domain.

We now reach our conclusion of this post.

Proposition 4. If is regular and of Krull dimension , any mapping to a basis of gives rise to a regular sequence (-sequence). Hence and therefore is Cohen-Macaulay.

Proof. As , once we have shown that is a regular sequence, we have . To show it being a regular sequence, first of all notice that is non-zero-divisor (because is an integral domain). For any , we see is a regular local ring of dimension , hence again an integral domain. Therefore are non-zero-divisors.

Reference / Further Readings

• Charles A. Weibel, An Introduction to Homological Algebra.
• M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra.
• David Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry.
• Winfred Bruns, Jürgen Herzog, Cohen-Macaulay Rings