# Why Cohen-Macaulay rings have become important in commutative algebra?

I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects.

I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm studying about Stanley-Reisner rings and I think I need to have a better general understanding about why I need to study CM rings.

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I don't know about historically, but one major reason in modern days is geometry. It turns out that if you form a "good" moduli space (insert some stability notion) of smooth varieties of some type, then they degenerate at the boundary to some singular varieties. It turns out that these will have at worst Cohen-Macaulay singularities, so understanding Cohen-Macaulay rings is extremely important in the study of classifying smooth varieties. – Matt Jul 30 '13 at 14:01
You said you're studying Stanley-Reisner rings, so I assume you've read this, but this seminal paper does immediately demonstrate the relevance of Cohen-Macaulay rings to combinatorial commutative algebra: dedekind.mit.edu/~rstan/pubs/pubfiles/27.pdf – Sam Hopkins Jul 30 '13 at 17:04
Serre's FAC refers to theorem of Cohen-Macaulay in Samuel's Alegre Locale(1953). It states that a system of parameters of a regular local ring is a regular sequence. Zariski-Samuel's book(1958) defines Cohen-Macaulay local ring(they call it Macaulay ring). – Makoto Kato Jul 30 '13 at 18:32
Wikipedia: They are named for Francis Sowerby Macaulay (1916), who proved the unmixedness theorem for polynomial rings, and for Cohen (1946), who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property. – Makoto Kato Jul 30 '13 at 21:22

I think there are many reasons. Here are a few.

# Practical reasons

Cohen-Macaulay rings are just plain easier to work with.

### Computations in local cohomology

For example, any number of computations in local cohomology modules become much easier in the Cohen-Macaulay case (see for example Bruns and Herzog's book on the topic). Explicitly, it's much easier to determine if a class in $z \in H^{\dim R}_{\mathfrak{m}}(R)$ is zero or not in the case that $R$ is Cohen-Macaulay.

### Duality

Both Grothendieck-local and Grothendieck-Serre duality work much better in Cohen-Macaulay rings. The dualizing complex (assuming it exists) is a complex whose first non-zero cohomology is the canonical module and which is equal to this (shifted) canonical module if and only if the ring is Cohen-Macaulay. Without this hypothesis one frequently needs to work in the derived category and do numerous computations with spectral sequences. It is convenient to not have to.

### Vanishing and exactness

If $R$ is Cohen-Macaulay and $I$ is a height-one ideal (and suppose the rings are quotients of Gorenstein/regular rings so they have dualizing complexes). Then we have a surjection of canonical modules $\text{Hom}_R(I, \omega_R) = \omega_R(I^{-1}) \to \omega_{R/I}$. This is surjective because the next term is zero when $R$ is Cohen-Macaulay. This sort of vanishing applies to more general situations and is really useful (there is a local dual version involving local cohomology). There are lots of other vanishing results that you can deduce from this kind too.

# Ubiquity of Cohen-Macaulay rings

Ok, if Cohen-Macaulay rings weren't so common, the above nice properties would be less interesting. But Cohen-Macaulay rings are really common. Here are some examples.

### Summands of regular (or Cohen-Macaulay) rings

If $R \subseteq S$ is a extension of rings and $R \to S$ splits as a map of $R$-modules, then if $S$ is Cohen-Macaulay, so is $R$ (the point is $H^i_m(R) \to H^i_{mS}(S)$ injects and the latter term is zero, at least after a little localization on $S$ if necessary). Lots of rings coming from representation theory for instance are summands of regular rings.

### Complete intersections

Complete intersection rings are Cohen-Macaulay.

### Rational/log terminal/F-regular singularities

A lot of classes of singularities which are most useful today are Cohen-Macaulay. One of their most useful properties is their vanishing properties (see above).

# Pithy quotes

"Life is really worth living in a Noetherian ring $R$ when all the local rings have the property that every s.o.p. is an $R$-sequence. Such a ring is called Cohen-Macaulay (C-M for short)."

[Page 887 of Hochster, Some applications of the Frobenius in characteristic 0 ]

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It would be nice to know who introduced this terminology and when (Rees, Serre, Auslander-Buchsbaum, or earlier)? – user26857 Aug 3 '13 at 13:12

I'm no expert on the evolution of Cohen-Macaulay rings, so I will leave that part of your question for those who actually know their history.

On a high level, Cohen-Macaulay rings are wonderful precisely because they lie within the intersection of algebraic geometry, algebraic topology, combinatorics, commutative algebra, and probably another four or five active research fields which I lack the perspective to mention. As such, there are plenty of "hooks" for people with diverse backgrounds to use what they know to prove something cool in a seemingly disparate field. I haven't touched commutative algebra with a ten-foot pole in my own research, but hey: I understand link conditions on simplicial complexes, so I have something to latch on to when I read about C-M rings.

For a precise sequence of as many as 9 answers to your specific question that were relevant as early as 1978, I would recommend the introduction of this paper by Melvin Hochster. It is called "Cohen Macaulay Rings and their Modules".

Update (Nov 24, 2014): A fairly comprehensive introduction to the importance of Cohen-Macaulay complexes can now also be found in Anders Björner's recent article dedicated to Richard Stanley on his 70th birthday.

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I think apart from commutative algebra, one main importance of the Cohen-Macauley is in Geometry. CM rings lie between the regular rings and other types of (bad) singularities. In a sense CM singularities are the best singularities and are much easier to deal with. One of the simplest properties of CM singularities is for example that they are equidimensional.

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Richard Stanley's description on How the Upper Bound Conjecture was Proved tells much about the history of how Cohen-Macaulay rings became important in combinatorics. Additional links can be found here.

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