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


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.



Why Cohen-Macaulay rings have become important in commutative algebra?
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](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183541144&page=record) ]


