How to think about CM rings?

Question

There are a few questions about CM rings and depth.

1. Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me. (currently I'm regarding it as a generalization of not-a-zero-divisor that's needed to carry out induction argument, e.g. as in $\operatorname{dim} \frac{M}{(a_1,\dotsc,a_n)M} = \operatorname{dim} M - n$ for $M$-regular sequence $a_1,\dotsc,a_n$; correct me if I'm wrong!) But I don't understand why the length of a maximal regular sequence is of interest. Is it merely due to some technical consideration in cohomology that we want many $\operatorname{Ext}$ groups to vanish?

2. What does CM rings mean geometrically? As I read from Eisenbud's book, there doesn't seem to be an exact geometric concept that corresponds to it. Nonetheless I would still like to know about any geometric intuition of CM rings. I know that it should be locally equidimensional. Some examples of CM rings come from complete intersection (I read this from wiki). But what else?

3. Why do we care about CM rings? If I understand it correctly, CM rings ⇔ unmixedness theorem holds for every ideal for a noetherian ring, which should mean every closed subschemes have equidimensional irreducible components (and there's no embedded components). This looks quite restrictive.

Answer 1

"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).": Hochster, "Some applications of the Frobenius in characteristic 0", 1978.

Section 3 of that paper is devoted to explaining what it "really means" to be Cohen–Macaulay. It begins with a long subsection on invariant theory, but then gets to some algebraic geometry that will interest you.

In particular, he points out that if $R$ is a standard graded algebra over a field, then it is a module-finite algebra over a polynomial subring $S$, and that $R$ is Cohen–Macaulay if and only if it is free as an $S$-module. Equivalently, the scheme-theoretic fibers of the finite morphism $\operatorname{Spec} R \to \operatorname{Spec} S$ all have the same length.

At the end of section 3, Hochster explains that the CM condition is exactly what is required to make intersection multiplicity "work correctly": If $X$ and $Y$ are CM, then you can compute the intersection multiplicity of $X$ and $Y$ without all those higher $\operatorname{Tor}$s that Serre had to add to the definition.

He gives lots of examples and explains "where Cohen–Macaulayness comes from" (or doesn't) in each one. The whole thing is eminently readable and highly recommended.

Answer 2

If I remember correctly, CM is equivalent to asking the dualizing complex (the generalization of the canonical line bundle) to be a sheaf, rather than a more general complex (while Gorenstein is asking for it to be in fact a line bundle). In other words, we're classifying singularities according to how reasonable a theory of volume forms they admit.

Answer 3

Geometrically, depth is measuring "dimension" via hypersurfaces. The set of zero-divisors is the union of the associated primes, so to say that an element x is a non-zero-divisor is to say that it is not contained in any associated prime. Thus the hypersurface $V(x)$ does not intersect $M$ in a component.

The other condition on a regular sequence is that $xM\neq M$, which amounts to saying that the hypersurface $V(x)$ must intersect $M$ somewhere.

So basically, you're cutting down $M$ by hypersurfaces. Since they can't intersect in a component, they actually do cut it down by some amount, and since they must intersect somewhere they aren't throwing everything away at once. This is a very loose description, but I don't have time right now to make it more precise.

If you look at quotients of polynomial rings you can actually see this at work. Here you can compute depth by drawing pictures: take an ideal $I$ in $k[x,y,z]$ say, and look at $V(I)$. Find some hypersurface (a plane in this case) that intersects $V(I)$ but not in a component. Then repeat on this intersection. Using this you should be able to find the classic example of a regular sequence that does not stay regular under permutation. You can also convince yourself that in a local ring, all permutations are regular.

In this interpretation, CM rings are exactly those for which the dimension can be measured by using hypersurfaces in this way.

I'm somewhat rushing to catch a flight (bad time to look at mathoverflow!), so there may be some mistakes but the idea is sound.

Answer 4

One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows:

Think first of the ring of symmetric polynomials in $n$ variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials.

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen-Macaulay comes close. A Cohen Macaulay ring can be described as a direct sum where each summand $S_i$ is of the form $\eta_i R$, where $R$ is a polynomial rings (whose variables are the elements of a system of parameters) and $\eta_i$ are elements. Being a direct sum is important here.

For graded rings such a description has remarkable combinatorial consequences.

Answer 5

One should care about CM rings and schemes for example because they have good duality properties; see for example Serre's duality theorem in Hartshorne III.7.

There are Serre's $S_n$ properties generalizing CM. $S_1$ means "no embedded components" (if X is reduced, this is automatic of course), and $S_2$ means "$S_1$ and X is saturated in codimension 2". Both of these properties have a clear geometric meaning. Now you can consider CM to be "this good, and even better".