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.



How to think about CM rings?
"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](https://doi.org/10.1090/S0002-9904-1978-14531-5).


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.


