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 $\mathrm{Spec}\ S R \to \mathrm{Spec}\ R$ 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$without all those higher$\mathrm{Tor}$s that Serre had to add to the definition. He gives lots of examples and explains "where Cohen-Macaulayness comes from" (or doesn't) in each one. The whole thing is eminently readable and highly recommended. 1 "Life is really worth living in a Noetherian ring R when all the local rings have the property that every s.o.p. is an R-sequence. Such a ring is called Cohen-Macaulay (C-M for short).": Hochster, 1978 Section 3 of that paper is devoted to explaining what it "really means" to be Cohen-Macaulay. It begins with a long subsection on invariant theory, but then gets to some algebraic geometry that will interest you. In particular, he points out that if$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$-module. Equivalently, the scheme-theoretic fibers of the finite morphism$\mathrm{Spec}\ S \to \mathrm{Spec}\ R$all have the same length. At the end of section 3, Hochster explains that the CM condition is exactly what is required to make intersection multiplicity "work correctly": If$X$and$Y$are CM, then you can compute the intersection multiplicity of$X$and$Y$without all those higher$\mathrm{Tor}\$s that Serre had to add to the definition.