## Return to Answer

2 fixed inadvertent swap of Spec R and Spec S in 3rd para.

"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 $\mathrm{Spec}\ S R \to \mathrm{Spec}\ R$ 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.

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.
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