So I’ve missed a few days of MaBloWriMo. But I do have a talk topic now (I was mistaken–it’s actually tomorrow)! I’ll be speaking about some applications of Sperner’s lemma. Notes will be up soon.

Today I want to talk about how depth (an “arithmetic” invariant) compares to dimension (a “geometric” invariant). It turns out that the geometric invariant wins out in size. When they turn out to be equal, then the relevant object is called Cohen-Macaulay. This is a condition I’d like to say more about in future posts.

0.5. Depth and dimension

Consider an ${R}$-module ${M}$, which is always assumed to be finitely generated. Let ${I \subset R}$ be an ideal with ${IM \neq M}$. We know that if ${x \in I}$ is a nonzerodivisor on ${M}$, then ${x}$ is part of a maximal ${M}$-sequence in ${I}$, which has length ${\mathrm{depth}_I M}$ necessarily. It follows that ${M/xM}$ has a ${M}$-sequence of length ${\mathrm{depth}_I M - 1}$ (because the initial ${x}$ is thrown out) which can be extended no further. In particular, we find

Proposition 15 Hypotheses as above, let ${x \in I}$ be a nonzerodivisor on ${M}$. Then$\displaystyle \mathrm{depth}_I (M/xM) = \mathrm{depth} M - 1.$

This is strikingly analogous to the dimension of the module ${M}$. Recall that ${\dim M}$ is defined to be the Krull dimension of the topological space ${\mathrm{Supp} M = V( \mathrm{Ann} M)}$ for ${\mathrm{Ann} M}$ the annihilator of ${M}$. But the “generic points” of the topological space ${V(\mathrm{Ann} M)}$, or the smallest primes in ${\mathrm{Supp} M}$, are precisely the associated primes of ${M}$. So if ${x}$ is a nonzerodivisor on ${M}$, we have that ${x}$ is not contained in any associated primes of ${M}$, so that ${\mathrm{Supp}(M/xM)}$ must have smaller dimension than ${\mathrm{Supp} M}$. That is,

$\displaystyle \dim M/xM \leq \dim M - 1.$ (more…)