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 -module
, which is always assumed to be finitely generated. Let
be an ideal with
. We know that if
is a nonzerodivisor on
, then
is part of a maximal
-sequence in
, which has length
necessarily. It follows that
has a
-sequence of length
(because the initial
is thrown out) which can be extended no further. In particular, we find
Proposition 15 Hypotheses as above, let
be a nonzerodivisor on
. Then
This is strikingly analogous to the dimension of the module . Recall that
is defined to be the Krull dimension of the topological space
for
the annihilator of
. But the “generic points” of the topological space
, or the smallest primes in
, are precisely the associated primes of
. So if
is a nonzerodivisor on
, we have that
is not contained in any associated primes of
, so that
must have smaller dimension than
. That is,