Beginning Dimension Theory

Recall the purpose of this development is to get some results on ring dimensions. All the hypothesis and notation from last time still hold (the important one to remember is that (R, \frak{m}) is a local ring).

Let’s introduce a new notation, which will disappear shortly. We call the characteristic polynomial of the \frak{m}-primary ideal \frak{q}, \chi_q^M(n)=l(M/\frak{q}^nM). An immediate corollary to the last post is that for large n, \chi_q(n)=l(R/\frak{q}^n) has degree \leq s where s is the least number of generators of \frak{q}.

Now we want to show that for our purposes the choice of \frak{m}-primary ideal doesn’t matter. The claim is that \deg \chi_q(n)=\deg \chi_m(n).

We know that there is some integer r such that \frak{q} contains \frak{m}^r. i.e. \frak{m}\supset \frak{q}\supset \frak{m}^r. Thus \frak{m}^n\supset \frak{q}^n \supset \frak{m}^{rn}. Thus for large n, we get \chi_m(n)\leq \chi_q(n)\leq \chi_m(rn). Since these are polynomials, we let n tend to \infty to get the claim.

Let’s denote the common degree d(R). Thus d(R) is the order of the pole at t=1 of the Hilbert function of G_\frak{m}(R).

Since this is short so far, we will very briefly start our first goal of showing that if \delta(R) is the least number of generators of an \frak{m}-primary ideal, and we impose Noetherian on R, then \delta(R)=d(R)=\dim R.

What we just showed above in this new notation is that \delta(R)\geq d(R). The way we will eventually show the equality is to get \delta(R)\geq d(R)\geq \dim R \geq \delta(R).

The next step is involved and needs the Artin-Rees Lemma, so I’ll hold off and do it next time.

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