algebra

The Artin-Rees Lemma

We have a somewhat bumpy road to traverse today. I'll start with the Artin-Rees lemma and see if we can get to a use of it to continue our set of inequalities we're trying to prove. First we'll need some new ideas. Suppose \$latex R\$ is any old ring (in particular, we are dropping graded… Continue reading The Artin-Rees Lemma

algebra

Applying the Hilbert Polynomial

Let's start applying to some specific situations now. Suppose \$latex R\$ is a Noetherian local ring with maximal ideal \$latex \frak{m}\$. Let \$latex \frak{q}\$ be an \$latex \frak{m}\$-primary ideal. Let \$latex M\$ be a finitely-generated \$latex R\$-module, and \$latex (M_n)\$ a stable \$latex \frak{q}\$-filtration of \$latex M\$. Don't panic from the set-up. I think I… Continue reading Applying the Hilbert Polynomial

algebra

Hilbert Polynomial II

My overall goal has not changed, but I definitely have a much clearer picture of where my posts are headed for right now. I recently was working on what happens to dimension when you intersect varieties, and I needed a commutative algebra result that sort of surprised me. So that is my first benchmark on… Continue reading Hilbert Polynomial II

Hilbert Polynomial I

I've been fiddling around on here for a few weeks trying to figure out what my next major set of posts should be about. I've finally settled. It turns out that algebraic geometry requires knowledge of a ridiculously large amount of commutative algebra. Now I usually try to avoid repeat posting when I know that… Continue reading Hilbert Polynomial I