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

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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, algebraic geometry

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