Let’s start applying to some specific situations now. Suppose is a Noetherian local ring with maximal ideal . Let be an -primary ideal. Let be a finitely-generated -module, and a stable -filtration of .
Don’t panic from the set-up. I think I haven’t talked about filtrations. All the stable -filtration means is that we have a chain of submodules such that for large .
The goal for the day is to prove three things.
1) has finite length for all .
Define and . We have a natural way to make into a finitely-generated graded -module. The multiplication in the ring comes from the following. If , then let the image in be denoted . We take . This does not depend on representative.
We’ll say is the n-th grade: . Now is an Artinian local ring and each is a Noetherian -module annihilated by . Thus they are all Noetherian -modules. So by the Artinian condition we get that each is of finite length. Thus .
2) For large , is a polynomial of degree where is the least number of generators of .
Suppose generate . Then in generate as an -algebra. But is an additive function on the filtration, so by last time we saw thatfor large there is some polynomial such that , and each has degree 1, so the polynomial is of degree .
Thus we get that . So from two posts ago, we get for large that is some polynomial of degree .
3) Probably the most important part is that the degree and leading coefficient of depends only on and and not on the filtration.
Let be some other stable -filtration with polynomial . Since any two stable -filtrations have bounded difference, there is an integer such that and for all . But this condition on the polynomials says that and , which means that . Thus they have the same degree and leading coefficient.
That seems to be enough for one day. Unfortunately, I haven't quite got to the right setting that I want yet.