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 this front. Lucky for me, there is a nice clean way to prove it using the Hilbert polynomial, so I can just continue this course for now.
Let’s now reconstruct the Hilbert polynomial in a different way. As before let be a finitely generated graded -module. Then is finitely generated as an -module.
Let be an additive funtion (in ) on the class of finitely generated -modules. We define the Poincare series of to be the generating funciton of . So we get a power series with coefficients in : .
By a remarkably similar argument to the last post we can check by induction that is a rational function in of the form where .
Let’s suggestively call the order of the pole at , .
We now simplify the situation by taking all . Then the main idea for today is that is a polynomial of degree . In fact, .
Our simplification gives that . So is the coefficient of . If we cancel factors of out of we can assume and that . Write . Then since we get that for all .
Thus we get a polynomial with non-zero leading term. Note the values at integers are integers, but the coefficients in general are only rationals.
Since was any additive function, this is a bit more general. But taking we get the Hilbert polynomial from last time.
Next time we’ll start using this to streamline some proofs about dimension.