Why is this theorem attributed to Serre?

2012-06-18T16:43:20Z

Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.

$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}\bigr)$. $P(M,t)$ is a rational function in $t$ of the form $f(t)/\prod_{i=1}^{s} (1-t^{k_i})$, where $f(t) \in \mathbf{Z}[t]$.

This theorem appears in the section of the book called Hilbert-Functions (page 116), so one understands that it could have possibly been discovered by Hilbert.

But why is the above theorem attributed to Serre? References about when Serre was credited to the above theorem would be helpful.

Answer by David Speyer for Why is this theorem attributed to Serre?

2012-06-19T16:22:25Z

I don't know what Atiyah-Macdonald were thinking, but I can tell you a theorem which is attributed to Serre (correctly, I think), and is relevant to this question.

Let $M$ be a finitely-generated graded $k[x_0, x_1, \ldots, x_n]$ module. Let $H^0(M)$, $H^1(M)$, ..., $H^n(M)$ be the local cohomology modules of $M$ with respect to the maximal ideal $\langle x_0,\ldots, x_n \rangle$. These are graded modules which satisfy the following properties:

Theorem: For all integers $d$, the function $$\dim M_d - \sum_{r=0}^n (-1)^r \dim H^r(M)_d$$ is polynomial in $d$.

Theorem (Serre vanishing) For $d$ sufficiently large, $H^r(M)_d=0$.

So Serre vanishing separates Hilbert's theorem into two parts: A certain function is a polynomial for all $d$, and that function is equal to the Hilbert function for large $d$.

I'm presenting this using the language of commutative algebra, which I don't think is the language Serre used. In sheaf cohomology language, let $\mathcal{M}$ be the sheaf on $\mathbb{P}^{n-1}$ corresponding to $M$ and let $\mathcal{H}^r(M) = \bigoplus_{d=-\infty}^{\infty} H^r(\mathbb{P}^{n-1}, \mathcal{M} \otimes \mathcal{O}(-d))$. Then the relation between sheaf cohomology and local cohomology is that $$\mathcal{H}^r(M) \cong H^{r+1}(M)$$ for $r \geq 1$ and there is a short exact sequence $$0 \to H^0(M) \to M \to \mathcal{H}^0(M) \to H^1(M) \to 0.$$ In this language, Serre vanishing says that, for $d$ large, $\mathcal{H}^r(M)_d=0$ for $r>0$ and $M_d \cong \mathcal{H}^0(M)_d$; this is how the result is usually stated. The first theorem in this language is that $\dim \sum_{r=0}^{n-1} (-1)^r \mathcal{H}^r(M)_d$ is a polynomial in $d$.

Why is this theorem attributed to Serre? - MathOverflow

Why is this theorem attributed to Serre?

Answer by David Speyer for Why is this theorem attributed to Serre?