Q: Why is this theorem attributed to J.-P. Serre? Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.


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*$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}\bigr)$. $P(M,t)$ 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. 


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


A: 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 $d$.
