Page $117$ of Atiyah, MacDonald's *[Introduction to Commutative Algebra](http://books.google.co.in/books?id=HOASFid4x18C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)* text has the following theorem. Let $P(M,t)$ denote the *[Poincare- series](http://en.wikipedia.org/wiki/Hilbert%25E2%2580%2593Poincar%25C3%25A9_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 J.-P. [Serre](http://en.wikipedia.org/wiki/Jean-Pierre_Serre)? References about when Serre was credited to the above theorem would be helpful.



Why is this theorem attributed to J.-P. Serre?
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](http://homepages.math.uic.edu/~bshipley/huneke.pdf) 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 $d$.


