David Eisenbud, Hilbert theorem

Question

I just started reading D. Eisenbud Commutative algebra with a view towards algebraic geometry and I wonder about a theorem on page 42:

If $M$ is a finitely generated graded module over $k[x_1,...,x_r]$ then $H_M(s)$ agrees, for large $s$, with a polynomial of degree $\leq r-1$, where $H_M(s):=\dim_k(M_s)$.

I don't understand what "then $H_M(s)$ agrees, for large $s$, with a polynomial of degree $\leq r-1$" means. Can someone explain this?

Answer 1

It means that there is a polynomial $p(n)=a_{r-1}n^r+...+a_0$, and a number $N$ such that $H_M(s)=p(s)$ for $s>N$.

Maybe more important to understand than that statement is the following equivalent one.

Define $\Delta f(n):=f(n+1)-f(n)$ and $\Delta^{m}f(n)=\Delta\Delta^{m-1}f(n)$.

Then the statement above is the same as $\Delta^{r}H_M(s)=0$ for all $s$ large.

Answer 2

It means that $H_M(s) = p(s)$ for some polynomial $p$ of degree $\leq r-1$ for $s$ sufficiently large.