# Hilbert Polynomial vs Hilbert Quasi-Polynomial

Let $R$ be an $\mathbb{N}$-graded ring with $R_0$ Artinian and $R = R_0[x_1,\dots,x_r]$, where the degree of $x_i$ is $d_i > 0$. Let $M$ a finitely generated $\mathbb{N}$-graded $R$-module with Hilbert function $H(M,n)$. Then it is known that there exists a unique polynomial $P_M(t)$ such that $H(M,n) = P_M(n)$ for large enough $n$. This result can be found e.g. in Matsumura's Commutative Ring Theory at pages 94-95.

In Bruns&Herzog Cohen-Macaulay Rings, a quasi-polynomial is defined to be a function $f:\mathbb{Z} \rightarrow \mathbb{C}$, such that $f$ is a periodic piecewise polynomial. Then Theorem 4.4.3 reads as follows:

Question: I am failing to see in what way the setting of this theorem is a generalization of the setting described in the first paragraph of this question above. This has to be a generalization, since now the statement in (a) involves a quasi-polynomial instead of a polynomial. One possibility that i see is that even though $R$ is concentrated in non-negative degrees, $M$ may now be non-zero in negative degrees as well. But there can be finitely many such negative degrees since $M$ is finitely generated and $R$ is positively graded. So i don't think that the existence of finitely many negative components of $M$ would affect the Hilbert polynomial.

Although I don't understand your question very well, let me just say that the Theorem 13.2 in Matsumura says something about the Hilbert series of a finitely generated positively graded $R$-module (this is why $f\in\mathbb Z[t]$). The corollary on the bottom of page 95 gives that the Hilbert function agrees with a polynomial only in the case that the variables have degrees $d_1=\dots=d_r = 1$.
Theorem 4.4.3 in Bruns and Herzog deals with $\mathbb Z$-graded modules and is focused on Hilbert (quasi)polynomial and Hilbert functions rather than on Hilbert series which is treated earlier in Proposition 4.4.1.
• My question can be alternatively phrased as: which feature of the context in B&H gives rise to a quasi-polynomial instead of a polynomial? As you mention the only difference between the contexts of Matsumura and B&H is that in the latter the module may be $\mathbb{Z}$-graded, while Matsumura considers nonnegatively graded modules. But as i mention in my question, $M$ can only be non-zero at finitely many negative degrees and so this should not affect its Hilbert function for large $n$. So i would still expect the Hilbert series of $M$ to be of polynomial type. Is it clearer now? – Manos Dec 26 '14 at 23:31
• No, i believe i have this distinction clear. Where i got confused is on page 95 in Matsumura. I didn't realize that he talks about "Hilbert Polynomial" in the case where $d_1=\cdots=d_r=1$. But when these degrees are not equal then we have a quasi-polynomial. Correct? – Manos Dec 26 '14 at 23:36