Definition 111.26.2. A graded module $M$ over a ring $A$ is an $A$-module $M$ endowed with a direct sum decomposition $\bigoplus \nolimits _{n \in {\mathbf Z}} M_ n$ into $A$-submodules. We will say that $M$ is locally finite if all of the $M_ n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and that $\varphi$ is a Euler-PoincarĂ© function on finite $A$-modules. This means that for every finitely generated $A$-module $M$ we are given an integer $\varphi (M) \in {\mathbf Z}$ and for every short exact sequence

$0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0$

we have $\varphi (M) = \varphi (M') + \varphi (M'')$. The Hilbert function of a locally finite graded module $M$ (with respect to $\varphi$) is the function $\chi _\varphi (M, n) = \varphi (M_ n)$. We say that $M$ has a Hilbert polynomial if there is some numerical polynomial $P_\varphi$ such that $\chi _\varphi (M, n) = P_\varphi (n)$ for all sufficiently large integers $n$.

## Comments (2)

Comment #6663 by bryce on

Hello Professor,

I think there is a typo: "we have $\phi(M) = \phi(M') + \phi(M')$" instead of $\phi(M) = \phi(M') + \phi(M'')$.

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