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

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 $n$.

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