The Stacks project

Definition 110.26.2. 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-Poincaré 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 " instead of .

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