The Stacks project

Definition 111.26.1. A numerical polynomial is a polynomial $f(x) \in {\mathbf Q}[x]$ such that $f(n) \in {\mathbf Z}$ for every integer $n$.

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-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

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 $n$.

Definition 111.26.3. A graded $A$-algebra is a graded $A$-module $B = \bigoplus _{n \geq 0} B_ n$ together with an $A$-bilinear map

that turns $B$ into an $A$-algebra so that $B_ n \cdot B_ m \subset B_{n + m}$. Finally, a graded module $M$ over a graded $A$-algebra $B$ is given by a graded $A$-module $M$ together with a (compatible) $B$-module structure such that $B_ n \cdot M_ d \subset M_{n + d}$. Now you can define homomorphisms of graded modules/rings, graded submodules, graded ideals, exact sequences of graded modules, etc, etc.