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 $n$.
111.26 Hilbert functions
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 $n$.
\[ 0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0 \]
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.
\[ B \times B \longrightarrow B, \ (b, b') \longmapsto bb' \]
Exercise 111.26.4. Let $A = k$ a field. What are all possible Euler-Poincaré functions on finite $A$-modules in this case?
Exercise 111.26.5. Let $A ={\mathbf Z}$. What are all possible Euler-Poincaré functions on finite $A$-modules in this case?
Exercise 111.26.6. Let $A = k[x, y]/(xy)$ with $k$ algebraically closed. What are all possible Euler-Poincaré functions on finite $A$-modules in this case?
Exercise 111.26.7. Suppose that $A$ is Noetherian. Show that the kernel of a map of locally finite graded $A$-modules is locally finite.
Exercise 111.26.8. Let $k$ be a field and let $A = k$ and $B = k[x, y]$ with grading determined by $\deg (x) = 2$ and $\deg (y) = 3$. Let $\varphi (M) = \dim _ k(M)$. Compute the Hilbert function of $B$ as a graded $k$-module. Is there a Hilbert polynomial in this case?
Exercise 111.26.9. Let $k$ be a field and let $A = k$ and $B = k[x, y]/(x^2, xy)$ with grading determined by $\deg (x) = 2$ and $\deg (y) = 3$. Let $\varphi (M) = \dim _ k(M)$. Compute the Hilbert function of $B$ as a graded $k$-module. Is there a Hilbert polynomial in this case?
Exercise 111.26.10. Let $k$ be a field and let $A = k$. Let $\varphi (M) = \dim _ k(M)$. Fix $d\in {\mathbf N}$. Consider the graded $A$-algebra $B = k[x, y, z]/(x^ d + y^ d + z^ d)$, where $x, y, z$ each have degree $1$. Compute the Hilbert function of $B$. Is there a Hilbert polynomial in this case?
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