The Stacks project

109.26 Hilbert functions

Definition 109.26.1. A numerical polynomial is a polynomial $f(x) \in {\mathbf Q}[x]$ such that $f(n) \in {\mathbf Z}$ for every integer $n$.

Definition 109.26.2. 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 $n$.

Definition 109.26.3. A graded $A$-algebra is a graded $A$-module $B = \bigoplus _{n \geq 0} B_ n$ together with an $A$-bilinear map

\[ B \times B \longrightarrow B, \ (b, b') \longmapsto bb' \]

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.

Exercise 109.26.4. Let $A = k$ a field. What are all possible Euler-Poincaré functions on finite $A$-modules in this case?

Exercise 109.26.5. Let $A ={\mathbf Z}$. What are all possible Euler-Poincaré functions on finite $A$-modules in this case?

Exercise 109.26.6. 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 109.26.7. Suppose that $A$ is Noetherian. Show that the kernel of a map of locally finite graded $A$-modules is locally finite.

Exercise 109.26.8. 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 109.26.9. 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 109.26.10. 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?


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 027W. Beware of the difference between the letter 'O' and the digit '0'.