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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Excercises 7}

\bigskip\noindent
A {\it numerical polynomial} is a polynomial $f(x) \in {\bbb Q}[x]$
such that $f(n) \in {\bbb Z}$ for every integer $n$. A {\it graded module}
$M$ over a ring $A$ is an $A$-module $M$ endowed with a direct sum
decomposition
$
\bigoplus\nolimits_{n \in {\bbb Z}} M_n
$
into $A$-submodules. We will say that $M$ is {\it locally finite} if all of
the $M_n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and
that $\varphi$ is a {\it Euler-Poincar\'e function} on finite $A$-modules. 
This means that for every finitely generated $A$-module $M$ we are given an
integer $\varphi(M) \in {\bbb Z}$ and for every short exact sequence 
$$
0
\longrightarrow
M'
\longrightarrow
M
\longrightarrow
M''
\longrightarrow
0
$$
we have $\varphi(M) = \varphi(M') + \varphi(M')$. The {\it 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
{\it Hilbert polynomial} if there is some numerical polynomial
$P_\varphi$ such that $\chi_\varphi(M,n) = P_\varphi(n)$ for all sufficiently
large integers $n$.

\medskip\noindent
A {\it graded $A$-algebra} is a graded $A$-module
$B = \bigoplus B_n$ 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 {\it 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 {\it
homomorphisms of graded modules/rings}, {\it graded submodules}, {\it graded
ideals}, {\it exact sequences of graded modules}, etc, etc.

\medskip\noindent
{\bf 1.} Let $A=k$ a field. What are all possible Euler-Poincar\'e functions
on finite $A$-modules in this case?

\medskip\noindent
{\bf 2.} Let $A ={\bbb Z}$. What are all possible Euler-Poincar\'e functions
on finite $A$-modules in this case?

\medskip\noindent
{\bf 3.} Let $A = k[x,y]/(xy)$ with $k$ algebraically closed. What are all
possible Euler-Poincar\'e functions on finite $A$-modules in this case?

\medskip\noindent
{\bf 4.} Suppose that $A$ is Noetherian. Show that the kernel of a map
of locally finite graded $A$-modules is locally finite.

\medskip\noindent
{\bf 5.} 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?

\medskip\noindent
{\bf 6.} 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?

\medskip\noindent
{\bf 7.} Let $k$ be a field and let $A=k$.  Let $\varphi(M)=\dim_k(M)$.
Fix $d\in {\bbb N}$. 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?

\medskip\noindent
To be continued.

\end
