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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Exercises 10}

\bigskip\noindent
A Noetherian ring $A$ is said to be {\it catenary}
if for any triple of prime ideals $\gp_1 \subset \gp_2 \subset \gp_3$
we have
$$
ht(\gp_3 / \gp_1) = ht(\gp_3/\gp_2) + ht(\gp_2/\gp_1).
$$

\medskip\noindent
{\bf 1.} Show that a Noetherian local domain of dimension $2$
is catenary.

\medskip\noindent
{\bf 2.} Let $A$ be a ring, and $\gm$ a maximal ideal. In $A[X]$
let $\tilde \gm_1 = (\gm,X)$ and $\tilde \gm_2 = (\gm,X-1)$. Show
that 
$$
A[X]_{\tilde \gm_1} \cong A[X]_{\tilde \gm_2}.
$$

\medskip\noindent
{\bf 3.} Find an example of a non Noetherian ring $R$ such that every
finitely generated ideal of $R$ is finitely presented as an $R$-module.
(A ring is said to be {\it coherent} if the last property holds.)

\medskip\noindent
{\bf 4.} Consider the domain
$$
{\bbb Q}[r,s,t]/(s^2-(r-1)(r-2)(r-3), t^2-(r+1)(r+2)(r+3)).
$$
Find a domain of the form ${\bbb Q}[x,y]/(f)$ with isomorphic
field of fractions.

\medskip\noindent
{\bf 5.} Give an example of a finite type ${\bbb Z}$-algebra $R$
with the following two properties:
\item{(a)} There is no ring map $R \to {\bbb Q}$.
\item{(b)} For every prime $p$ there exists a maximal ideal
$\gm \subset R$ such that $R/\gm = \cong {\bbb F}_p$.

\medskip\noindent
{\bf 6.} For $f \in {\bbb Z}[x,u]$ we define $f_p(x)
= f(x,x^p) \bmod p \in {\bbb F}_p[x]$. Give an example
of an $f \in {\bbb Z}[x,u]$ such that the following two
properties hold:
\item{(a)} There exist infinitely many $p$ such that $f_p$
does not have a zero in ${\bbb F}_p$.
\item{(b)} For all $p>>0$ the polynomial $f_p$ either has
a linear or a quadratic factor.

\medskip\noindent
{\bf 7.} For $f \in {\bbb Z}[x,y,u,v]$ we define $f_p(x,y)
= f(x,y,x^p,y^p) \bmod p \in {\bbb F}_p[x,y]$. Give an ``interesting''
example of an $f$ such that $f_p$ is reducible for all $p>>0$.
For example, $f=xv-yu$ with $f_p=xy^p-x^py=xy(x^{p-1}-y^{p-1})$ is
``uninteresting''; any $f$ depeding only on $x,u$ is ``uninteresting'',
etc.


\end
