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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Exercises 11}

\bigskip\noindent
Let $A$ be a ring. Recall that a {\it finite locally free} $A$-module
$M$ is a module such that for every $\gp \in \Spec A$ there exists an
$f\in A$, $f \not \in \gp$ such that $M_f$ is a finite free $A_f$-module.

\medskip\noindent
{\bf 1.} Let $A$ be a ring.
\item{(a)} Suppose that $M$ is a finite locally free $A$-module, and
suppose that $\varphi : M \to M$ is an endomorphism. Define/construct
the {\it trace}  and {\it determinant} of $\varphi$ and prove that your
construction is ``functorial in the triple $(A,M,\varphi)$''.
\item{(b)} Show that if $M,N$ are finite locally free $A$-modules,
and if $\varphi : M \to N$ and $\psi : N \to M$ then
$Trace(\varphi \circ \psi) = Trace(\psi \circ \varphi)$ and
$Det(\varphi \circ \psi) = Det(\psi \circ \varphi)$.
\item{(c)} In case $M$ is finite locally free show that
$Det$ defines a multiplicative map $End_A(M) \to A$.

\medskip\noindent
{\bf 2.} Now suppose that $B$ is an $A$-algebra which is finite
locally free as an $A$-module, in other words $B$ is a finite locally
free $A$-algebra. 
\item{(a)} Define $Trace_{B/A}$ and $Norm_{B/A}$ using
$Trace$ and $Det$ as defined above.
\item{(b)} Let $b\in B$ and let $\pi : \Spec B \to \Spec A$ be
the induced morphism. Show that $\pi(V(b)) = V(Norm_{B/A}(b))$.
(Recall that $V(f) = \{ \gp \mid f \in \gp\}$.)
\item{(c)} (Base change.) Suppose that $i:A \to A'$ is a ring map. Set
$B' = B\otimes_A A'$. Indicate why $i(Norm_{B/A}(b))$ equals
$Norm_{B'/A'}(b\otimes 1)$.
\item{(d)} Compute $Norm_{B/A}(b)$ when
$B = A\times A\times A \times\ldots\times A$
and $b = (a_1,\ldots,a_n)$.
\item{(e)} Compute the norm of $y-y^3$ under the finite flat
map ${\bbb Q}[x] \to {\bbb Q}[y]$, $x \to y^n$. (Hint: use
the ``base change''
$A = {\bbb Q}[x] \subset A' = {\bbb Q}(\zeta_n)(x^{1/n})$.)

\medskip\noindent
{\bf Remark.} Let $h \in {\bbb Z}[y]$ be a monic polynomial of degree $d$.
Then:
\item{(a)} The map $A = {\bbb Z}[x] \to B ={\bbb Z}[y]$,
$x \mapsto h$ is finite locally free of rank $d$.
\item{(b)} For all primes $p$ the map
$A_p = {\bbb F}_p[x]\to B_p = {\bbb F}_p[y]$,
$y \mapsto h(y) \bmod p$ is finite locally free of rank $d$.

\medskip\noindent
{\bf 3.} Let $h,A,B,A_p,B_p$ be as in the remark. For $f \in {\bbb Z}[x,u]$ we
define $f_p(x) = f(x,x^p) \bmod p \in {\bbb F}_p[x]$. For
$g \in {\bbb Z}[y,v]$ we define
$g_p(y) = g(y,y^p) \bmod p \in {\bbb F}_p[y]$. 
\item{(a)} Give an example of a $h$ and $g$ such that 
there does not exist a $f$ with the property
$$
f_p  =  Norm_{B_p/A_p}(g_p).
$$
\item{(b)} Show that for any choice of $h$ and $g$ as above
there exists a nonzero $f$ such that for all $p$ we have 
$$
Norm_{B_p/A_p}(g_p)\ \ \hbox{divides}\ \ f_p .
$$
If you want you can restrict to the case $h=y^n$, even with $n=2$,
but it is true in general.
\item{(c)} Discuss the relevance of this to Exercises 6 \& 7 of the previous
set.

\medskip\noindent
{\bf 4.} Unsolved problems. They may be really hard or they may be easy.
I don't know.
\item{(a)} Is there any $f \in {\bbb Z}[x,u]$ such that $f_p$ is
irreducible for an inifinite number of $p$?
\item{(b)} Let $f \in {\bbb Z}[x,u]$ nonzero, and suppose 
$\deg_x(f_p) = dp+d'$ for all large $p$. (In other words $\deg_u(f)=d$
and the coefficient $c$ of $u^d$ in $f$ has $\deg_x(c)=d'$.) Suppose we
can write $d = d_1 + d_2$ and $d' = d'_1 + d'_2$ with $d_1,d_2 > 0$
and $d'_1, d'_2 \geq 0$ such that for all sufficiently large $p$
there exists a factorization
$$
	f_p = f_{1,p} f_{2,p}
$$
with $\deg_x(f_{1,p}) = d_1p+d'_1$. Is it true that $f$ comes about via a
norm construction as in Exercise 4? (More precisely, are there a $h$ and
$g$ such that $Norm_{B_p/A_p}(g_p)$ divides $f_p$ for all $p >> 0$.)
\item{(c)} Analogous question to the one in (b) but now with
$f \in {\bbb Z}[x_1,x_2,u_1,u_2]$ irreducible and just assuming that
$f_p(x_1,x_2) = f(x_1,x_2,x_1^p,x_2^p) \bmod p$ factors for all
$p >> 0$.



\end
