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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Excercises 6}

\bigskip\noindent
{\bf 1.} Suppose that $k$ is a field having a primitive $n$th root
of unity $\zeta$. This means that $\zeta^n = 1$, but $\zeta^m\not=1$ for 
$0<m<n$.
\item{(a)} Show that the characteristic of $k$ is prime to $n$.
\item{(b)} Suppose that $a \in k$ is an element of $k$ which is not
an $d$th power in $k$ for any divisor $d$ of $n$, $in\geq d>1$. Show that
$k[x]/(x^n-a)$ is a field. (Hint: Consider a splitting field for
$x^n-a$ and use Galois theory.)

\medskip\noindent
{\bf 2.} Let $\nu : k[x]\setminus \{0\}  \to {\bbb Z}$ be a map
with the following properties: $\nu(fg) = \nu(f) + \nu(g)$ whenever
$f$, $g$ not zero, and  $\nu(f+g) \geq min(\nu(f),\nu(g))$ whenever 
$f$, $g$, $f+g$ are not zero, and $\nu(c)=0$ for all $c\in k^*$.
\item{(a)} Show that if $f$, $g$, and $f+g$ are nonzero and
$\nu(f) \not= \nu(g)$ then we have equality $\nu(f+g) = min(\nu(f),\nu(g))$.
\item{(b)} Show that if $f=\sum a_i x^i$, $f\not=0$, then
$\nu(f) \geq min(\{i\nu(x)\}_{a_i\not=0})$. When does equality hold?
\item{(c)} Show that if $\nu$ attains a negative value then
$\nu(f) = -n \deg(f)$ for some $n\in {\bbb N}$.
\item{(d)} Suppose $\nu(x) \geq 0$. Show that
$\{f \mid f=0,\ or\ \nu(f) > 0\}$ is a prime ideal of $k[x]$.
\item{(e)} Describe all possible $\nu$.

\medskip\noindent
{\bf 3.} Let $(A,\gm,k)$ be a local ring and let $k \subset k'$ be
a finite field extension. Show there exists a flat, local map of
local rings $A \to B$ such that $\gm_B = \gm B$ and $B/\gm B$ is
isomorphic to $k'$ as $k$-algebra. (Hint: first do the case where
$k \subset k'$ is generated by a single element.)

\medskip\noindent
{\bf Remark.} The same result holds for arbitrary field extensions
$k \subset K$.

\medskip\noindent
{\bf 4.} Let $R$ be a ring and let $M$ be a finitely presented $R$
module. Recall this means that there is an exact sequence
$$
R^{\oplus r} \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0.
$$
This is called a presentation of $M$. Note that the map $R^{\oplus n} \to M$
is given by a sequence of elements $x_1,\ldots,x_n$ of $M$. The elements $x_i$
are generators of $M$. The map $R^{\oplus r} \to R^{\oplus n}$ is given by a
$n \times r$ matrix $A$ with coefficients in $R$. The columns of $A$ are said
to be the relations. Any vector $(r_i) \in R^{\oplus r}$ such that
$\sum r_i x_i =0$ is a linear combination of the columns of $A$.

\smallskip\noindent
Of course any module has a lot of different presentations.
We define $Pres(M)$ to be the collection of matrices you can obtain in this
way (meaning all matrices $A$ of any size $n \times r$ that occur in some
presentation of $M$).
\item{(a)} Show that if $A\in Pres(M)$ has size $n\times r$ then
the matrix $\tilde A$ of size $n \times (r+1)$ obtained from $A$ by
adding a column of zeros occurs in $Pres(M)$. (Hint: this corresponds
to adding a trivial relation.)
\item{(b)} Show that if $A\in Pres(M)$, then any $\tilde A \in Pres(M)$,
where $\tilde A$ is obtained from $A$ by replacing $i$th column vector
$A_i$ by $A_i + \sum_{j\not=i} r_j A_j$ for any $r_j \in R$. (Hint: This
corresponds to replacing a relation by itself plus a linear combination of
other relations.)
\item{(c)} Show that if $A\in Pres(M)$ has size $n\times r$ then
the matrix $\tilde A$ of size $(n+1) \times (r+1)$ obtained from $A$ 
by setting
$$
\eqalign{
\tilde a_{ij}      & = a_{ij},\ i<n+1, j<r+1,\cr
\tilde a_{n+1 j}   & = 0,\ j<r+1,\cr 
\tilde a_{i r+1}   & = 0,\ i<n+1,\cr
\tilde a_{n+1 r+1} & = 1}
$$
occurs in $Pres(M)$. (Hint: This corresponds to adding $x_{n+1}=0$ and
the trivial relation $x_{n+1}=0$.)
\item{(d)} Show that if $A\in Pres(M)$, then $\tilde A \in Pres(M)$
where $\tilde A$ is obtained from $A$ by replacing the $j$th row
by a sum consisting of itself and a linear combination of other
rows (with coefficients in $R$). (Hint: This corresponds to replacing
$x_j$ by $x_j + \sum_{i\not = j} r_i x_i$ and adjusting the relations
accordingly.)

\smallskip\noindent
We say that matrices $A$ and $A'$ with coefficients in $R$
are obtained from each other by a sequence of elementary moves if
there is a sequence of matrices $A=A_0, A_1, A_2,\ldots, A_n=A'$ such
that for each $0\leq \ell < n$ the pair $(A_\ell, A_{\ell+1})$ is the
pair $(A,\tilde A)$ or $(\tilde A, A)$ for one of the operations on
matrices described in (a)-(d) above.

\item{(e)} Show that any two matrices in $Pres(M)$ are obtained from
each other by a sequence of elementary moves. (Hint: First show this 
holds if $A$, $A'$ in $Pres(M)$ are matrices of relations among the same
set of generators.)
\item{(f)} Let $k$ be an integer. Suppose that $A$, $A'$ are obtained from
each other by a sequence of elementary moves. Say $A$ has size $n \times r$
and $A'$ has size $n' \times r'$. Show that the ideal generated by the 
$(n-k)\times (n-k)$ minors of $A$ agrees with the ideal generated by the
$(n'-k)\times (n'-k)$ minors of $A'$. [[Convention: If $k\geq n$ then we say
the ideal generated by the $(n-k)\times (n-k)$-minors is $R$. In other words,
the determinant of a matrix of size $0 \times 0$, $-1 \times -1$, etc
is defined to be $1$.]]

\smallskip\noindent
This defines the $k$th fitting ideal of $M$. Notation $Fit_k(M)$.

\item{(g)} Show that
$Fit_0(M) \subset Fit_1(M) \subset Fit_2(M) \subset \ldots$.
(Hint: Use that a determinant can be computed by expanding along a column.)
\item{(h)} Show that $M=(0)$ if $Fit_0(M)=R$.
\item{(i)} Show that $M$ if $Fit_0(M)=(0)$ and $Fit_1(M)=R$, then $M$ is
locally free of rank $1$. (This is slightly tricky.)

\end
