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\centerline{\Kopfont Schemes}

\smallskip
\centerline{Exercises 5}

\bigskip\noindent
{\bf Schemes -- invertible sheaves are important}

\bigskip\noindent
Feel free to quote results from Hartshorne or EGA.

\bigskip\noindent
An invertible ${\cal O}_X$-module on a locally ringed space $(X,{\cal O}_X)$
is a sheaf of ${\cal O}_X$-modules ${\cal L}$ such that every point
has an open neighbourhood $U \subset X$ such that ${\cal L}|_U$
is isomorphic to ${\cal O}_U$ as ${\cal O}_U$-module.
We say that ${\cal L}$ is trivial if it is isomorphic to 
${\cal O}_X$ as a ${\cal O}_X$-module.

\bigskip\item{\bf 1.} General facts.
\itemitem{\bf (a)} Show that an invertible ${\cal O}_X$-module on 
a scheme $X$ is quasi-coherent.
\itemitem{\bf (b)} Suppose $X\to Y$ is a morphism of ringed spaces,
and ${\cal L}$ an invertible ${\cal O}_Y$-module.
Show that $f^\ast {\cal L}$ is an invertible ${\cal O}_X$ module.

\medskip\item{\bf 2.} Algebra.
\itemitem{\bf (a)} Show that an invertible ${\cal O}_X$-module on 
an affine scheme $\Spec A$ corresponds to an $A$-module $M$ which is
(i) finite, (ii) projective, (iii) locally free of rank 1,
and hence (iv) flat, and (v) finitely presented. (Feel free to
quote things from last semesters course; or from algebra books.)
\itemitem{\bf (b)} Suppose that $A$ is a domain and that $M$ is
a module as in (a). Show that $M$ is isomorphic as an $A$-module
to an ideal $I \subset A$ such that $IA_\gp$ is principal for
every prime $\gp$.

\medskip\item{\bf 3.} Simple examples.
\itemitem{\bf (a)} Let $k$ be a field. Let $A = k[x]$.
Show that $X=\Spec A$ has only trivial invertible ${\cal O}_X$-modules.
\itemitem{\bf (b)} Let $A$ be the ring
$$
A = \{ f\in k[x] \mid f(0)=f(1) \}.
$$
Show that $X = \Spec A$ has a nontrivial invertible ${\cal O}_X$-module,
unless $k={\bbb F}_2$. (Hint: Think about $\Spec A$ as identifying
$0$ and $1$ in ${\bbb A}^1_k=\Spec k[x]$.)
\itemitem{\bf (c)} Same question for the ring $A = k[x^2,x^3] \subset
k[x]$ (except now $k = {\bbb F}_2$ works as well).

\medskip\item{\bf 4.} Higher dimensions.
\itemitem{\bf (a)} Prove that every invertible sheaf on two dimensional
affine space is trivial. More precisely, let 
${\bbb A}^2_k = \Spec k[x,y]$ where $k$ is a field.
Show that every invertible sheaf on ${\bbb A}^2_k$ is trivial.
(Hint: One way to do this is to consider the corresponding
module $M$, to look at $M \otimes_{k[x,y]} k(x)[y]$, and
then use 3(a) to find a generator for this; then you still have to think.
Another way to is to use 2(b) and use what we know about ideals of the
polynomial ring: primes of height one are generated by an irreducible
polynomial; then you still have to think.)
\itemitem{\bf (b)} Prove that every invertible sheaf on any open
subscheme of two dimensional affine space is trivial. More precisely, let 
$U \subset {\bbb A}^2_k$ be an open subscheme where $k$ is a field.
Show that every invertible sheaf on $U$ is trivial. Hint: Show that every
invertible sheaf on $U$ extends to one on ${\bbb A}^2_k$. Not easy;
but you can find it in Hartshorne.
\itemitem{\bf (c)} Find an example of a nontrivial
invertible sheaf on a punctured cone over a field. More
precisely, let $k$ be a field and let $C = \Spec k[x,y,z]/(xy-z^2)$.
Let $U = C \setminus \{ (x,y,z) \}$. Find a nontrivial
invertible sheaf on $U$. Hint: It may be easier to compute the
group of isomorphism classes of invertible sheaves on $U$ than to
just find one. Note that $U$ is covered by the opens 
$\Spec k[x,y,z,1/x]/(xy-z^2) $ and $\Spec k[x,y,z,1/y]/(xy-z^2)$
which are ``easy'' to deal with.


\bye
