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\centerline{\Kopfont Schemes}

\smallskip
\centerline{Exercises 6}

\bigskip\noindent
{\bf Schemes -- Divisors}

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

\bigskip\noindent
{\bf Definitions.} Throughout, let $X$ be a Noetherian, integral and
separated scheme.
\item{\bf (a)} A Weil divisor is a formal linear combination
$\Sigma n_i[Z_i]$ of prime divisors $Z_i$ with integer coefficients.
\item{\bf (b)} A prime divisor is a closed subscheme $Z \subset X$,
which is integral with generic point $\xi \in Z$ such that
${\cal O}_{X,\xi}$ has dimension $1$. We will use the notation 
${\cal O}_{X,Z} = {\cal O}_{X,\xi}$
when $\xi \in Z \subset X$ is as above. Note that ${\cal O}_{X,Z} \subset
K(X)$ is a subring of the function field of $X$.
\item{\bf (c)} The Weil divisor associated to a rational function
$f \in K(X)^\ast$ is the sum $\Sigma v_Z(f)[Z]$. Here $v_Z(f)$ is
defined as follows
\itemitem{(c1)} If $f \in {\cal O}_{X,Z}^\ast$ then $v_Z(f)=0$.
\itemitem{(c2)} If $f \in {\cal O}_{X,Z}$ then 
$$
v_Z(f)=\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(f)).
$$
\itemitem{(c3)} If $f = {a \over b}$ with $a,b \in {\cal O}_{X,Z}$
then 
$$
v_Z(f)=\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(a)) -
\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(b)).
$$
\item{\bf (d)} An effective Cartier divisor on {\it any} scheme $S$
is a closed subscheme $D \subset S$ such that every point $d\in D$
has an affine open neighbourhood $\Spec A = U \subset S$ in $S$
so that $D \cap U = \Spec A/(f)$ with $f \in A$ a nonzero divisor.
\item{\bf (e)} The Weil divisor $[D]$ associated to an effective
Cartier divisor $D \subset X$ of our Noetherian integral separated
scheme $X$ is defined as the sum $\Sigma v_Z(D)[Z]$ where
$v_Z(D)$ is defined as follows
\itemitem{(e1)} If the generic point $\xi$ of $Z$ is not in $D$
then $v_Z(D)=0$.
\itemitem{(e1)} If the generic point $\xi$ of $Z$ is in $D$
then 
$$
v_Z(D)=\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(f))
$$
where $f \in {\cal O}_{X,Z}={\cal O}_{X,\xi}$ is the nonzero divisor
which defines $D$ in an affine neighbourhood of $\xi$ (as in definition
{\bf (d)} above).
\item{\bf (f)} Let $S$ be {\it any} scheme. The sheaf of total quotient
rings ${\cal K}_S$ is the sheaf of ${\cal O}_S$-algebras which is
the sheafification of the pre-sheaf ${\cal K}'$ defined as follows.
For $U \subset S$ open we set ${\cal K}'(U) = S_U^{-1}{\cal O}_S(U)$
where $S_U \subset {\cal O}_S(U)$ is the multiplicative subset
consisting of sections $f \in {\cal O}_S(U)$ such that the germ
of $f$ in ${\cal O}_{S,u}$ is a nonzero divisor for every $u\in U$.
In particular the elements of $S_U$ are all nonzero divisors.
Thus ${\cal O}_S$ is a subsheaf of ${\cal K}_S$, and we get a
short exact sequence
$$
0 \to {\cal O}_S^\ast \to {\cal K}_S^\ast \to
{\cal K}_S^\ast/{\cal O}_S^\ast \to 0.
$$
\item{\bf (g)} A Cartier divisor on {\it any} scheme $S$ is a global
section of the quotient sheaf ${\cal K}_S^\ast/{\cal O}_S^\ast$.
\item{\bf (h)} The Weil divisor associated to a Cartier divisor
$\tau \in \Gamma(X, {\cal K}_X^\ast/{\cal O}_X^\ast)$ over our 
Noetherian integral separated scheme
$X$ is the sum $\Sigma v_Z(\tau)[Z]$ where $v_Z(\tau)$ is defined
as by the following recipe
\itemitem{(h1)} If the germ of $\tau$ at the generic point $\xi$
of $Z$ is zero -- in other words the image of $\tau$ in the stalk
$({\cal K}^\ast/{\cal O}^\ast)_\xi$ is ``zero'' -- then $v_Z(\tau)=0$.
\itemitem{(h2)} Find an affine open neighbourhood $\Spec A = U \subset X$
so that $\tau|_U$ is the image of a section $f \in {\cal K}(U)$
and moreover $f = a/b$ with $a,b \in A$. Then we set
$$
v_Z(f)=\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(a)) -
\length_{{\cal O}_{X,Z}}({\cal O}_{X,Z}/(b)).
$$

\medskip\noindent
{\bf Remarks.} (a) On a Noetherian integral separated scheme $X$ the 
sheaf ${\cal K}_X$ is constant with value the function field $K(X)$.

\noindent (b) To make sense out of the definitions above one needs
to show that
$$
\length_{\cal O}({\cal O}/(ab)) =
\length_{\cal O}({\cal O}/(a)) +
\length_{\cal O}({\cal O}/(b))
$$
for any pair $(a,b)$ of nonzero elements of a Noetherian 1-dimensional
local domain ${\cal O}$. This will be done in the lectures.

\goodbreak\bigskip\item{\bf 1.} Describe how to assign a Cartier divisor
to an effective Cartier divisor.

\medskip\noindent
The following questions have some logical dependencies; if you point them
out then you won't have to do all of them.

\medskip\item{\bf 2.} Give an example of a Weil divisor
(on a Noetherian integral separated scheme) which is not
the Weil divisor associated to a rational function.

\medskip\item{\bf 3.} Give an example of a Weil divisor
(on a Noetherian integral separated scheme) which is not
the Weil divisor associated to any effective Cartier divisor.

\medskip\item{\bf 4.} Give an example of a Weil divisor
(on a Noetherian integral separated scheme) which is not
the Weil divisor associated to any Cartier divisor.

\medskip\item{\bf 5.} Give an example of a Weil divisor $D$
(on a Noetherian integral separated scheme) which is not
the Weil divisor associated to any Cartier divisor but
such that $nD$ is the Weil divisor associate to a Cartier
divisor for some $n>1$.

\medskip\item{\bf 6.} Give an example of a Weil divisor $D$
(on a Noetherian integral separated scheme) which is not
the Weil divisor associated to any Cartier divisor and
such that $nD$ is NOT the Weil divisor associate to a Cartier
divisor for any $n>1$.

\medskip\item{\bf 7.}
Give an example of a Cartier divisor
which is not the difference of (the Cartier divisors associated
to) two effective Cartier divisors.\footnote{$\dagger$}{I do not know how to do this one myself, but I think this happens.}

\bye
