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\hbox to\hsize{\hfil \bf Homework 9: Math 953 Spring 2005\hfil}
\vskip\baselineskip
\hbox to\hsize{\hfil Due April 18, 2005\hfil}
\noindent (1) (Cf. Problem 3.14)
Give an example of a scheme $X$ over a field $k$ such that
the closed points of $X$ are not dense in $X$. If $X$ is locally of finite
type over $k$, show that the closed points are dense.
[Hint: Justify that it is enough to show that the closed points
are dense in every open set of an open affine cover. Apply the ring version of
the Nullstellensatz from Problem set 3 to verify the density
in affine opens.]
\noindent (2) Let $U$ and $V$ be affine open subsets
of a scheme $X$. Show that $U\cap V$ need not be affine.
[Hint: Show that $W={\bf A}^2-\{p\}$
is not affine, if $p$ is the origin (or more generally any
closed point). Do this by computing $\Shf{O}_Y(W)$, where
$Y={\bf A}^2$. Now take $X$ to be the affine plane with the origin doubled
to give the example, with $U$ and $V$ neighborhoods
of each origin. It is worth noting that
$U\cap V$ in fact {\it is} affine if $X$ is a separated $S$-scheme, where $S$ is affine.
The way to see this is to note $U\!\times_S\!V$ is an
open affine subset of $X\times _SX$, and that
$U\cap V = U\!\times_X\! V=(U\!\times_S\!V)\cap \Delta(X)$
is a closed subscheme of $U\!\times_S\!V$, hence affine.]
\noindent (3) Let $S$ be a scheme, let $X$ and $Y$ be $S$-schemes,
and let $f$ and $g$ be $S$-morphisms of $X$ to $Y$.
\itemitem{(a)} If $Y$ is separated over $S$, show that
$\{x\in X: f(x) = g(x)\}$ is closed in $X$.
[Hint: Look at the morphism $h: X\to Y\!\times_S\!Y$
induced by $f$ and $g$; essentially,
$h$ is $(f,g)$.]
\itemitem{(b)} Show that (a) can fail if $Y$ is not separated.
\itemitem{(c)} Let $a$ be an element of a ring $A$.
Show that $D(a)$ is not dense in $\hbox{Spec}(A)$ if and only if
$a^rb=0$ for some $r\ge 1$ and some $b\in A$ which is not nilpotent.
\itemitem{(d)} Let $a$ be an element of a ring $A$ and let $B$ be a ring. If
$D(a)$ is dense in $\hbox{Spec}(A)$ and if $\hbox{Spec}(A)$ is
reduced, show that $A\to A_a$ is injective. Conclude that two morphisms
from $\hbox{Spec}(A)$ to $\hbox{Spec}(B)$ are the same
if their restrictions to $D(a)$ are the same.
\itemitem{(e)} Show that $X=\{x\in X: f(x) = g(x)\}$ can occur,
even though $f$ and $g$ are not the same morphism.
\itemitem{(f)} Do Problem 4.2 on p. 105.
\bye