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\centerline{\Kopfont Schemes}

\smallskip
\centerline{Exercises 2}

\bigskip\noindent
{\bf Schemes -- examples are important}

\bigskip\noindent
Let ${\cal C}$ be the category of locally ringed spaces.
An affine scheme is an object in ${\cal C}$ isomorphic in ${\cal C}$ to
a pair of the form $(\Spec A, {\cal O}_A)$. A scheme is an
object $(X, {\cal O}_X)$ of ${\cal C}$ such that every point $x\in X$ 
has an open neighbourhood $U \subset X$ such that the pair
$(U, {\cal O}_X|_U)$ is an affine scheme.

\bigskip\item{\bf 1.} Suppose that $X$ is a scheme whose underlying 
topological space has 2 points. Show that $X$ is an affine scheme.

\medskip\item{\bf 2.} Give an example of an affine scheme $(X, {\cal O}_X)$
and an open $U \subset X$ such that $(U, {\cal O}_X|U)$ is not an affine
scheme.

\medskip\item{\bf 3.} Given an example of a pair of affine schemes
$(X, {\cal O}_X)$, $(Y, {\cal O}_Y)$, an open subscheme $(U, {\cal O}_X|_U)$
of $X$ and a morphism of schemes $(U, {\cal O}_X|_U) \to (Y, {\cal O}_Y)$
that does not extend to a morphism of schemes $(X, {\cal O}_X) \to
(Y, {\cal O}_Y)$.

\medskip\item{\bf 4.} Give an example of a scheme $X$, a field $K$, and a
morphism of ringed spaces $\Spec K \to X$ which is NOT a morphism of schemes.

\medskip\item{\bf 5.} Do all the exercises in Hartshorne, Chapter II,
Sections 1 and 2...\ \ Just kidding!

\medskip\noindent
{\bf Remark.} When $(X, {\cal O}_X)$ is a ringed space and $U \subset X$
is an open subset then $(U, {\cal O}_X|_U)$ is a ringed space. Notation:
${\cal O}_U = {\cal O}_X|_U$. There is a canonical morphisms of ringed spaces
$$
	j : (U, {\cal O}_U) \longrightarrow (X, {\cal O}_X).
$$
If $(X, {\cal O}_X)$ is a locally ringed space, so is $(U, {\cal O}_U)$ and
$j$ is a morphism of locally ringed spaces. If $(X, {\cal O}_X)$ is a scheme
so is $(U, {\cal O}_U)$ and $j$ is a morphism of schemes. We say that
$(U, {\cal O}_U)$ is an {\it open subscheme} of $(X, {\cal O}_X)$ and that
$j$ is an {\it open immersion}. More generally, any morphism
$j' : (V, {\cal O}_V) \to (X, {\cal O}_X)$ that is {\it isomorphic} to a
morphism $j : (U, {\cal O}_U) \to (X, {\cal O}_X)$ as above is called an
open immersion.

\bye
