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\title{Brief Article}
\author{The Author}
\date{}

\begin{document}
\maketitle

  \lipsum*{1}
\begin{exercise}\label{ExI.5}
\begin{compactenum}[(a) ]
\item\label{ExI.5.a}
Let $X$ bet the two-element set $\{0,1\}$, and make $X$ a topological space with the discrete topology.  A sheaf on $X$ is thus a collection of four sets with certain maps between them; describe the relations among these objects.  ($X$ is actually $\Spec R$ for some rings $R$; can you find one?)
\item\label{ExI.5.b}
Do the same in the case where the topology of $X=\{0,1\}$ has as open sets only $\emptyset,\{0\}$, and $\{0,1\}$.  Again, this space may be realized as $\Spec R$.
\end{compactenum}
\begin{proof}
Ad (a): A sheaf on the discrete space $X$ consists of sets (objects in a category) $A_{e},A_{0},A_{1},A_{X}$, with maps (from that category)
$$\xymatrix{
A_{e}\ar[r]\ar[d]\ar[dr]&A_{0}\ar[d]\\
A_{1}\ar[r]&A_{X}
}$$
making the diagram commute.  We have seen a space $\Spec R$ of the form $X$, in Ex.\ref{ExI.1}.\ref{ExI.1.c}, namely the ring of dual numbers over $\cmplx$.

Ad (b): In this case, the diagram in question is
$$\xymatrix{
A_{e}\ar[dr]\ar[d]\\
A_{0}\ar[r]&A_{X}
}$$
and $X$ is (isomorphic to) $\Spec R$, with $R=K[x]_{(x)}$, from Ex. \ref{ExI.4}\ref{ExI.4.b}.
\end{proof}
\end{exercise}
\lipsum*{2}

Where in the world is Ex. \ref{ExI.5}?
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