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\begin{document}
\title[]{Stacks HWX - running list}\maketitle
\begin{enumerate}
\item Show that the sheaf axiom of Hartshorne is equivalent to our sheaf axiom using coproducts.
\item Verify that $p^{-1}$ and $p_*$ of a sheaf is a sheaf.
\item Let $F$ be a presheaf and let $p\colon X' \to X$ and $q\colon Y' \to Y$ be two morphisms such $F$ satisfies the sheaf axiom with respect to every base change of $p$ and $q$. Prove that $F$ satisfies the sheaf with respect to $p \times q \colon X' \times Y' \to X \to Y$.
\item Adjoint functor is fully faithful if and only if the unit (or counit) is an isomorphism. (Hint: Yoneda's lemma.)
\item \textbf{Diagonal}.
\begin{enumerate}
\item Prove that that the diagonal is an isomorphism if and only if $f$ is etale.
\end{enumerate}
\item A functor on $X_{zar}$ with an open cover by schemes is a scheme.
\item Show, explicitly, that the map $[G/G]^{ps} \to \star$ is an equivalence of categories.
\item Stackify the stack $B_{\G_m}$ by hand.
\item Let $R \to X \times X$ be an equivalence relation. Show that the diagonal $\bigtriangleup \colon X \to X \times X$
\end{enumerate}
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