\documentclass[12pt, reqno]{amsart}

\input{frontmatter/packages.tex}
\input{frontmatter/standardMacros.tex}
\input{frontmatter/extraMacros.tex}
\input{frontmatter/theoremStyle.tex}



\begin{document}
\title[]{Stacks HW3 - \'etale morphisms and sheaves}\maketitle
\begin{enumerate}

\item Think of lots of examples and non examples of \'etale morphisms, and work out the details as explicitly as you can.

\item Show that the sheaf axiom of Hartshorne is equivalent to our sheaf axiom.

%\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 Let $C$ be a category and let  $D \colon I \to \Fun(C^{op},\Sets)$ be a diagram. 
\begin{enumerate}
\item Show that $\varprojlim D$ is the functor $X \mapsto \varprojlim
  D(i)(X)$.
\item Show that $\varinjlim D$ is the functor $X \mapsto \varinjlim D(i)(X)$.
\end{enumerate}

\item Let $C$ be a category and let  $D \colon I \to \widetilde{\Sch}$
  be a diagram, where $\widetilde{\Sch}$ is the subcategory of sheaves in $\Fun(C^{op},\Sets)$.
\begin{enumerate}
\item Show that $\varprojlim D$ (in the category of sheaves) is the functor $X \mapsto \varprojlim
  D(i)(X)$.
\item Show that $\varinjlim D$ (in the category of sheaves)  is the \emph{sheafication} of the functor $X \mapsto \varinjlim D(i)(X)$.
\end{enumerate}

\item Let $F$ be a sheaf, $F'$ be a presheaf, and $f\colon F' \to F$ be an
  injection (as presheaves). Show that the sheafication of $F'$ is
  isomorphic to $F$ if and only if every section of $F$ is locally in
  the image of $f$. 


% \item When is a morphism of fields \'etale?
% \item When is a morphism of DVR's \'etale?
% \item When is a morphism $\calO_K[S] \to \calO_L[S']$ \'etale?
% \item Suppose that $S \to \Spec K$ is \'etale. What can $S$ look like?
% \item Etale morphisms of curves. What do they look like?




% \item Make a flow chart of the different descriptions of \'etale, etc.
% \item Use the infintesmal description of etale, etc, to prove that a composition of etale morphisms is etale. Etc.
% \item Show that being etale is local on the source.
% \item Show that all of these properties are stable under base change.
% \item Show that one can check blankness etale locally.
% \item Show that you only need Artin rings to check the inf description.
% \item Write out the equations for a morphism of curves that looks like ...
% \item A problem that gets them comfortable with the Jacobian criteria of etaleness.


% \item 2. Suppose $p:D\to C$ is a fibered category ''fibered in groupoids'' (i.e. for any object $X$ of $C$, every morphism in $D(X)$ is an isomorphism). Show that every arrow of $D$ is cartesian.



\end{enumerate}
\end{document}

