\documentclass[12pt, reqno]{amsart}

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



\begin{document}
\title[]{Stacks HW2 - limits, colimits, group objects}\maketitle
\begin{enumerate}

\item Let $C$ be a category and let $h \colon C \to
  \Fun(C^{\op},\Sets)$ be the Yoneda embedding. Show that for any
  arrows $X \to Y$ and $Z \to Y$ in $C$, there is a natural
  isomorphism 
\[
h_{X\times_Y Z} \to h_X \times_{h_Y} h_Z
\]
of functors, where $h_X \times_{h_Y} h_Z$ is the functor
\[
h_X \times_{h_Y} h_Z \colon W \mapsto h_X(W) \times_{h_Y(W)} h_Z(W).
\]

\item Let $G$ be an object of a category $C$. Show that the
  functor of points
\[
h_G \colon C^{\op} \to \Sets
\]
factors through the forgetful functor from groups
\[
\xymatrix{
& \ar[d] {\Groups}\\
h_G \colon C^{\op} \ar[r] \ar[ru]& \Sets
}
\]
if and only if $G$ is a group object of $C$.


\item Let  $\star$ be one of $\Spec k$ (with $k$ a field) or $\Spec
  \Z$. Let 
$$G = \coprod_{\g \in \Z/2\Z} \star$$
be the group object corresponding to $\Z/2\Z$. Work out explictily the
group structure. In other words, work out the maps in terms of rings,
and show that $G$ represents the functor 
\[
\Sch^{\op} \to \Groups, \,  W \mapsto \Z/2\Z
\]


\item
  \begin{enumerate}
  \item Let $n \geq 1$ be an integer and let
    \[
    \GL_n \colon (\Sch)^{\op} \to \Sets
    \]
    be the functor sending a scheme $Y$ to the set
    $\GL_n(\Gamma(Y,\calO_Y))$. Prove that $\GL_n$ is a representable
    functor.
  \item Let $X$ represent the functor $\GL_n$. Prove that the group
    structures on the sets $\GL_n(\Gamma(Y,\calO_Y))$ induce the
    structure of a group scheme on $X$. (I.e. use the previous
    exercise, noting that there is one detail to check.)

  \end{enumerate}


\item Give an example of a category $C$, a subcategory $C'$, and a
  diagram $D \colon I \to C'$ such that the limit (or colimit, your
  choice) in $C'$ is not the limit in $C$. 

\item Use the functor of points to define a map $\A^1 \to \A^1$ given
  by the formula $z \mapsto z^2$. Compare this with how one would
  define such a map with locally ringed spaces.

\item \textbf{Monomorphisms}.
  \begin{enumerate}
  \item Show that a morphism $X \to Y$ is a monomorphism if and only
    if for every $T \in C$, the map of sets $X(T) \to Y(T)$ is
    injective.
  \item Show that a map of schemes which is injective topologically may not be a monomorphism.
  \item Show that a map of schemes which is surjective topologically may not be an epimorphism.
  \end{enumerate}


\end{enumerate}
\end{document}

