%! TEX root = AG2.tex
% vim: tw=50
% 16/10/2024 10AM

\begin{remark*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item If $\mathcal{F}$ is a \gls{sheaf}, then
      $\mathcal{F}(\emptyset) = 0$ since the empty
      cover is a cover of $\emptyset$.
    \item Properties (1) and (2) of the definition
      of a \gls{sheaf} together can be stated by
      saying
      \[
        0 \to F(U) \stackrel{\alpha}{\to}
        \bigoplus_{i \in I} \mathcal{F}(U_i)
        \twofuncs{\beta_1}{\beta_2}
        \bigoplus_{i, j \in I} \mathcal{F}(U_i
        \cap U_j)
      \]
      is \emph{exact}, for all $U \subseteq X$
      open and open covers $\{U_i\}$ of $U$.

      Here,
      \begin{align*}
        \alpha(S)
        &= (S|_{U_i})_{i \in I} \\
        \beta_1((S_i)_{i \in I})
        &= (S_i|_{U_i \cap U_r})_{i, j \in I} \\
        \beta_2((S_i)_{i \in I})
        &= (S_j|_{U_i \cap U_r})_{i, j \in I}
      \end{align*}
      Exactness means:w
      \begin{enumerate}[(1)]
        \item $\alpha$ is injective (this is
          property
          (1) of a \gls{sheaf})
        \item $\beta_1 \circ \alpha = \beta_2
          \circ \alpha$ (obvious)
        \item For any $(S_i) \in \bigoplus_{i \in
          I} \mathcal{F}(U_i)$ with
          $\beta_1((S_i)) = \beta_2((S_i))$, there
          exists an $s \in \mathcal{F}(U)$ with
          $\alpha(S) = (S_i)$. (this is property
          (2) of a \gls{sheaf})
      \end{enumerate}
  \end{enumerate}
\end{remark*}

\begin{remark*}
  $\alpha$ is the \emph{equalizer} of $\beta_1,
  \beta_2$ (from category theory).
\end{remark*}

This definition works, even if e.g. we a set
rather than abelian group.

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $X$ any topological space
      \[
        \mathcal{F}(U) = \{\text{continuous
        functions $f : U \to \Rbb$}\}
      \]
      is a \gls{sheaf}.
    \item $X = \Cbb$ with the Euclidean topology.
      Then for
      \[
        \mathcal{F}(U) = \{f : U \to \Cbb \mid
        \text{$f$ bounded and holomorphic}\}
      \]
      gluing fails because one may not preserve
      boundedness.
    \item Let $G$ be a group, and set
      $\mathcal{F}(U) = G$ for all $U \subseteq X$.
      $\rho_{UV} = \id$. This is a \gls{pres},
      known as the \emph{constant presheaf}. If we
      give $G$ the discrete topology, set
      \[
        \mathcal{F}'(U) = \{f : U \to G
        \text{continuous}\}
      .\]
      These are locally constant functions.
      Obviously a \gls{sheaf}, called the
      \emph{constant sheaf}.
    \item If $X$ is a variety, denote by
      $\mathcal{O}_X(U)$ the set of regular
      functions $f : U \to k$. $\mathcal{O}_X$ is
      a \gls{sheaf}, called the \emph{structure
      sheaf} of $X$. See Part II \courseref{AG}
      for definitions of these.
  \end{enumerate}
\end{example*}

\begin{fcdefn}[Stalk / germ]
  Let $\mathcal{F}$ be a \gls{pres} on $X$, $p
  \in X$. Then the \emph{stalk} of $f$ at $p$ is
  \[
    \mathcal{F}_p \defeq \{(U, s) \st \text{$U$ an
    open neighbourhood of $p$, $s \in
    \mathcal{F}(U)$}\} / \equiv
  \]
  where $(U, s) \equiv (V, t)$ if there exists $W
  \subseteq U \cap V$ a neighbourhood of $p$ such
  that $s|_W = t|_W$.

  The equivalence class of $(U, s) \in
  \mathcal{F}_p$ is written as $s_p$, and is the
  \emph{germ} of $s$ at $p$.
\end{fcdefn}

Note that given a morphism $f : \mathcal{F} \to
\mathcal{G}$, we obtain $f_p : \mathcal{F}_p \to
\mathcal{G}_p$ via $f_p(U, s) = (U, f_U(s))$.

Note that a morphism of \glspl{sheaf} is just a
morphism of \glspl{pres}.

\begin{proposition*}
  Let $f : \mathcal{F} \to \mathcal{G}$ be a
  morphism of \glspl{sheaf}. Then $f$ is an
  isomorphism if and only if $f_p$ is an
  isomorphism for all $p \in X$.
\end{proposition*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[$\Rightarrow$]
    \item[$\Rightarrow$] Obvious.
    \item[$\Leftarrow$] Assume $f_p$ is an
      isomorphism for all $p$. We will show $f_U :
      \mathcal{F}(U) \to \mathcal{G}(U)$ is an
      isomorphism for all $U$, and can then define
      the inverse to $f$ by $(f^{-1})_U =
      (f_U)^{-1}$.

      $f_U$ is injective: Suppose $s \in
      \mathcal{F}(U)$, $f_U(s) = 0$. Then for all
      $p \in U$, $f_p((U, s)) = (U, f_u(s)) = (U,
      0) = 0 \in \mathcal{G}_p$. Thus $s_p = 0$
      since $f_p$ is injective. Thus there exists
      an open neighbourhoord $V_p \subseteq U$ of
      $p$ such that $s|_{V_p} = 0$. But
      $\{V_p\}_{p \in U}$ covers $U$, so by (1) in
      the definition of a \gls{sheaf}, we get $s =
      0$.

      $f_U$ is surjective: Let $p \in
      \mathcal{G}(U)$. Then for all $p \in U$,
      there exists $s_p \in \mathcal{F}_p$ such
      that $f_p(s_p) = t_p$, i.e. there exists an
      open neighbourhood of $p \in U$ and a germ
      $(V_p, \tilde{s}_p)$ representating $s_p$
      such that $(V_p, f_{V_p}(\tilde{s}_p)) = (U,
      t) = t_p$. Shrinking $V_p$ if necessary, we
      can assume $f_{V_p}(\tilde{s}_p) = t|_{V_p}$
      on $V_p \cap V_q$. Then
      \[
        f_{V_p \cap V_q}(\tilde{s}_p |_{V_p \cap
        V_q} - \tilde{s}_q|_{V_p \cap V_q})
        = t|_{V_p \cap V_q} - t|_{V_p \cap V_q} =
        0
      .\]
      Since we have shown $f_{V_p \cap V_q}$ is
      injective, we get
      \[
        \tilde{s}_p|_{V_p \cap V_q} =
        \tilde{s}_q|_{V_p \cap V_q}
      ,\]
      and by (2) in the definition of a
      \gls{sheaf}, there exists $s \in
      \mathcal{F}(U)$ such that $s|_{V_p} =
      \tilde{s}_p$ for all $p$. Now
      \[
        f_U(s)|_{V_p} = f_{V_p}(s|_{V_p}) =
        f_{V_p}(\tilde{s}_p) = t|_{V_p}
      .\]
      Thus $f_U(s) - t = 0$ by (1) in the
      definition of a \gls{sheaf}, i.e. $f_U(s) =
      t$. Thus $f_U$ is surjective.
      \qedhere
  \end{enumerate}
\end{proof}

\begin{remark*}
  If instead $f_p : \mathcal{F}_p \to
  \mathcal{G}_p$ is injective for all $p$, then
  $f_U : \mathcal{F}(U) \to \mathcal{G}(U)$ is
  still injective.

  If instead $f_p : \mathcal{F}_p \to
  \mathcal{G}_p$ is surjective for all $p$, we
  need not have $f_U : \mathcal{F}(U) \to
  \mathcal{G}(U)$ surjective (we will see examples
  later).
\end{remark*}

\textbf{Sheafification:} Given a \gls{pres}
$\mathcal{F}$, there exists a \gls{sheaf}
$\mathcal{F}^+$ and a morphism $\theta :
\mathcal{F} \to \mathcal{F}^+$ satisfying the
following universal property:

For any sheaf $\mathcal{G}$ and a morphism
$\varphi : \mathcal{F} \to \mathcal{G}$, there
exists a unique morphism $\varphi^+ :
\mathcal{F}^+ \to \mathcal{G}$ such that
\[
  \begin{tikzcd}
    F \ar[r, "\theta"] \ar[rd, "\varphi"]
    & F^+ \ar[d, "\varphi^+"] \\
    & \mathcal{G}
  \end{tikzcd}
\]
commutes. The pair $(\mathcal{F}^+, \theta)$ is
unique up to unique isomorphism.

Also, $\mathcal{F}_p \cong \mathcal{F}_p^+$ via
$\theta_p$ for all $p \in X$.
