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In other words: the $k$-algebra homomorphisms
$A(X) \to L$ correspond to $L$-valued points, i.e.
points with coordinates in $L$.

Could work over $\Zbb$. Take an ideal $I \subseteq
\Zbb[x_1, \ldots, x_n]$, and set
\[
  A = \Zbb[x_1, \ldots, x_n] / I
.\]
Ring homomorphisms $A \to \Zbb$ are $1-1$
correspondence with $(a_1, \ldots, a_n) \in
\Zbb^n$ such that $f(a_1, \ldots, a_n) = 0
~\forall f \in I$. Maps $A \to \Fbb_p$ give
``solutions'' $\bmod p$, or maps $A \to \Qbb$ give
rational solutions.

\textbf{What we want:} Given an extension of $A$,e
we want to define a gadget $X = \mathrm{Spec} A$
(spectrum of $A$), and an $R$-valued point of $X$
is a ring homomorphism $A \to R$. We write the set
of $R$-valued points as
\[
  X(R) \defeq \Hom_{\text{Ring}}(A, R)
.\]
Morphisms $\mathrm{Spec} B \to \mathrm{Spec} A$
should be the same as ring homomorphisms $A \to
B$.

\begin{definition*}[Category of affine schemes]
  % Definition 0
  The category of affine schemes is the opposite
  category of rings.
\end{definition*}

\textbf{Reminder:} All of our rings have $1$ and
are commutative, and ring homomorphisms $\varphi :
A \to B$ satisfy $\varphi(1) = 1$.

\begin{definition*}
  % Definition 1
  A \emph{scheme} is a geometric object which is
  locally an affine scheme.
\end{definition*}

\textbf{Analogy:} A manifold is something which
locally looks like an open subset of $\Rbb^n$.

\begin{fcdefn}[Spectrum]
  \glssymboldefn{Spec}%
  Let $A$ be a ring. Then
  \[
    \mathrm{Spec} = \{p \subseteq A \st
    \text{$P$ a prime ideal}\}
  .\]
\end{fcdefn}

\textbf{Note:} In general, if we have an
$L$-valued point of $X = Z(I) \subseteq \Abb^n$,
we get a ring homomorphism $\varphi : A(X) \to L$
has image an integral subdomain of $L$, so $\ker
\varphi$ is prime.

\begin{fcdefn}[$V(I)$]
  \glssymboldefn{V}%
  For $I \subseteq A$ an ideal, define
  \[
    V(I) = \{ P \in \Spec A \st P \supseteq I\}
  .\]
\end{fcdefn}

\begin{proposition*}
  The sets $\V(I)$ form the closed sets of a
  topology on $\Spec A$, called the \emph{Zariski
  topology}.
\end{proposition*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $\V(A) = \emptyset$
    \item $\V(0) = \Spec A$
    \item If $\{I_j\}_{j \in J}$ is a collection
      of ideals, then
      \[
        \bigcap_{j \in J} \V(I_j)
        = \V \left( \sum_{r\in J} I_j \right) 
      .\]
      (easy!)
    \item $\V(I_1) \cup \V(I_2) = \V(I_1 \cap
      I_2)$:
      \begin{enumerate}[$\subseteq$]
        \item[$\subseteq$] If $P \supseteq I_1$ or
          $P \supseteq I_2$, then $P \supseteq I_1
          \cap I_2$.
        \item[$\supseteq$] If $P \supseteq I_1
          \cap I_2$, then $P \supseteq I_1$ or $P
          \supseteq I_2$.
      \end{enumerate}
      See Atiyah + MacDonald, Prop 1.11 ii) [Try
      to prove it for yourself!] \qedhere
  \end{enumerate}
\end{proof}

\begin{example*}
  $A = k[x_1, \ldots, x_n]$ with $k$ algebraically
  closed. For $I \subseteq A$, the maximal ideals
  of $A$ corresponding to points of $Z(I)$ are
  precisely the maximal ideals containing $I$.
\end{example*}

\newpage
\section{Sheaves}

Fix a topological space $X$.

\begin{fcdefn}[Presheaf]
  \glsnoundefn{pres}{presheaf}{presheaves}
  A \emph{preasheaf} $\mathcal{F}$ on $X$
  consists of data:
  \begin{enumerate}[(1)]
    \item For every open set $U \subseteq X$, an
      abelian group $\mathcal{F}(U)$.
    \item Whenever $V \subseteq U \subseteq X$
      open, there is a restriction homomorphism
      $\rho_{UV} : \mathcal{F}(U) \to
      \mathcal{F}(V)$ such that $\rho_{UU} =
      \id_{F(U)}$ and if $W \subseteq V
      \subseteq U \subseteq X$, then $\rho_{UW}
      = \rho_{VW} \circ \rho_{UV}$.
  \end{enumerate}
\end{fcdefn}

\begin{remark*}
  This is precisely a contravariant functor from
  the category of open sets [i.e. objects are open
  sets $U \subseteq X$, morphisms are inclusions
  $V \subseteq U$] to the category of abelian
  groups.

  Can replace the category of abelian groups with
  your favourite category.
\end{remark*}

\begin{fcdefn}[Morphism of presheaves]
  If $\mathcal{F}, \mathcal{M}$ are \glspl{pres} on
  $X$, then a \emph{morphism} of \glspl{pres} $f :
  \mathcal{F} \to \mathcal{M}$ is data of, for
  each $U \subseteq X$ open, a group homomorphism
  $f_U : \mathcal{F}(U) \to \mathcal{M}(U)$ such
  that whenever $V \subseteq U$, we have a
  commutative diagram
  \[
    \begin{tikzcd}
      s \ar[d, mapsto]
      &
      \mathcal{F}(U) \ar[r, "f_U"]
      \ar[d, "\rho_{UV}^{\mathcal{F}}"]
      & \mathcal{M}(U) \ar[d,
      "\rho_{UV}^{\mathcal{M}}"] \\
      s|_V
      &
      \mathcal{F}(V) \ar[r, "f_V"]
      & \mathcal{M}(V)
    \end{tikzcd}
  .\]
\end{fcdefn}

\begin{example*}
  $\mathcal{F}(U) = \{f : U \to \Rbb
  \text{ continuous}\}$. $\rho_{UV} :
  \mathcal{F}(U) \to \mathcal{F}(V)$ is
  restriction of functions.
\end{example*}

\begin{fcdefn}[Sheaf]
  \glsnoundefn{sheaf}{sheaf}{sheaves}
  A \gls{pres} $\mathcal{F}$ on $X$ is a sheaf if it
  satisfies:
  \begin{enumerate}[(1)]
    \item If $U \subseteq X$ is covered by
      $\{U_i\}$ ($U, U_i \subseteq X$ open) and $s
      \in \mathcal{F}(U)$ such that $s|_{U_i} =
      \rho_{UU_i}(s) = 0$, then $s = 0$.
    \item If $U, \{U_i\}$ are as is (1), and $s_i
      \in \mathcal{F}(U_i)$ for each $i$ such that
      $s_i|_{U_i \cap U_j} = s_j |_{U_i \cap U_j}$
      for all $i, j$, then there exists $s \in
      \mathcal{F}(U)$ such that $s|_{U_i} = s_i$
      for all $i$ (gluing axiom).
  \end{enumerate}
\end{fcdefn}
