\documentclass{lecture}

\begin{document}

\section{Gluing spaces with functions}

We present a general technique to construct spaces with functions by
,,patching together`` other spaces with functions ,,along open subsets``. This
will later be used to argue that, in order to define a structure of variety on a
topological sapce (or even a set), it suffices to give one atlas.

\begin{theorem}[Gluing theorem]
    Let $(X_i, \mathcal{O}_{X_i})_{i \in I}$ be a family of spaces with functions. For
    all pair $(i, j)$, assume that the following has been given
    \begin{enumerate}[(a)]
        \item an open subset $X_{ij} \subseteq X_i$
        \item an isomorphism of spaces with functions
            \[
            \varphi_{ji}\colon (X_{ij}, \mathcal{O}_{X_{ij}})
            \to (X_{ji}, \mathcal{O}_{X_{ji}})
            \]
    \end{enumerate}
    subject to the following compatibility conditions
    \begin{enumerate}[(1)]
        \item for all $i$, $X_{ii} = X_i$ and $\varphi_{ii} = \text{id}_{X_i}$
        \item for all pair $(i, j)$, $\varphi_{ij} = \varphi_{ji}^{-1}$
        \item for all triple $(i, j, k)$, $\varphi_{ji}(X_{ik} \cap X_{ij}) = X_{jk} \cap X_{ji}$
            and $\varphi_{kj} \circ \varphi_{ji} = \varphi_{ki}$
            on $X_{ik} \cap X_{ij}$.
    \end{enumerate}

    Then there exists a space with functions $(X, \mathcal{O}_X)$ equipped with a family of
    open sets $(U_i)_{i \in I}$
    and isomorphisms of spaces with functions
    \begin{enumerate}[(A1)]
        \item $\varphi_i \colon (U_i, \mathcal{O}_X|_{U_i}) \to (X_i, \mathcal{O}_{X_i})$,
    \end{enumerate}
    such that $\bigcup_{i \in  I} U_i = X$ and, for all pair $(i, j)$,
    \begin{enumerate}[(A1)]
        \setcounter{enumi}{1}
        \item $\varphi_i(U_i \cap U_j) = X_{ij}$, and
        \item $\varphi_j \circ \varphi_i^{-1} = \varphi_{ji}$ on $X_{ij}$.
    \end{enumerate}
    Such a familiy $(U_i, \varphi_i)_{i \in I}$ is called
    an atlas for $(X, \mathcal{O}_X)$.

    Moreover, if $(Y, \mathcal{O}_Y)$ is a space with functions equipped with an atlas
    $(V_i, \psi_i)_{i \in I}$ satisfying conditions (A1), (A2) and (A3), then
    the isomorphisms $\psi_i^{-1} \circ \varphi_i \colon U_i \to V_i$ induce
    an isomorphism $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$.
\end{theorem}

\begin{proof}
    Uniqueness up to canonical isomorphism: Let $(U_i, \varphi_i)_{i \in I}$
    and $(V_i, \psi_i)_{i \in I}$ be two atlases modelled on the same gluing data,
    then for all pair $(i, j)$,
    \begin{salign*}
    \psi_j^{-1} \circ \varphi_j \Big|_{U_i \cap U_j}
    &= \psi_j^{-1} \circ \underbrace{(\varphi_j \circ \varphi_i^{-1})}_{= \varphi_{ji}}
    \circ \varphi_i \Big|_{U_i \cap U_j} \\
    &= \psi_j^{-1} \circ \underbrace{(\psi_j \circ \psi_i^{-1})}_{= \varphi_{ji}}
    \circ \varphi_i \Big|_{U_i \cap U_j} \\
    &= \psi_i^{-1} \circ \varphi_i \Big|_{U_i \cap U_j}
    \end{salign*}
    so there is a well-defined map
    \begin{salign*}
        f\colon X = \bigcup_{i \in  I} U_i &\to \bigcup_{i \in  I} V_i = Y \\
        (x \in U_i) &\mapsto (\psi_i^{-1} \circ \varphi_i(x) \in V_i)
    \end{salign*}
    which induces an isomorphism
    of spaces with functions.

    Existence: Define $\tilde{X} \coloneqq \bigsqcup_{i \in I} X_i$ and let the
    topology be the final topology with respect to the canonical maps
    $(X_i \to \tilde{X})_{i \in I}$. Then define
    $X \coloneqq \tilde{X} / \sim $ where
    $(i, x) \sim (j, y)$ in $\tilde{X}$ if $x = \varphi_{ij}(y)$. Conditions
    (1), (2) and (3) show that $\sim $ is reflexive, symmetric and transitive.
    We equip $X$ with the quotient topology and denote by
    \[
    p\colon \tilde{X} \to X
    \] the canonical continuous projection. Let $U_i \coloneqq p(X_i)$. Since
    $p^{-1}(U_i) = \bigsqcup_{j \in I} X_{ji}$
    is open in $\tilde{X}$, $U_i$ is open in $X$. Moreover,
    $\bigcup_{i \in  I} U_i = X$, so we have an open covering of $X$. We
    put $p_i \coloneqq p|_{X_i}$ and we define a sheaf on $X$ by setting
    \[
    \mathcal{O}_X(U) \coloneqq \{ f \colon U \to k  \mid \forall i \in I, f \circ p_i
    \in \mathcal{O}_{X_i}(p_i^{-1}(U)) \} 
    \] for all open sets $U \subseteq X$. This defines a sheaf on $X$, with
    respect to which $(X, \mathcal{O}_X)$ is a space with functions.
    Finally, $p_i\colon X_i \to U_i$ is a homeomorphism and, by construction
    $\mathcal{O}_{U_i} \simeq (p_i)_{*} \mathcal{O}_{X_i}$ via pullback by $p_i$.
    We have thus constructed a space with functions $(X, \mathcal{O}_X)$,
    equipped with an open covering $(U_i)_{i \in I}$ and local charts
    \[
    \varphi_i \coloneqq p_i^{-1}\colon (U_i, \mathcal{O}_X|_{U_i})
    \stackrel{\sim }{\longrightarrow }
    (X_i, \mathcal{O}_{X_i})
    .\] It remains to check that
    $\varphi_i(U_i \cap U_j) = X_{ij}$ and
    $\varphi_j \circ \varphi_i^{-1} = \varphi_{ji}$ on $X_{ij}$, but
    this follows from the construction of
    $\displaystyle{X = \bigsqcup_{i \in I} X_i / \sim }$ and
    the definition of the $\varphi_i$'s as $p|_{X_i}^{-1}$.
\end{proof}

\begin{bsp}[]
    Take $k = \R$ or $\mathbb{C}$ equipped with either the Zariski or the usual topology. Consider
    the spaces with functions $X_1 = k$, $X_2 = k$ and the open sets
    $X_{12} = k \setminus \{0\} \subseteq X_1$ and
    $X_{21} = k \setminus \{0\} \subseteq X_2$. Finally, set
    \begin{salign*}
        \varphi_{21}\colon X_{12} &\to X_{21} \\
        t &\mapsto \frac{1}{t}
    .\end{salign*}
    Since this is an isomorphism of spaces with functions, we can glue
    $X_1$ and $X_2$ along $X_{12} \xlongrightarrow[\varphi_{21}]{\sim } X_{21} $
    and define a space with functions $(X, \mathcal{O}_X)$ with
    an atlas modelled on $(X_1, X_2, \varphi_{21})$. We will now identify this
    space $X$ with the projective line $k \mathbb{P}^{1}$. By definition, the latter
    is the set of $1$-dimensional vector subspaces (lines) of $k^2$:
    \begin{salign*}
    k \mathbb{P}^{1} \coloneqq (k^2 \setminus \{0\}) / k^{\times }
    .\end{salign*}
    Then, we have a covering
    $U_1 \cup U_2 = k \mathbb{P}^{1}$, where
    $U_1 = \{ [x_1 : x_2]  \mid x_1 \neq 0\} $
    and $U_2 = \{ [x_1 : x_2 ]  \mid x_2 \neq 0\} $, and we can define charts
    \begin{salign*}
        \varphi_1\colon U_1 &\xlongrightarrow{\sim } k \\
        [x_1 : x_2 ] &\longmapsto x_2 / x_1 \\
        [1:w] & \longmapsfrom w
    \end{salign*}
    and $\varphi_2\colon U_2 \to k$ likewise. Then, on the intersection
    \[
    U_1 \cap U_2 = \{ [x_1 : x_2 ]  \mid x_1 \neq 0, x_2 \neq 0\}
    \] we have a commutative diagram
    \[
    \begin{tikzcd}
        U_1 \cap U_2 \arrow{d}{\varphi_1} \arrow{dr}{\varphi_2} & \\
        X_1 \arrow{r}{\varphi_{21}} & X_2
    \end{tikzcd}
    \] with $\varphi_i(U_1 \cap U_2)$ open in $X_i$. In view of
    the gluing theorem, we can use this to set up a bijection
    $k \mathbb{P}^{1} \to X$ where $\displaystyle{X \coloneqq (X_1 \sqcup X_2) / \sim_{\varphi_{12}}}$
    and define a topology and a sheaf of regular functions on
    $k \mathbb{P}^{1}$ via this identification. Note that this was done without putting 
    a topology on $k \mathbb{P}^{1}$: the latter is obtained using the bijection
    $k \mathbb{P}^{1} \to X$ constructed above. We now spell out the notion of regular functions
    thus obtained on $k \mathbb{P}^{1}$.
\end{bsp}

\begin{satz}
    With the identification
    \[
    k \mathbb{P}^{1} = X_1 \sqcup X_2 / \sim
    \] constructed above, a function $f\colon U \to k$ defined on 
    an open subset $U \subseteq k \mathbb{P}^{1}$ is an element of $\mathcal{O}_X(U)$ if
    and only if, for each local chart $\varphi_i \colon U_i \to k$, the function
    \[
    f \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U) \to k
    \] is regular on the open set $\varphi_i(U_i \cap U) \subseteq k$.
\end{satz}

\begin{definition}[]
    Let $k$ be a field. An \emph{algebraic $k$-prevariety} is a space
    with functions $(X, \mathcal{O}_X)$ such that
    \begin{enumerate}[(i)]
        \item $X$ is quasi-compact.
        \item $(X, \mathcal{O}_X)$ is locally isomorphic to an affine variety.
    \end{enumerate}
\end{definition}

\begin{bem}[]
    Saying that $(X, \mathcal{O}_X)$ is locally isomorphic to an affine variety means
    that for $x \in X$, it exists an open neighbourhood $x \in U$ such that
    $(U, \mathcal{O}_X|_U)$ is isomorphic to an open subset of an affine variety. Since
    such an open set is a union of principal open sets, which are themselves affine, one can
    equivalently ask that $(U, \mathcal{O}_U)$ be affine. Thus:
\end{bem}

\begin{satz}
    A space with functions $(X, \mathcal{O}_X)$ is an algebraic prevariety, if and only if
    there exists a finite open covering
    \[
    X = U_1 \cup \ldots \cup U_n
    \] such that $(U_i, \mathcal{O}_X|_{U_i})$ is an affine variety.
\end{satz}

\begin{bem}[]
    As a consequence of the gluing theorem, in order to either construct an algebraic
    prevariety or put a structure of an algebraic prevariety on a set, it suffices to either
    define $X$ from certain gluing data $(X_i, X_{ij}, \varphi_{ij})_{(i,j)}$ satisfying
    appropriate compatibility conditions, or find a covering
    $(U_i)_{i \in I}$ of a set $X$ and local charts $\varphi_i \colon U_i \to X_i$ such that
    $X_{ij} = \varphi_i (U_i \cap U_j)$ is open in $X_i$ and
    $\varphi_j \circ \varphi_i^{-1}$ is an isomorphism of spaces with functions.

    In practice, $X$ is sometimes given as a topological space, and
    $(U_i)_{i \in I}$ is an open covering, with local charts $\varphi_i\colon U_i \to X_i$ that
    are homeomorphisms. So the condition that $X_{ij}$ be open in $X_i$ is automatic
    in this case and one just has to check that
    \[
    \varphi_{j} \circ \varphi_i^{-1} \colon X_{ij} \to X_{ji}
    \] induces an isomorphism of spaces with functions. In the present context where
    $X_i$ and $X_j$ are affine varieties, this means a map
    \[
    X_{ij} \subseteq k^{n} \to X_{ji} \subseteq k^{m}
    \] between locally closed subsets of $k^{n}$ and $k^{m}$ whose components are regular functions.
\end{bem}

\begin{bsp}[Projective sets]
    We have already seen that projective spaces $k \mathbb{P}^{n}$ are algebraic pre-varieties.
    Let $P \in k[x_0, \ldots, x_n]_d$ be a homogeneous polynomial of degree $d \ge 0$. Although
    $P$ cannot be evaluated at a point
    $[x_0 : \ldots : x_n] \in k \mathbb{P}^{n}$, the condition
    $P(x_0, \ldots, x_n) = 0$ can be tested, because for $\lambda \in k^{x}$,
    \begin{salign*}
    P(x_0, \ldots, x_n) = 0 \iff 0 = \lambda ^{d} P(x_0, \ldots, x_n)
    = P(\lambda x_0, \ldots, \lambda x_n)
    .\end{salign*}
    We use this to define the following \emph{projective sets}
    \[
    \mathcal{V}_{k \mathbb{P}^{n}}(P_1, \ldots, P_m)
    = \{ [x_0 : \ldots : x_n] \in k \mathbb{P}^{n}  \mid  P_i(x_0, \ldots, x_n) = 0 \quad \forall i\} 
    \] for homogeneous polynomials in $(x_0, \ldots, x_n)$.

    We claim that these projective sets are the clsoed sets of a topology on
    $k \mathbb{P}^{n}$, called the Zariski topology. A basis for that topology
    is provided by the principal open sets
    $D_{k \mathbb{P}^{n}} (P)$ where $P$ is a homogeneous polynomial. By definition, a regular
    function on a locally closed subset of $k \mathbb{P}^{n}$ is locally given by the restriction
    of a ration fraction of the form
    \[
    \frac{P(x_0, \ldots, x_n)}{Q(x_0, \ldots, x_n)}
    \] where $P$ and $Q$ are homogeneous polynomials of the same degree.
    This defines a sheaf of regular functions on any given locally closed subset
    $X$ of $k \mathbb{P}^{n}$.
\end{bsp}

\begin{satz}
    A Zariski-closed subset $X$ of $k \mathbb{P}^{n}$ equipped with its
    sheaf of regular functions, is an algebraic pre-variety. The same holds
    for all open subsets $U \subseteq X$.
\end{satz}

\begin{proof}
    Consider the open covering
    \begin{salign*}
        X &= \bigcup_{i = 0} ^{n} X \cap U_i \\
          &= \bigcup_{i = 0}^{n} \{ [x_0 : \ldots : x_n ] \in X  \mid x_i \neq 0\} 
    .\end{salign*}
    Then the restriction to $X \cap U_i$ of the local chart
    \begin{salign*}
        \varphi_i \colon U_i &\longrightarrow k^{n} \\
        x = [x_0 : \ldots : x_n] &\longmapsto
        \underbrace{\left( \frac{x_0}{x_i}, \ldots, \hat{\frac{x_i}{x_i}}, \ldots, \frac{x_n}{x_i} \right)}_{w = (w_0, \ldots, \hat{w}_i, \ldots, w_n)}
    \end{salign*}
    sends an $x$ such that $P_1(x) = \ldots = P_m(x) = 0$ to a $w$ such that
    $Q_1(w) = \ldots = Q_m(w) = 0$ where, for all $j$,
    \begin{salign*}
        Q_j(w) &= P_j(w_0, \ldots, w_{i-1}, 1, w_{i+1}, \ldots, w_n) \\
               &= P_j(x_0, \ldots, x_{i-1}, x_i, x_{i+1}, \ldots, x_n)
    \end{salign*}
    is the dehomogeneisation of $P_j$. So
    $\varphi_i(X \cap U_i) = \mathcal{V}_{k^{n}}(Q_1, \ldots, Q_m) \eqqcolon X_i$
    is an algebraic subset of $k^{n}$, in particular an affine variety. It remains
    to check that $\varphi_i|_{X \cap U_i}$ pulls back regular functions on $X_i$ to
    regular functions on $X \cap U_i$, and similarly for $(\varphi_i|_{X \cap U_i})^{-1}$.
    But if $f$ and $g$ are polynomials in $(w_0, \ldots, \hat{w}_i, \ldots, w_n)$,
    \begin{salign*}
        \left(\varphi_i^{*} \frac{f}{g}\right)(x)
        &= \frac{f(\varphi_i(x))}{g(\varphi_i(x))} \\
        &= \frac{f\left( \frac{x_0}{x_i}, \ldots, \hat{\frac{x_i}{x_i}}, \ldots, \frac{x_n}{x_i} \right) }{g\left( \frac{x_0}{x_i}, \ldots, \hat{\frac{x_i}{x_i}}, \ldots, \frac{x_n}{x_i} \right) }
    \end{salign*}
    which can be rewritten as a quotient of two homogeneous polynomials of the same
    degree by multiplying the numerator and denominator
    by $x_i^{r}$ with $r \ge \text{max}(\text{deg}(f) , \text{deg}(g))$. The computation
    is similar but easier for $\left( \varphi_i |_{X \cap U_i} \right)^{-1}$.
\end{proof}

\begin{definition}
    A space with functions $(X, \mathcal{O}_X)$ which is isomorphic to a
    Zariski-closed subset of $k \mathbb{P}^{n}$ is called a
    \emph{projective $k$-variety}.
\end{definition}

\end{document}