add rav 10

sose2022/galois/vortrag_affin.tex

@@ -841,5 +841,62 @@ Wir benötigen noch zwei Lemmata aus der kommutativen Algebra:
     ersten Absatz.
 \end{proof}
 
+\begin{bsp}[]
+    Sei $A = \mathbb{C}[X]$ und $B = A[Y]/(Y^2 - X)$.  
+    Dann ist $\{1, Y\}$ eine Basis
+    von $B$ über $A$.
+    \begin{proof}
+    $\{1, Y\} $ ist ein Erzeugendensystem von $B$ als $A$-Modul, denn
+    $Y^2 = X$ in $B$.
+    Außerdem folgt für $f, g \in \mathbb{C}[X]$ mit $f + gY = 0$ bereits, dass
+    $f + gY \in (Y^2 - X)$ also existiert ein $h \in \mathbb{C}[X]$ mit $f + gY = h(Y^2 - X)$.
+    Angenommen $g \neq 0$. Dann
+    ist $1 = \text{deg}_Y (f + gY) = \text{deg}_Y(h) + \text{deg}_Y(Y^2 -X) = \text{deg}_Y(h) + 2$.
+    Widerspruch. Also ist $f \in (Y^2 - X)$, aber da $f \in \mathbb{C}[X]$ folgt $f = 0$. Also
+    sind $\{1, Y\}$ linear unabhängig.
+    \end{proof}
+
+    $B$ ist also endlich frei und insbesondere projektiv über $A$. Allerdings ist
+    mit $f = Y^2 - X$
+    \[
+    \Omega_{B / A} = \Omega_{(A[Y] / (f)) / A} \simeq A[Y]/(f, f') = A[Y]/(Y^2 - X, 2Y)
+    = \mathbb{C}[X,Y]/(Y^2 - X, 2Y)
+    .\] Nach Ersetzen von $A$ durch $A_X = \mathbb{C}[X, X^{-1}]$
+    ist $Y^2$ eine Einheit in $B$ und damit auch $Y$. Also ist
+    \[
+    \Omega_{B / A} = B / (2Y) = 0
+    .\] Also $A \to B$ endlich étale. Es ist $[ B : A ] = 2$ und
+    sei $\psi\colon \spec B \to \spec A$ die induzierte Abbildung. Dann ist
+    $\spec A \simeq D(X) \subseteq \spec \mathbb{C}[X]$. Sei $\lambda \in \mathbb{C}^{\times}$
+    und $x_{\lambda} = \mathfrak{p}_{\lambda}$ das von $(X - \lambda)$ erzeugte Maximalideal.
+    Dann folgt
+    \begin{align*}
+    \psi^{-1}(x_{\lambda})
+    &\simeq \spec \kappa(x_{\lambda}) \otimes_A B \\
+    &= \spec A/(X - \lambda) \otimes_{A} A[Y]/(Y^2-X) \\
+    &\simeq \spec A[Y]/(Y^2 - X, X - \lambda) \\
+    &\simeq \spec \mathbb{C}[Y]/(Y^2 - \lambda) \\
+    &= \{ Y - \lambda, Y + \lambda \} 
+    .\intertext{Aber für den generischen Punkt $\xi \in \spec A$ folgt}
+        \psi^{-1}(\xi) &\simeq \spec \kappa(\xi) \otimes_A B \\
+                  &\simeq \spec \mathbb{C}(X) \otimes_A A[Y] / (Y^2 - X) \\
+                  &\simeq \spec \mathbb{C}(X)[Y]/(Y^2 - X)
+    .\end{align*}
+    Das Polynom $(Y^2 - X)$ ist irreduzibel in $\mathbb{C}(X)$ also
+    ist $\mathbb{C}(X)[Y]/(Y^2 - X)$ eine Körpererweiterung vom Grad $2$ von $\mathbb{C}(X)$.
+    Insbesondere ist die Faser $\psi^{-1}(\xi)$ einelementig, also ist
+    $\psi$ keine topologische Überlagerung.
+
+    Es ist weiter
+    \begin{align*}
+        B \otimes_A B &= B \otimes_A A[Z]/(Z^2-X) \\
+                      &= B[Z]/(Z^2-X) \\
+                      &= B[Z] / (Z - Y) (Z + Y) \\
+                      &\simeq B[Z] / (Z - Y) \times B[Z] / (Z + Y)\\
+                      &\simeq B \times B
+    .\end{align*}
+    Also wird $A \to B$ durch Basiswechsel mit $A \to B$ total zerlegbar.
+\end{bsp}
+
 
 \end{document}

sose2022/metaethik/axiomatische_ethik.tex

@@ -18,7 +18,7 @@
 
 %Die Philosophie ist eine hoffnungslose Wissenschaft.
 
-Nach Moore ist eine Definition des Wortes \emph{gut} unmöglich. gut
+Nach Moore ist eine Definition des Wortes \emph{gut} unmöglich. Gut
 wird von ihm als abstraktes, einfaches Grundwort vorausgesetzt, dessen Existenz er postuliert.
 Dieses Prinzip findet sich
 in der Mathematik, speziell der Mengenlehre wieder. Hier wird der Begriff der \emph{Menge}

ws2022/rav/lecture/rav.tex

@@ -24,5 +24,6 @@ Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.un
 \input{rav7.tex}
 \input{rav8.tex}
 \input{rav9.tex}
+\input{rav10.tex}
 
 \end{document}

ws2022/rav/lecture/rav10.tex

@@ -0,0 +1,378 @@
+\documentclass{lecture}
+
+\begin{document}
+
+\section{Examples of algebraic varieties}
+
+\begin{aufgabe}[]
+    Let $f\colon X \to Y$ be a morphism of algebraic pre-varieties. Assume
+    \begin{enumerate}[(i)]
+        \item $Y$ is a variety.
+        \item There exists an open covering $(Y_i)_{i \in I}$ of $Y$ such that the open subset
+            $f^{-1}(Y_i)$ is a variety.
+    \end{enumerate}
+    Show that $X$ is a variety.
+\end{aufgabe}
+
+\begin{aufgabe}[]
+    Let $X$ be a topological space. Assume that there exists a covering $(X_i)_{i \in I}$ of
+    $X$ by irreducible open subsets such that for all $(i,j)$, $(X_i \cap X_j) \neq \emptyset$.
+    Show that $X$ is irreducible.
+\end{aufgabe}
+
+\subsection{Grassmann varieties}
+
+Let $0 \le p \le n$ be integers. The Grassmannian $\text{Gr}(p, n)$ is the set
+of $p$-dimensional linear subspaces of $k^{n}$. In order to endow this set with a structure
+of algebraic prevariety, there are various possibilities:
+
+\begin{enumerate}[(i)]
+    \item To a $p$-dimensional linear subspace $E \subseteq k^{n}$, we associate the line
+        $\Lambda^{p} E \subseteq \Lambda^{p} k^{n} \simeq k^{\binom{n}{p}}$, which
+        defines a point in the projective space $k\mathbb{P}^{\binom{n}{p}-1}$.
+
+        Claim: The map $\text{Gr}(p, n) \to k\mathbb{P}^{\binom{n}{p} -1}$
+        is an injective map whose image is a Zariski-closed subset of $k\mathbb{P}^{\binom{n}{p} -1}$.
+
+        This identifies $\text{Gr}(n, p)$ canonically to a projective variety. In particular
+        one obtains in this way a structure of \emph{algebraic variety}
+        on $\text{Gr}(p, n)$.
+    \item For the second approach, recall that $\text{GL}(n, k)$ acts transitively on
+        $\text{Gr}(p, n)$. But the identification of $k^{n}$ to $(k^{n})^{*}$
+        via the canonical basis of $k^{n}$ enables one to define, for all $E \in \text{Gr}(p, n)$,
+        a canonical complement $E^{\perp} \in \text{Gr}(n-p, n)$, i.e.
+        an $(n-p)$-dimensional linear subspace such that $E \oplus E^{\perp} = k^{n}$.
+        
+        So the stabiliser of $E \in \text{Gr}(p, n)$ for the action of
+        $\text{GL}(n, k)$ is conjugate to the subgroup
+        \begin{salign*}
+        \text{P}(p, n) \coloneqq
+        \left\{ g \in \text{GL}(n, k)  \middle \vert
+            \begin{array}{l}
+                g = \begin{pmatrix} A & B \\ 0 & C \end{pmatrix} \\
+                \text{with } A \in \text{GL}(p, k), B \in \text{Mat}(p \times (n-p), k),\\
+                \text{and } C \in \text{GL}(n-p, k) 
+            \end{array}
+        \right\} 
+        .\end{salign*}
+        This shows that the Grassmannian $\text{Gr}(p, n)$ is a homogeneous space
+        under $\text{GL}(n, k)$ and that
+        \begin{salign*}
+        \text{Gr}(p, n) \simeq \text{GL}(n, k) / \text{P}(p, n)
+        \end{salign*}
+        which is useful if one knows that, given an affine algebraic group $G$ and
+        a closed subgroup $H$, the homogeneous space $G / H$ is an algebraic variety. We
+        will come back to this later on.
+    \item The third uses the gluing theorem. In particular, it also constructs
+        a standard atlas on $\text{Gr}(p, n)$, like the one we had on
+        $k\mathbb{P}^{n-1} = \text{Gr}(1, n)$.
+        The idea is that, in order to determine a $p$-dimensional subspace of $k^{n}$,
+        it suffices to give a basis of that subspace, which is a family of $p$ vectors
+        in $k^{n}$. Geometrically, this means that the subspace in question is seen
+        as the graph of a linear map $A\colon k^{p} \to k^{n}$.
+
+        Take $E \in \text{Gr}(p, n)$ and let $(v_1, \ldots, v_p)$ be a basis of $E$ over $k$.
+        Let $M$ be the $(n \times p)$-matrix representing the coordinates
+        of $(v_1, \ldots, v_p)$ in the canonical basis of $k^{n}$. Since $M$ has rank $p$,
+        there exists a $(p \times p)$-submatrix of $M$ with non-zero determinant: We set
+        \begin{salign*}
+            J &\coloneqq \{ \text{indices } j_1 < \ldots < j_p \text{ of the rows of that submatrix}\} \\
+            M_J &\coloneqq \text{the submatrix in question}
+        .\end{salign*}
+        Note that if $M' \in \text{Mat}(n \times p, k)$ corresponds to a basis
+        $(v_1', \ldots, v_p')$, there exists a matrix $g \in \text{GL}(p, k)$ such that
+        $M' = Mg$. But then $(M')_J = (Mg)_J = M_J g$, so
+        \[
+        \text{det }(M')_J = \text{det } (M_J g) = \text{det}(M_J) \text{det}(g)
+        ,\]
+        which is non-zero if and only if $\text{det}(M_J)$ is non-zero. As a consequence,
+        given a subset $J \subseteq \{1, \ldots, n\} $ of cardinal $p$, there is a well-defined
+        subset
+        \begin{salign*}
+        G_J \coloneqq \left\{ E \in \text{G}(p, n)  \mid 
+                \exists M \in \text{Mat}(n \times p, k), E = \text{im }M \text{ and }
+                \text{det}(M_J) \neq 0
+        \right\} 
+        .\end{salign*}
+        Moreover, if $M$ satisfies the conditions $E = \text{im }M$ and
+        $\text{det}(M_J) \neq 0$, then
+        $(M M_J^{-1})_J = I_p$ and $\text{im}(MM^{-1}_J) = \text{im }M = E$.
+        In fact, if $E \in G_J$, there is a unique matrix $N \in \text{Mat}(n \times p, k)$,
+        such that $E = \text{im }N$ and $N_J = I_p$, for if $N_1, N_2$ are two
+        such matrices, the columns of $N_2$ are linear combinations of those of $N_1$,
+        thus $\exists g \in \text{GL}(p, k)$ such that $N_2 = N_1g$. But then
+        \[
+        I_p = (N_2)_J = (N_1g)_J = (N_1)_J g = g
+        .\] 
+        So, there is a well-defined map
+        \begin{salign*}
+            \hat{\varphi}_J: G_J &\longrightarrow \operatorname{Hom}(k^{J}, k^{n}) \\
+            E &\longmapsto N \text{ such that } E = \text{im }N \text{ and } N_J = I_p
+        \end{salign*}
+        whose image can be identified to the subspace
+        $\text{Hom}(k^{J}, k^{J^{c}})$, where $J^{c}$ is the complement of $J$ in
+        $\{1, \ldots, n\} $, via the map $N \mapsto N_{J^{c}}$. Conversely, a
+        linear map $A \in \text{Hom}(k^{J}, k^{J^{c}})$ determines a rank $p$ map
+        $N \in \text{Hom}(k^{J}, k^{n})$ such that $N_J = I_p$ via the formula
+        $N(x) = x + Ax$.
+
+        Geometrically, this means that the $p$-dimensional subspace
+        $\text{im }N \subseteq k^{n}$ is equal to the graph of $A$.
+        This also means that we can think of $G_J$ as the set
+        \begin{salign*}
+        \{E \in \text{Gr}(p, n)  \mid E \cap k^{J^{c}} = \{0_{k^{n}}\} \} 
+        .\end{salign*}
+        The point is that $\text{im } \hat{\varphi}_J = \text{Hom}(k^{J}, k^{J^{c}})$
+        can be canonically identified with the affine space $k^{p(n-p)}$ and that we
+        have a bijection
+        \begin{salign*}
+            \varphi_J \colon G_J &\xlongrightarrow{\simeq} \text{Hom}(k^{J}, k^{J^{c}})
+        \simeq k^{p(n-p)} \\
+            E &\longmapsto A  \mid \text{gr}(A) = E \\
+            \text{gr}(A) &\longmapsfrom A
+        .\end{salign*}
+        Note that the matrix $N \in \text{Mat}(n \times p, k)$
+        such that $\text{im }N = E$ and $N_J = I_p$
+        is row-equivalent to $\begin{pmatrix} I_p \\ A \end{pmatrix} $
+        with $A \in \text{Mat}((n-p) \times p, k)$.
+
+        Now, if $E \in G_{J_1} \cap G_{J_2}$, then, for all
+        $M \in \text{Mat}(p \times n, k)$ such that $\text{im } M = E$,
+        $\hat{\varphi}_{J_1}(E) = M M_{J_1}^{-1}$ and
+        $\hat{\varphi}_{J_2}(E) = M M_{J_2}^{-1}$. So
+        \begin{salign*}
+        \text{im } \hat{\varphi}_{J_1}
+        &= \left\{ N \in \text{Hom}(k^{J_1}, k^{n})  \mid N_{J_1} = I_p,
+            \text{im } N_{J_1} = E \text{ and }
+            \text{det}(N_{J_2})  \neq 0
+        \right\}  \\
+        &= \{ N \in \text{im } \hat{\varphi}_{J_1}  \mid \text{det}(N_{J_2})  \neq 0\}
+        \end{salign*}
+        which is open in $\text{im } \hat{\varphi}_{J_1} \simeq \text{im } \varphi_{J_1}$.
+
+        Moreover, for all $N \in \text{im }\hat{\varphi}_{J_1}$,
+        \[
+        \hat{\varphi}_{J_2} \circ \hat{\varphi}_{J_1}^{-1}(N) = N N_{J_2}^{-1}
+        \] and, by Cramer's formulae, this is a regular function
+        on $\text{im }\hat{\varphi}_{J_1}$.
+
+        We have therefore constructed a covering
+        \[
+        \text{Gr}(p, n) = \bigcup_{J \subseteq \{1, \ldots, n\}, \# J = p } G_J
+        \] 
+        of the Grassmannian $\text{Gr}(p, n)$ by subsets $G_J$
+        that can be identified to the affine variety $k^{p(n-p)}$ via bijective
+        maps $\varphi_J\colon G_j \to k^{p(n-p)}$ such that,
+        for all $(J_1, J_2)$, $\varphi_{J_1}(G_{J_1} \cap G_{J_2})$ is open
+        in $k^{p(n-p)}$ and the map
+        $\varphi_{J_2} \circ \varphi_{J_1}^{-1}\colon \varphi_{J_1}(G_{J_1} \cap G_{J_2}) \to \varphi_{J_2}(G_{J_1} \cap G_{J_2})$
+        is a morphism of affine varieties. By the gluing theorem,
+        this endows $\text{Gr}(p, n)$ with a structure of algebraic prevariety.
+\end{enumerate}
+
+\subsection{Vector bundles}
+
+\begin{definition}[]
+    A \emph{vector bundle} is a triple
+    $(E, X, \pi)$ consisting of two algebraic varieties $E$ and $X$, and
+    a morphism $\pi\colon E \to X$ such that
+    \begin{enumerate}[(i)]
+        \item for $x \in X$, $\pi^{-1}(\{x\} )$ is a $k$-vector space.
+        \item for $x \in X$, there exists an open neighbourhood $U$ of $x$
+            and an isomorphism of algebraic varieties
+            \[
+            \Phi\colon \pi^{-1}(U) \xlongrightarrow{\simeq} U \times \pi^{-1}(\{x\} )
+            \] such that
+            \begin{enumerate}[(a)]
+                \item $\text{pr}_1 \circ \Phi = \pi |_{\pi^{-1}(U)}$ and
+                \item for $y \in U$, $\Phi|_{\pi^{-1}(\{y\})}\colon \pi^{-1}(\{y\})
+                    \to \{y\} \times \pi^{-1}(\{x\})$ is
+                    an isomorphism of $k$-vector spaces.
+            \end{enumerate}
+    \end{enumerate}
+    A morphism of vector bundles is a morphism of algebraic varieties $f\colon E_1 \to E_2$
+    such that $\pi_2 \circ f = \pi_1$ and $f$ is $k$-linear in the fibres.
+\end{definition}
+
+\begin{bem}
+    In practice, one often proves that a variety $E$ is a vector bundle over $X$ by
+    finding a morphism $\pi\colon E \to X$ and an open covering
+    \[
+    X = \bigcup_{i \in  I} U_i
+    \] such that $E|_{U_i} \coloneqq \pi^{-1}(U_i)$ is isomorphic to
+    $U_i \times k^{n_i}$ for some integer $n_i$, in such a way that, on $U_i \cap U_j$,
+    the morphism
+    \[
+    \Phi_j \circ \Phi_i^{-1}\Big|_{\Phi_i(\pi^{-1}(U_i \cap U_j))}\colon
+    (U_i \cap U_j) \times k^{n_i} \longrightarrow
+    (U_i \cap U_j) \times k^{n_j}
+    \] is an isomorphism of algebraic varieties such that the following diagram commutes
+    and $\Phi_j \circ \Phi_i^{-1}$ is linear fibrewise:
+    \[
+    \begin{tikzcd}
+        (U_i \cap U_j) \times k^{n_i} \arrow{dr}{\text{pr}_1} \arrow{rr}{\Phi_j \circ \Phi_i^{-1}}
+        & & (U_i \cap U_j) \times k^{n_j} \arrow{dl}{\text{pr}_1}\\
+        & U_i \cap U_j & \\
+    \end{tikzcd}
+    .\] In particular $k^{n_i} \simeq k^{n_j}$ as $k$-vector spaces, so
+    $n_i = n_j$ if $U_i \cap U_j \neq \emptyset$, and
+    $\Phi_j \circ \Phi_i^{-1}$ is necessarily of the form
+    \[
+        (x, v) \longmapsto (x, g_{ji}(x) \cdot v)
+    \] for some morphism of algebraic varieties
+    \[
+    g_{ji}\colon U_i \cap U_j \longrightarrow \text{GL}(n, k)
+    .\]
+    These maps $(g_{ij})_{(i, j) \in I \times I}$ then
+    satisfy for $x \in U_i \cap U_j \cap U_l$
+    \[
+    g_{lj}(x) g_{ji}(x) = g_{li}(x)
+    \] and for $x \in U_i$, $g_{ii}(x) = \text{I}_n$.
+\end{bem}
+
+\begin{satz}
+    If $\pi\colon E \to X$ is a morphism of algebraic varieties and
+    $X$ has an open covering $(U_i)_{i \in I}$ over which $E$ admits
+    local trivialisations
+    \[
+    \Phi_i \colon E|_{U_i} = \pi^{-1}(U_i) \xlongrightarrow{\simeq} U_i \times k^{n}
+    \]
+    with $\text{pr}_1 \circ \Phi_i = \pi|_{\pi^{-1}(U_i)}$
+    such that the isomorphisms
+    \[
+    \Phi_j \circ \Phi_i^{-1} \colon (U_i \cap U_j) \times k^{n}
+    \longrightarrow (U_i \cap U_j) \times k^{n}
+    \] are
+    linear in the fibres, then for all $x \in X$, $\pi^{-1}(\{x\})$ has
+    a well-defined structure of $k$-vector space and the local trivialisations
+    $(\Phi_i)_{i \in I}$ are linear in the fibres. In particular,
+    $E$ is a vector bundle.
+\end{satz}
+
+\begin{proof}
+    For $x \in U_i$ and $a, b \in \pi^{-1}(\{x\})$, let
+    \[
+    a + \lambda b \coloneqq \Phi_i^{-1}(x, \text{pr}_2 (\Phi_i(a)) + \lambda \text{pr}_2 (\Phi_i(b)))
+    .\]
+    By using the linearity in the fibres of $\Phi_j \circ \Phi_i^{-1}$, one verifies
+    that this does not depend on the choice of $i \in I$.
+\end{proof}
+
+\begin{bem}[]
+    Assume given an algebraic prevariety $X$ obtained by gluing affine varieties
+    $(X_i)_{i \in I}$ along isomorphisms $\varphi_{ji}\colon X_{ij} \xrightarrow{\simeq} X_{ji}$
+    defined on open subsets $X_{ij} \subseteq X_i$,
+    such that $X_{ii} = X_i$, $\varphi_{ii} = \text{Id}_{X_i}$
+    %, $\varphi_{ji}(X_{ij})$ is open in $X_{ji}$
+    and
+    $\varphi_{lj} \circ \varphi_{ji} = \varphi_{li}$ on $X_{ij} \cap X_{il} \subseteq X_i$.
+
+    Recall that such an $X$ comes equipped with a canonical
+    map $p \colon \bigsqcup_{i \in I} \to X$ such that
+    $p_i \coloneqq p|_{X_i}\colon X_i \to X$ is an isomorphism onto an affine open subset
+    $U_i \coloneqq p_i(X_i) \subseteq X$ and, if we set $\varphi_i = p_i^{-1}$,
+    we have $\varphi_j \circ \varphi_i^{-1} = \varphi_{ji}$
+    on $\varphi_i(U_i \cap U_j)$.
+
+    Let us now consider the vector bundle $X_i \times k^{n}$ on each of the affine varieties
+    $X_i$ and assume that an isomorphism of algebraic prevarieties of the form
+    \begin{salign*}
+        \Phi_{ji}\colon X_{ij} \times k^{n} &\longrightarrow X_{ji} \times k^{n} \\
+        (x, v) &\longmapsto (\varphi_{ji}(x), h_{ji}(x) \cdot v)
+    \end{salign*}
+    has been given, where $h_{ij}\colon X_{ij} \to \text{GL}(n, k)$
+    is a morphism of algebraic varieties, in such a way that the following compatibility
+    conditions are satisfied:
+    \begin{salign*}
+    \Phi_{ii} = \text{Id}_{X_{ii} \times k^{n}}
+    \end{salign*}
+    and, for all $(i, j, l)$ and all $(x, v) \in (X_{ij} \cap X_{il}) \times k^{n}$
+    \[
+    \Phi_{lj} \circ \Phi_{ji}(x, v) = \Phi_{li}(x, v)
+    .\] 
+    Then there is associated to this gluing data an algebraic vector bundle
+    $\pi\colon E \to X$, endowed with 
+    local trivialisations $\Phi_i \colon E|_{U_i} \xrightarrow{\simeq} U_i \times k^{n}$,
+    where as earlier $U_i = p(X_i) \subseteq X$,
+    in such a way that, for all $(i, j)$ and all $(\xi, v) \in (U_i \cap U_j) \times k^{n}$,
+    \[
+    \Phi_j \circ \Phi_i^{-1}(\xi, v) =
+    (\xi, g_{ji}(\xi) \cdot v)
+    \] where $g_{ji}(x) = h_{ji}(\varphi_i(\xi)) \in \text{GL}(n, k)$, so
+    $g_{ii} = \text{I}_n$ on $U_i$, and, for all $(i, j, l)$ and
+    all $\xi \in U_i \cap U_j \cap U_l$,
+    \begin{salign*}
+        g_{lj}(\xi) g_{ji}(\xi) &= h_{lj}(\varphi_j(\xi)) h_{ji}(\varphi_i(\xi)) \\
+    &= h_{lj}(\varphi_{ji}(\varphi_i(\xi))) h_{ji}(\varphi_i(\xi)) \\
+    &= h_{li}(\varphi_i(\xi)) \\
+    &= g_{li}(\xi)
+    .\end{salign*}
+
+    Indeed, we can simply set
+    \begin{salign*}
+    E \coloneqq \left( \bigsqcup_{i \in I} X_i \times k^{n} \right) / \sim 
+    \end{salign*}
+    where $(x, v) \sim (\varphi_{ji}(x), h_{ji}(x) \cdot v)$, and, by the 
+    gluing theorem, this defines an algebraic prevariety, equipped
+    with a morphism $\pi\colon E \to X$ induced
+    by the first projection $\text{pr}_1\colon \bigsqcup_{i \in I} (X_i \times k^{n})
+    \to \bigsqcup_{i \in I} X_i$.
+    The canonical map $\hat{p}\colon \bigsqcup_{ i \in I} (X_i \times k^{n}) \to E$
+    makes the following diagram commute
+    \[
+    \begin{tikzcd}
+        \bigsqcup_{i \in I} (X_i \times k^{n}) \arrow{d}{\text{pr}_1}
+        \arrow{r}{\hat{p}} & E \arrow{d}{\pi} \\
+        \bigsqcup_{i \in I} X_i \arrow{r}{p} & X \\
+    \end{tikzcd}
+    \]
+    and it induces an isomorphism of prevarieties
+    \[
+    \hat{p}|_{X_i \times k^{n}}\colon X_i \times k^{n}
+    \xrightarrow{\simeq} E|_{p(X_i)}
+    = \pi^{-1}(p(X_i))
+    \]
+    such that $\pi \circ \hat{p}|_{X_i \times k^{n}} = p|_{X_i} \circ \text{pr}_1$.
+    Since $p|_{X_i}$ is an isomorphism between $X_i$ and the open subset
+    $U_i = p(X_i) \subseteq X$ with inverse $\varphi_i$, the
+    isomorphism $\hat{p}|_{X_i \times k^{n}}$
+    induces a local trivialisation
+    \begin{salign*}
+        \Phi_i \colon E|_{U_i} &\longrightarrow U_i \times k^{n} \\
+        w &\longmapsto (\pi(w), v)
+    \end{salign*}
+    where $v$ is defined as above by $\hat{p}(x, v) = w$. Note that $p(x) = \pi(w)$ in this
+    case, and that $\pi^{-1}(\{\pi(w)\}) \simeq k^{n}$
+    via $\Phi|_{\pi^{-1}(\{\pi(w)\})}$. As the isomorphism of algebraic prevarieties
+    \[
+    \Phi_j \circ \Phi_i^{-1}\colon (U_i \cap U_j) \times k^{n}
+    \longrightarrow (U_i \cap U_j) \times k^{n}
+    \] 
+    thus defined is clearly linear fibrewise, we have indeed constructed in this way
+    a vector bundle $\pi\colon E \to X$, at least in the category of algebraic prevarieties.
+
+    Note that if the prevariety $X$ obtained via the gluing of the $X_i$ is
+    a variety, then we can show that $E$ is actually a variety
+    (because the product variety $U_i \times k^{n}$ is separated). The rest of the verifications,
+    in particular the fact that for all $(\xi, v) \in U_i \cap U_j \times k^{n}$
+    \[
+    \Phi_j \circ \Phi_i^{-1}(\xi, v) = (\xi, h_{ji}(\varphi_i(\xi)) \cdot v)
+    \] is left to the reader.
+\end{bem}
+
+\begin{aufgabe}[]
+    Consider the set
+    \[
+    E \coloneqq \{ (\rho, v) \in k \mathbb{P}^{1} \times k\mathbb{P}^{2}  \mid v \in \rho\}
+    \] and the canonical map $\pi\colon E \to k\mathbb{P}^{1}$.
+
+    Show that $E$ is a vector bundle on $k\mathbb{P}^{1}$ and compute
+    its ,,cocycle of transition functions`` $g_{10}$ on the standard atlas
+    $(U_0, U_1)$ of $k\mathbb{P}^{1}$ with
+    \begin{salign*}
+        \varphi_{10}\colon k \setminus \{0\} &\longrightarrow k \setminus \{0\} \\
+        t &\longmapsto \frac{1}{t}
+    .\end{salign*}
+\end{aufgabe}
+
+\end{document}