diff --git a/sose2022/galois/vortrag_affin.pdf b/sose2022/galois/vortrag_affin.pdf index 9884ea5..7d2563b 100644 Binary files a/sose2022/galois/vortrag_affin.pdf and b/sose2022/galois/vortrag_affin.pdf differ diff --git a/sose2022/galois/vortrag_affin.tex b/sose2022/galois/vortrag_affin.tex index 08495b4..2bad0cb 100644 --- a/sose2022/galois/vortrag_affin.tex +++ b/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} diff --git a/sose2022/lie/skript.xopp b/sose2022/lie/skript.xopp index a8048ee..0ded2f1 100644 Binary files a/sose2022/lie/skript.xopp and b/sose2022/lie/skript.xopp differ diff --git a/sose2022/metaethik/axiomatische_ethik.pdf b/sose2022/metaethik/axiomatische_ethik.pdf index ece92e4..210b746 100644 Binary files a/sose2022/metaethik/axiomatische_ethik.pdf and b/sose2022/metaethik/axiomatische_ethik.pdf differ diff --git a/sose2022/metaethik/axiomatische_ethik.tex b/sose2022/metaethik/axiomatische_ethik.tex index b3b9a75..813cc95 100644 --- a/sose2022/metaethik/axiomatische_ethik.tex +++ b/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} diff --git a/ws2022/rav/lecture/rav.pdf b/ws2022/rav/lecture/rav.pdf index b5ee4b4..4f9f7d2 100644 Binary files a/ws2022/rav/lecture/rav.pdf and b/ws2022/rav/lecture/rav.pdf differ diff --git a/ws2022/rav/lecture/rav.tex b/ws2022/rav/lecture/rav.tex index 3345c7f..ce49d5f 100644 --- a/ws2022/rav/lecture/rav.tex +++ b/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} diff --git a/ws2022/rav/lecture/rav10.pdf b/ws2022/rav/lecture/rav10.pdf new file mode 100644 index 0000000..6520926 Binary files /dev/null and b/ws2022/rav/lecture/rav10.pdf differ diff --git a/ws2022/rav/lecture/rav10.tex b/ws2022/rav/lecture/rav10.tex new file mode 100644 index 0000000..92c4d13 --- /dev/null +++ b/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}