From 92ca413742e0910f060765d6dcba9b4e47efd47a Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Mon, 4 May 2026 14:59:20 +0200 Subject: [PATCH] =?UTF-8?q?Working=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/ltex.dictionary.de-DE.txt | 7 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 4 + 03.tex | 4 +- 04.tex | 24 +- 05.tex | 274 +++++++++++++++++++- gfx/stdPreamble.tex | 4 +- material/221109-Skript.pdf | Bin 279003 -> 0 bytes 7 files changed, 291 insertions(+), 26 deletions(-) delete mode 100644 material/221109-Skript.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 9563e82..0b9103a 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -141,3 +141,10 @@ Prägarbe Verklebeeigenschaft quadratintegrierbaren quadratintegrierbare +Moduln +Linearsystem +Linearsysteme +Garbenmorphismen +Garbenmorphismus +Kokern +Garbenaxiom diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 8885fcf..d50d2ff 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -75,3 +75,7 @@ {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QEs ergibt also Sinn, zu sagen: „Die meromorphe Differenzialform \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q hat bei \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q eine Polstelle \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-ter Ordnung“.\\E$"} {"rule":"DOPPELTE_SATZZEICHEN","sentence":"^\\QDifferenzialformen?.\\E$"} {"rule":"KOMMA_ZWISCHEN_HAUPT_UND_NEBENSATZ_2","sentence":"^\\QWenn \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q eine Stammfunktion besitzt (= wenn es eine Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gibt mit \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q), dann ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QWenn in der Situation aus Beispiel \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Funktionen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gegeben sind, dann ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"KLEINSCHREIBUNG_KEIN_NAME","sentence":"^\\QÄhnlich wie in der Vorlesung „Lineare Algebra“ wollen wir in diesem Abschnitt Kern, Bild und Kokern für Garbenmorphismen definieren.\\E$"} +{"rule":"DE_CASE","sentence":"^\\Q[Beweis \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q] Wir zeigen die Kontraposition: Angenommen, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q wäre als Garbenmorphismus nicht injektiv.\\E$"} +{"rule":"DE_CASE","sentence":"^\\Q[Beweis \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q] Es sei \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q ein injektiver Garbenmorphismus, sei \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/03.tex b/03.tex index 472d33d..9681abc 100644 --- a/03.tex +++ b/03.tex @@ -384,7 +384,7 @@ kennen. \end{proof} Eine quantitative Version des Hauptsatzes der Algebra findet sich in -Satz~\vref{satz:4-4-2}. +Satz~\ref{satz:4-4-2}. \sideremark{Vorlesung 4} @@ -444,7 +444,7 @@ Topologie der Faser. \item Die Abbildung $f : ℂ → ℂ^*$, $z ↦ \exp(z)$ ist eine unverzweigte Überlagerung. - \item Die Quotientenabbildung $π : ℂ → ℂ/Γ$ aus Beispiel~\vref{bsp:2-1-8} + \item Die Quotientenabbildung $π : ℂ → ℂ/Γ$ aus Beispiel~\ref{bsp:2-1-8} („komplexe Tori“) eine unverzweigte Überlagerung. \end{enumerate} \end{bsp} diff --git a/04.tex b/04.tex index 825b3db..06c9c12 100644 --- a/04.tex +++ b/04.tex @@ -97,7 +97,7 @@ Anschaulich ist ein Divisor eine diskrete Menge von Punkten mit Vielfachheiten. Untergruppe von $\Div(X)$. \end{bemerkung} -\begin{bemerkung}[Hauptdivisoren haben Grad 0] +\begin{bemerkung}[Hauptdivisoren haben Grad 0]\label{rem:4-2-5}% Nicht jeder Divisor ist ein Hauptdivisor! Wenn $X$ kompakt ist, dann hat jede meromorphe Funktion genau so viele Nullstellen wie Polstellen, also ist für jeden Hauptdivisor $D$ der Grad $\deg(D) = 0$. @@ -340,15 +340,15 @@ Quotienten $\sI_x / \sI_x²$. \end{beobachtung} -\subsubsection{Kotangentialräume und Differenzialformen im $ℂ^n$} +\subsubsection{Kotangentialräume und Differenzialformen in $ℂ$} -\begin{bsp}[Proberechnung: Kotangentialräume im $ℂ^n$]\label{bsp:4-3-19}% +\begin{bsp}[Proberechnung: Kotangentialräume im $ℂ$]\label{bsp:4-3-19}% Es sei $X ⊂ ℂ$ offen und es sei $p ∈ X$. Die Menge der Keime von holomorphen Funktionen ist dann \begin{align*} - 𝒪_{X,p} &= \left\{ \text{Potenzreihen $\sum_{i=0}^{∞} a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\} \\ - m_p &= \left\{ \text{Potenzreihen $\sum_{i=1}^{∞} a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\} \\ - m²_p &= \left\{ \text{Potenzreihen $\sum_{i=2}^{∞} a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\}. + 𝒪_{X,p} &= \left\{ \text{Potenzreihen $\sum_{i=0}^∞ a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\} \\ + m_p &= \left\{ \text{Potenzreihen $\sum_{i=1}^∞ a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\} \\ + m²_p &= \left\{ \text{Potenzreihen $\sum_{i=2}^∞ a_i (z-p)ⁱ$ mit positivem Konvergenzradius} \right\}. \end{align*} Der komplexe Kotangentialraum $m_p / m²_p$ ist dann ein eindimensionaler Vektorraum, erzeugt von der Restklasse der Funktion $z - p$. Allgemeiner: @@ -357,7 +357,7 @@ Quotienten $\sI_x / \sI_x²$. m²_p$. \end{bsp} -\begin{bsp}[Proberechnung: Differenzialformen $ℂ^n$]\label{bsp:4-3-20}% +\begin{bsp}[Proberechnung: Differenzialformen in $ℂ$]\label{bsp:4-3-20}% In der Situation aus Beispiel~\ref{bsp:4-3-19} sei $f ∈ 𝒪_X(U)$ irgendeine holomorphe Funktion. Dann ist \[ @@ -366,7 +366,7 @@ Quotienten $\sI_x / \sI_x²$. Also gilt die Gleichheit von Differenzialformen $df = f' · dz$. \end{bsp} -\begin{bsp}[Proberechnung: Differenzialformen $ℂ^n$] +\begin{bsp}[Proberechnung: Differenzialformen in $ℂ$] In der Situation aus Beispiel~\ref{bsp:4-3-20} sei eine weitere Funktion $g ∈ 𝒪_X(U)$ gegeben mit $g'(p) ≠ 0$ für alle $p ∈ U$, dann gilt $dg = g'·dz$, also @@ -531,9 +531,9 @@ Quotienten $\sI_x / \sI_x²$. \[ ω_1(p) = α(p) · ω_2(p) \] - gilt. Ich behaupte: $α$ ist meromorph eine meromorphe Funktion auf $X$. Dazu - betrachte ich einen Atlas $(\varphi_i : U_i → V_i)_{i ∈ I}$ und schreibe die - durch $ω_1$ und $ω_2$ auf $V_i$ induzierten Differenzialformen aus + gilt. Ich behaupte: $α$ ist eine meromorphe Funktion auf $X$. Dazu betrachte + ich einen Atlas $(\varphi_i : U_i → V_i)_{i ∈ I}$ und schreibe die durch $ω_1$ + und $ω_2$ auf $V_i$ induzierten Differenzialformen aus \[ f_{i,1}·dz \quad\text{und}\quad @@ -671,7 +671,7 @@ Integrale über Wege auf $ℂ$ auf Integrale über Wege auf Riemannschen Fläche übertragen. \begin{fakt}[Wegintegrale] - In der Situation von Konstruktion~\ref{kons:4-5-3} gilt folgendes. + In der Situation von Konstruktion~\ref{kons:4-5-3} gilt Folgendes. \begin{enumerate} \item Der Wert des Integrals hängt nur von $ω$ und der Homotopieklasse des Weges ab. diff --git a/05.tex b/05.tex index 1b159d1..dba424f 100644 --- a/05.tex +++ b/05.tex @@ -32,7 +32,7 @@ eine Abelsche Gruppe zu („auf $U$ differenzierbare/holomorphe/meromorphe Funktionen“). Wenn $V ⊆ U$ eine offene Teilmenge ist, dann kann ich Elemente der Gruppe auf $V$ einschränken. -\begin{definition}[Prägarbe]\label{def:5-1-1}% +\begin{defn}[Prägarbe]\label{def:5-1-1}% Sei $(X, τ)$ ein topologischer Raum. Eine \emph{Prägarbe}\index{Prägarbe} $ℱ$ von abelschen Gruppen ist eine Abbildung \[ @@ -49,10 +49,10 @@ der Gruppe auf $V$ einschränken. \item Für jede Kette $W ⊆ V ⊆ U$ von offenen Mengen gilt $ρ_{U,W} = ρ_{V,W} ◦ ρ_{U,V}$. \end{itemize} -\end{definition} +\end{defn} \begin{notation}[Prägarben] - In der Situation von Definition~\ref{def:5-1-1} sei $U ⊆ X$ offen. Elemente + In der Situation von Definition~\ref{def:5-1-1} sei $U ⊆ X$ offen. Elemente von $ℱ(U)$ werden oft als \emph{Schnitte der Prägarbe $ℱ$ über der Menge $U$}\index{Schnitte!in Prägarben} bezeichnet. Die Abbildungen $ρ_{U,V}$ heißen \emph{Einschränkungsabbildungen}\index{Einschränkungsabbildung}. @@ -61,7 +61,7 @@ der Gruppe auf $V$ einschränken. Jede $𝒞^∞$-Funktion ist differenzierbar und jede holomorphe Funktion ist $𝒞^∞$. Der formal richtige Begriff ist der „Morphismus von Prägarben“. -\begin{definition}[Morphismus von Prägarben] +\begin{defn}[Morphismus von Prägarben] Sei $(X,τ)$ ein topologischer Raum und $ℱ$, $𝒢$ seien Prägarben. Ein \emph{Morphismus}\index{Morphismus!Prägarben} $α : ℱ → 𝒢$ von Prägarben ist für jedes $U ∈ τ$ eine Abbildung @@ -75,7 +75,7 @@ $𝒞^∞$. Der formal richtige Begriff ist der „Morphismus von Prägarben“ ℱ(V) \ar[r, "α(V)"'] & 𝒢(V) \end{tikzcd} \] -\end{definition} +\end{defn} \section{Garben} @@ -87,7 +87,7 @@ $U_1$, $U_2$ und $𝒞^∞$-Funktionen $f_1$ auf $U_1$ und $f_2$ auf $U_2$, soda $f_1|_{U_1 ∩ U_2} = f_2|_{U_1 ∩ U_2}$, dann verkleben die Funktionen zu einer $𝒞^∞$-Funktion $f$ auf $U_1 ∪ U_2$. -\begin{definition}[Garbe] +\begin{defn}[Garbe] Es sei $ℱ$ eine Prägarbe auf einem topologischen Raum $X$. Nenne $ℱ$ eine \emph{Garbe}\index{Garbe}, wenn für alle offenen Mengen $U$ und alle Überdeckungen $X = \bigcup_i U_i$ Folgendes gilt. @@ -103,7 +103,7 @@ $𝒞^∞$-Funktion $f$ auf $U_1 ∪ U_2$. gilt, dann gibt es ein $s ∈ ℱ(U)$, sodass für jeden Index $i$ die Gleichheit $s|_{U_i} = s_i$ gilt. \end{enumerate} -\end{definition} +\end{defn} \begin{bemerkung}[Verklebeeigenschaft] Man nennt \ref{il:5-2-1-2} die @@ -125,7 +125,7 @@ $𝒞^∞$-Funktion $f$ auf $U_1 ∪ U_2$. Wir haben in den Definitionen~\ref{def:4-3-1} und \ref{def:4-3-10} Keime von Funktionen betrachtet. Das geht mit Garben ganz genau so. -\begin{definition}[Halm] +\begin{defn}[Halm] Sei $ℱ$ eine Garbe auf $X$ und $p$ ein Punkt. Nenne zwei Schnitte, die in der Nähe von $p$ definiert sind, \emph{äquivalent}, wenn sie auf einer Umgebung übereinstimmen: @@ -137,7 +137,7 @@ Funktionen betrachtet. Das geht mit Garben ganz genau so. \] Äquivalenzklassen heißen \emph{Keime}\index{Keim!einer Garbe}, die Menge der Äquivalenzklassen heißt \emph{Halm}\index{Halm}. -\end{definition} +\end{defn} \begin{beobachtung} Es sei $ℱ$ eine Garbe, sodass alle Halme gleich $0$ sind. Dann ist $ℱ$ die @@ -146,4 +146,260 @@ Funktionen betrachtet. Das geht mit Garben ganz genau so. 0$, also $ℱ(U) = 0$. \end{beobachtung} +\sideremark{Vorlesung 7} + +Gegeben eine Riemannsche Fläche $X$, dann ist die wichtigste Garbe $𝒪_X$, die +Garbe der holomorphen Funktionen. Die meisten anderen Garben, die uns noch so +interessieren, haben eine interessante Eigenschaft: Man kann die Schnitte mit +Funktionen multiplizieren. + +\begin{defn}[$𝒪_X$-Modulgarbe] + Es sei $X$ eine Riemannsche Fläche. Eine + \emph{$𝒪_X$-Modulgarbe}\index{Modulgarbe} ist eine Garbe $ℱ$ und für jedes $U + ∈ τ$ eine Modulverknüpfung + \[ + 𝒪_X(U) ⨯ ℱ(U) \longrightarrow ℱ(U), + \] + die mit Einschränkung verträglich ist. +\end{defn} + +\begin{bsp}[$𝒪_X$-Modulgarbe] + Beispiele für Modulgarben auf einer Riemannschen Fläche $X$ sind die Garben + $ℳ_X$ der meromorphen Funktionen und $Ω_X¹$, die Garbe der holomorphen + Differenzialformen. +\end{bsp} + + +\section{Modulgarben und Linearsysteme zu Divisoren} + +\begin{bsp}[$𝒪_X$-Modulgarbe zu Divisoren]\label{bsp:5-2-8}% + Es sei $X$ eine Riemannsche Fläche und $D ∈ \Div(X)$ ein Divisor. Dann + definiere eine Garbe $𝒪_X(D)$ wie folgt: Für jede offene Menge $U$ ist + \[ + 𝒪_X(D)(U) = \{ f ∈ ℳ_X(U) \::\: \ddiv(f) ≥ -D|_U\}. + \] + Das bedeutet Folgendes. + \begin{itemize} + \item Wenn ein Punkt $p$ mit positive Vielfachheit $n$ auftaucht, dann heißt + das, dass $f$ bei $p$ eine Polstelle von Ordnung höchstens $n$ haben darf. + + \item Wenn ein Punkt $p$ mit negativer Vielfachheit $-n$ auftaucht, dann + heißt das, dass $f$ bei $p$ eine Nullstelle von Ordnung mindestens $n$ haben + muss. + \end{itemize} +\end{bsp} + +\begin{beobachtung}[Modulstruktur] + Wenn in der Situation aus Beispiel~\ref{bsp:5-2-8} Funktionen $f ∈ 𝒪_X(D)(U)$ + und $g ∈ 𝒪_X(U)$ gegeben sind, dann ist $g·f ∈ 𝒪_X(D)(U)$. Also ist + $𝒪_X(D)(U) ⊂ ℳ_X(U)$ ein $𝒪_X(U)$-Untermodul. +\end{beobachtung} + +\begin{beobachtung}[Lokal freie Garben]\label{beob:5-2-9}% + Sei $p ∈ X$ ein Punkt. Dann wähle eine Kartenumgebung + \[ + \varphi : U → V + \] + und einen Punkt $p ∈ U$ so, dass $D|_U = n · p$ ist\footnote{Der Wert $n = 0$ + ist erlaubt!}. Dann liegt die Funktion + \[ + (z - \varphi(p))^{-n} ◦ \varphi : U → \mathbb C + \] + in $𝒪_X(D)(U)$ und ist sogar eine Basis von $𝒪_X(D)(U)$ als $𝒪_X(U)$-Modul. + Man sagt: $𝒪_X(D)$ ist eine \emph{lokal freie} + $𝒪_X$-Modulgarbe\index{Modulgarbe!lokal frei} vom Rang $1$. Alternativ sagt + man: $𝒪_X(D)$ ist eine \emph{invertierbare}\index{Modulgarbe!invertierbar} + $𝒪_X$-Modulgarbe. +\end{beobachtung} + +\begin{bemerkung}[Achtung! Mißverständnis!] + Die Menge $U$ aus Beobachtung~\ref{beob:5-2-9} war sehr speziell gewählt. + Beobachtung~\ref{beob:5-2-9} bedeutet \emph{nicht}, dass für jede offene Menge + $U$ der Modul $𝒪_X(D)(U)$ eine einelementige Basis besitzt. Als + Gegenbeispiel betrachte $X = ℙ¹$ und $D = -1 · ∞ ∈ \Div(ℙ¹)$. Dann ist + $𝒪_X(D)(ℙ¹) = 0$, denn Funktionen in $𝒪_X(D)(ℙ¹)$ dürfen keine Polstellen + haben (und sind daher holomorph, also konstant), müssen aber eine Nullstelle + haben. +\end{bemerkung} + +\begin{beobachtung}[Lineare Äquivalenz von Divisoren und Isomorphie von Modulgarben, I] + Es sei $X$ eine Riemannsche Fläche, es sei $D ∈ \Div(X)$ ein Divisor und $U ⊆ + X$ offen. Weiter sei $f ∈ ℳ_X(U)$ und $g ∈ 𝒪_X(D)(U)$. Dann ist $f·g ∈ + 𝒪_X(D + \ddiv(f))(U)$. Man rechnet sogar nach: Multiplikation mit $f$ + liefert einen Isomorphismus von $𝒪_X(U)$-Moduln + \[ + 𝒪_X(D)(U) → 𝒪_X(D + \ddiv(f))(U). + \] + Wir merken uns: Wenn zwei Divisoren $D_1$ und $D_2$ linear äquivalent sind, + dann sind die $𝒪_X$-Modulgarben $𝒪_X(D_1)$ und $𝒪_X(D_2)$ isomorph. +\end{beobachtung} + +\begin{hsg}[Lineare Äquivalenz von Divisoren und Isomorphie von Modulgarben, I] + Es sei $X$ eine Riemannsche Fläche, und es seien $D_1$, $D_2 ∈ \Div(X)$ zwei + Divisoren. Wenn die $𝒪_X$-Modulgarben $𝒪_X(D_1)$ und $𝒪_X(D_2)$ isomorph, + dann sind die Divisoren $D_1$ und $D_2$ linear äquivalent. + + Betrachten Sie erst einmal den Fall, dass $D_1 ≥ 0$ ist. Benötigen Sie zum + Beweis der linearen Äquivalenz eine meromorphe Funktion? Betrachten Sie eine + Isomorphie $\varphi : 𝒪_X(D_1) → 𝒪_X(D_2)$. Beobachten Sie, dass $1 ∈ + 𝒪_X(D_1)(X)$ ist und nehmen Sie $\varphi(1)$. +\end{hsg} + +\begin{bemerkung}[Grundlegendes Ziel für den Rest der Vorlesung] + Ich hatte schon erwähnt, wie wichtig der Körper $ℳ_X(X)$ ist, wenn wir die + Riemannsche Fläche $X$ verstehen wollen. Wir versuchen den Körper $ℳ_X(X)$ zu + verstehen, indem wir erst einmal Divisoren $D$ betrachten und die komplexen + Unterräume + \[ + L(D) := 𝒪_X(D)(X) ⊂ ℳ(X) + \] + verstehen ---auf diese Weise teilen wir uns den Körper $ℳ_X(X)$ in + endlich-dimensionale Vektorräume auf. Nenne $L(D)$ das + \emph{Linearsystem}\index{Linearsystem} von $D$. Die Dimension wird mit + \[ + l(D) := \dim_{ℂ} L(D) + \] + notiert. +\end{bemerkung} + +\begin{bsp}[Erste Linearsysteme] + Ist $X$ eine zusammenhängende und kompakte Riemannsche Fläche, dann ist + $l(0)=1$, denn + \[ + L(0)=𝒪_X(0)(X)=\{\text{holomorphe Funktionen auf }X\}=\text{konstant}. + \] + Wenn $D$ ein Divisor ist mit $\deg D < 0$, dann ist $l(D)=0$. Wenn nämlich $f + ∈ L(D)$ eine nicht-triviale meromorphe Funktion wäre, dann wäre $\ddiv(f) ≥ + -D$, also $\ddiv(f) \le D$, und damit + \[ + \deg \ddiv(f) \le \deg D < 0, + \] + Widerspruch, denn wir haben in Bemerkung~\ref{rem:4-2-5} gesehen, dass $\deg + \operatorname{div}(f)=0$ ist. +\end{bsp} + + +\section{Garbenmorphismen} + +In einem großen der Teil der Vorlesung „Riemannsche Flächen“ geht es um die +Räume $L(D)$. Wir studieren diese Räume, indem wir die Garbe $𝒪_X(D)$ mit +anderen Garben vergleichen. Genauer: Wenn es einen Morphismus +\[ + ℱ → 𝒪_X(D) +\] +von Garben gibt, dann gibt es einen zugehörigen Morphismus +\[ + ℱ(X) → L(D) +\] +von abelschen Gruppen, die oft auch Vektorräume sind. Ähnlich wie in der +Vorlesung „Lineare Algebra“ wollen wir in diesem Abschnitt Kern, Bild und Kokern +für Garbenmorphismen definieren. + +\begin{bemerkung}[Vorbemerkung] + Wenn $α : ℱ → 𝒢$ ein Garbenmorphismus ist und $p ∈ X$ ein Punkt, dann + induziert $α$ einen Morphismus zwischen den Halmen + \[ + α_p : ℱ_p → 𝒢_p. + \] +\end{bemerkung} + +\begin{defn}[Injektivität und Surjektivität von Garbenmorphismen] + Es sei $X$ ein topologischer Raum und $α : ℱ → 𝒢$ ein Morphismus von Garben. + Nenne $α$ + injektiv/surjektiv\index{injektiv!Garbenmorphismus}\index{surjektiv!Garbenmorphismus}, + wenn für alle $p ∈ X$ gilt: + \[ + α_p : ℱ_p → 𝒢_p + \] + ist injektiv/surjektiv. +\end{defn} + +Es ist vielleicht nicht ganz offensichtlich, was die Worte „injektiv“ und +„surjektiv“ in diesem Kontext bedeuten. Um das zu erklären, sind die Begriffe +„Kern“ und „Bildgarbe“ nützlich. + +\begin{defn}[Kern und Bild]\label{def:5-4-3}% + Es sei $X$ ein topologischer Raum und $α : ℱ → 𝒢$ ein Morphismus von Garben. + Wir definieren \emph{Kern}\index{Kern!Garbenmorphismus} und + \emph{Bildgarbe}\index{Bildgarbe} durch + \begin{align*} + \ker(α)(U) & := \{\, σ ∈ ℱ(U) \::\: \forall\, p ∈ U : σ_p ∈ \ker(α_p) \,\}, \\ + \img(α)(U) & := \{\, σ ∈ 𝒢(U) \::\: \forall\, p ∈ U : σ_p ∈ \operatorname{im}(α_p) \,\}. + \end{align*} +\end{defn} + +\begin{hsg} + Man rechne nach, dass $\ker(α)$ und $\img(α)$ aus Definition~\ref{def:5-4-3} + tatsächlich Garben sind. +\end{hsg} + +Der Begriff „injektiv“ ist relativ leicht zu verstehen. + +\begin{lemma}[Injektiver Garbenmorphismus] + Es sei $X$ ein topologischer Raum und $α : ℱ → 𝒢$ ein Morphismus von Garben. + Dann sind die folgenden Aussagen äquivalent. + \begin{enumerate} + \item\label{il:5-4-5-1} Der Garbenmorphismus $α$ ist injektiv. + + \item\label{il:5-4-5-2} Für jede offene Menge $U$ ist $α(U)$ injektiv als + Morphismus von Abelschen Gruppen. + \end{enumerate} +\end{lemma} +\begin{proof} + Vorüberlegung: Wenn $σ ∈ ℱ(U)$ ein Schnitt ist, sodass für alle Punkte $p ∈ U$ + gilt, dass $σ_p ∈ ℱ_p$ gleich null ist, dann folgt aus dem Garbenaxiom, dass + $σ$ gleich null ist. Wenn $ℋ$ eine Garbe ist, sodass für alle $p ∈ X$ gilt + $ℋ_p =0$, dann ist $ℋ = 0$. + \begin{description} + \item[Beweis \ref{il:5-4-5-1} $⇒$ \ref{il:5-4-5-2}] Es sei $α$ ein + injektiver Garbenmorphismus, sei $U ∈ τ$ und $σ ∈ \ker(α(U))$. Dann gilt + für alle $p ∈ U$: + \[ + σ_p ∈ \ker(α_p) = 0. + \] + Also ist nach der Vorüberlegung bereits $σ = 0$. + + \item[Beweis \ref{il:5-4-5-2} $⇒$ \ref{il:5-4-5-1}] Wir zeigen die + Kontraposition: Angenommen, $α$ wäre als Garbenmorphismus \emph{nicht} + injektiv. Das bedeutet: Es gibt einen Punkt $p ∈ X$, sodass $\ker(α_p) \ne + 0$ ist. Also existiert eine offene Menge $U = U(p)$ und ein Schnitt $σ ∈ + ℱ(U)$, sodass + \[ + σ_p \ne 0 \text{ in } ℱ_p + \quad\text{und}\quad + α_p(σ_p) = (α(U)(σ))_p = 0 \text{ in } 𝒢_p. + \] + Nach Verkleinern von $U$ können wir ohne Beschränkung der Allgemeinheit + annehmen, dass $σ \ne 0$ aber $α(U)(σ)=0$ ist. Dann ist $σ$ aber ein + nicht-trivialer Schnitt in $\ker(α)$. + \end{description} +\end{proof} + +\begin{bemerkung}[Achtung! Mißverständnis!] + Es sei $X$ ein topologischer Raum und $α : ℱ → 𝒢$ ein Morphismus von Garben. + Dann sind die folgenden Aussagen im Allgemeinen \emph{überhaupt nicht} + äquivalent. + \begin{enumerate} + \item\label{il:5-4-6-1} Der Garbenmorphismus $α$ ist surjektiv. + + \item\label{il:5-4-6-2} Für jede offene Menge $U$ ist $α(U)$ surjektiv als + Morphismus von Abelschen Gruppen. + \end{enumerate} + Für ein einfaches Beispiel sei $X = ℂ^*$. Dann ist + \[ + \exp : 𝒪_X → 𝒪_X^* + \] + ein Morphismus von Garben, gegeben durch + \[ + 𝒪_X(U) → 𝒪_X^*(U), \qquad f ↦ \exp ◦ f. + \] + Dieser Garbenmorphismus ist surjektiv, denn lokal hat jede Funktion aus + $𝒪_X^*$ einen Logarithmus. Die Abbildung + \[ + 𝒪_X(X) → 𝒪_X^*(X), \qquad f ↦ \exp ◦ f + \] + ist aber überhaupt nicht surjektiv, denn die Funktion $\Id ∈ 𝒪_X^*(X)$ hat + keinen Logarithmus! +\end{bemerkung} + % !TEX root = RiemannscheFlaechen diff --git a/gfx/stdPreamble.tex b/gfx/stdPreamble.tex index dd58c7d..b960691 100644 --- a/gfx/stdPreamble.tex +++ b/gfx/stdPreamble.tex @@ -10,8 +10,6 @@ \usepackage[T1]{fontenc} \usepackage{newunicodechar} \usepackage{mathtools} -\usepackage{varioref} -\usepackage[arrow,curve,matrix]{xy} % Graphics Packages \usepackage{colortbl} @@ -273,7 +271,7 @@ \newcommand{\cC}{\mathcal C} \newcommand{\cD}{\mathcal D} \newcommand{\cE}{\mathcal E} -\newcommand{\cM}{\mathcal M} +\newcommand{\cM}{\sM} \newcommand{\cN}{\mathcal N} \newcommand{\cV}{\mathcal V} diff --git a/material/221109-Skript.pdf b/material/221109-Skript.pdf deleted file mode 100644 index 9a70e5197f75efdd120ee038a483cbcd1a5548f8..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 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