From 301a97829b57d6b54be06798329930d9f7bbde00 Mon Sep 17 00:00:00 2001 From: Christian Merten Date: Fri, 22 Apr 2022 12:32:10 +0200 Subject: [PATCH] add abstract, update titlepage --- sose2022/bachelorarbeit/arbeit.cls | 2 +- sose2022/bachelorarbeit/arbeit.pdf | Bin 464200 -> 662609 bytes sose2022/bachelorarbeit/arbeit.tex | 98 ++++++++++++++++++++++++----- 3 files changed, 85 insertions(+), 15 deletions(-) diff --git a/sose2022/bachelorarbeit/arbeit.cls b/sose2022/bachelorarbeit/arbeit.cls index 91569d1..9aba5c6 100644 --- a/sose2022/bachelorarbeit/arbeit.cls +++ b/sose2022/bachelorarbeit/arbeit.cls @@ -4,7 +4,7 @@ \RequirePackage[utf8]{inputenc} \RequirePackage[T1]{fontenc} \RequirePackage{textcomp} -\RequirePackage[german]{babel} +\RequirePackage[german, english]{babel} \RequirePackage{amsmath, amssymb, amsthm} \RequirePackage{mdframed} \RequirePackage{fancyhdr} diff --git a/sose2022/bachelorarbeit/arbeit.pdf b/sose2022/bachelorarbeit/arbeit.pdf index 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-9,6 +11,13 @@ %\newcommand{\smallbullet}{\tikz \draw (0,0) circle (1.5pt);} +\renewenvironment{abstract}{ + \vspace*{\fill} + \begin{center}% + \bfseries\abstractname + \end{center}}% + {\vfill} + \newcommand{\com}[1]{#1^{\text{\scalebox{0.7}{\textbullet}}}} \newcommand{\K}{\mathcal{K}} %\newcommand{\colim}{\underset{\longrightarrow}{\text{colim }}} @@ -46,19 +55,72 @@ \begin{document} -\maketitle +\begin{titlepage} + \begin{center} + \vspace*{3cm} + \textsc{Bachelorarbeit} \\[1cm] + { + \LARGE + \textbf{Auflösung unbeschränkter Komplexe} + } + + \vspace{2cm} + { + \large + \emph{Christian Merten} + } + + { + \vspace{1cm} + Heidelberg, den \today. + } + + \vfill + + \small + \textsc{Betreuung durch}\\[5mm] + \begin{tabular}{c} + \emph{Prof. Dr. Alexander Schmidt} \\ + \emph{Dr. Marius Leonhardt} \\ + \emph{Dr. Katharina Hübner} + \end{tabular} + \end{center} +\end{titlepage} + +\clearpage + +\selectlanguage{german} + +\pagenumbering{gobble} \begin{abstract} - Wir zeigen die Existenz der abgeleiteten $R\com{\operatorname{Hom}}(\com{N}, -)$, - $R\com{\operatorname{Hom}}(-, \com{N}) $ und $- \otimes_A^{L} \com{N}$, indem - für jeden Komplex von $R$-Moduln für einen Ring $R$, eine K-injektive und K-projektive - Auflösung konstruiert wird. + Wir geben eine Einführung in die Konstruktion von derivierten Kategorien und + abgeleiteten Funktoren. Danach konstruieren wir verschiedene Auflösungen + unbeschränkter Komplexe von Moduln über einem Ring und wenden dies + letztendlich an, um die Existenz der abgeleiteten Funktoren von Hom und Tensorprodukt + zu zeigen und das klassische Adjunktionsresultat auf die abgeleiteten Funktoren + zu übertragen. \end{abstract} +\selectlanguage{english} +\begin{abstract} + We give an introduction to the construction of derived categories and derived + functors. Then we construct various resolutions of unbounded + complexes of modules over a ring, which we finally apply to show + the existence of the derived functors of Hom and tensorproduct and to transfer the + classic adjunction to the derived functors. +\end{abstract} + +\selectlanguage{german} + +\clearpage + \tableofcontents \newpage +\pagenumbering{arabic} + \section{Einleitung} %Aus der kommutativen Algebra ist für einen kommutativen Ring $A$ und $A$-Moduln @@ -71,7 +133,7 @@ $N$ die Adjunktion \[ - \otimes_A N \dashv \operatorname{Hom}_A(N, -) \] bekannt. In der klassischen homologischen Algebra definiert man -die Funktoren $\operatorname{Ext}_A^{i}(N, -)$ und $\operatorname{Tor}_A^{i}(-, N)$, als +die Funktoren $\operatorname{Ext}_A^{i}(N, -)$ und $\operatorname{Tor}_A^{i}(-, N)$ als Ableitungen der Funktoren $\operatorname{Hom}_A(N, -)$ und $- \otimes_A N$. Nun stellt sich die Frage, ob zwischen diesen ein analoges Adjunktionsresultat gilt. @@ -93,10 +155,11 @@ folgte, dass $\operatorname{Ext}_A^{i}(N, -)$ linksexakt sei. Die exakte Folge und $\operatorname{Ext}_{\Z}^{0}(\Z / 2 \Z, \Z) = \operatorname{Hom}_{\Z}(\Z / 2 \Z, \Z) = 0$, also ist $\operatorname{Ext}_{\Z}^{1}(\Z / 2 \Z, \Z) \to \operatorname{Ext}^{1}_{\Z}(\Z / 2 \Z, \Z)$ nicht injektiv. -Es liegt also nahe, dass der klassische Ableitungsbegriff unvollständig ist. Um einen -neuen und allgemeineren Ableitungsbegriff zu finden, -betrachten wir wie klassische Ableitungen gebildet -werden. +Es liegt also nahe, dass der klassische Ableitungsbegriff unvollständig ist. +Im \ref{sec:derived-cat}. Abschnitt wird deshalb ein allgemeinerer Ableitungsbegriff +dargestellt. +Um diesen zu finden, +betrachtet man die Bildung von klassischen Ableitungen. %Konstruktion eines neuen Ableitungsbegriffs führt zum Begriff der derivierten Kategorie: %Sei $\mathcal{A}$ eine abelsche Kategorie und sei $\mathcal{K}(\mathcal{A})$ die Homotopiekategorie %von $\mathcal{A}$, das heißt die Kategorie, deren Objekte Komplexe aus $\mathcal{A}$ und @@ -180,8 +243,9 @@ sehr schwer zu erfüllen und hängt durch $\mathcal{J}$ von dem konkreten Funkto hat das in seiner Arbeit ,,Resolution of unbounded complexes`` \cite{spaltenstein} für zahlreiche Funktoren unter wenigen Voraussetzungen -an die beteiligten Kategorien gelöst. Die vorliegende Arbeit stellt sein Argument dar und -wendet dies auf den Homfunktor und das Tensorprodukt an. +an die beteiligten Kategorien gelöst. Die vorliegende Arbeit stellt im +\ref{sec:resolutions}. Abschnitt sein Argument dar, um dies im \ref{sec:application}. +Abschnitt auf den Homfunktor und das Tensorprodukt anzuwenden. Dafür erweitern wir die Funktoren $\operatorname{Hom}_A(N, -)$ und $- \otimes_A N$ in natürlicher Weise zu Funktoren $\mathcal{K}(A\text{-Mod}) \to \mathcal{K}(A\text{-Mod})$, sodass die Erweiterungen, @@ -275,7 +339,7 @@ Es wird schnell klar, dass ein exakter K-injektiver Komplex bereits der Nullkomp damit die Bedingung (1) erfüllt ist. Der schwierige Teil ist Bedingung (2) nachzuweisen. Zuänchst bemerken wir, dass nach unten beschränkte Komplexe mit injektiven Objekten -K-injektiv sind. Klassisch ist ist bekannt, dass +K-injektiv sind. Klassisch ist bekannt, dass jeder nach unten beschränkte Komplex eine Auflösung durch einen solchen Komplex hat. Für einen beliebigen (unbeschränkten) Komplex $\com{M} \in \mathcal{K}(\mathcal{A})$ funktioniert die klassische, induktive Konstruktion jedoch nicht. @@ -326,6 +390,8 @@ zurückführen. \section{Derivierte Kategorien und abgeleitete Funktoren} +\label{sec:derived-cat} + Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien und sei $F\colon \mathcal{A} \to \mathcal{B}$ ein additiver Funktor. Das Ziel ist es in natürlicher Weise für jedes Objekt $X \in \mathcal{A}$ einen Komplex $\text{R}F(X)$ zu definieren, dessen Kohomologiegruppen, @@ -1040,6 +1106,8 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \section{K-injektive und K-projektive Auflösungen} +\label{sec:resolutions} + Sei $\com{Y} \in \mathcal{K}$. Um die Bedingungen von \ref{satz:existence-derived-functors} für $\com{\operatorname{Hom}}(\com{Y}, -)$ (bzw. $\com{\operatorname{Hom}}(-, \com{Y})$) @@ -2179,6 +2247,8 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \section{Ableitungen und Adjunktion} +\label{sec:application} + Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (links-)Moduln. \subsection{Abgeleitete $\com{\operatorname{Hom}}$ Funktoren}

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