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b/sose2022/bachelorarbeit/arbeit.tex @@ -1,7 +1,187 @@ \documentclass{../../lecture} +\author{Christian Merten} +\title{Auflösung unbeschränkter Komplexe} +\usepackage{tikz-cd} +\usepackage{amssymb} + +\newcommand{\com}[1]{#1^{\bullet}} +\newcommand{\K}{\mathcal{K}} + \begin{document} +\maketitle + +\section{Einleitung} + +\section{Derivierte Kategorien und abgeleitete Funktoren} + +\section{K-injektive und K-projektive Auflösungen} + +Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die +Komplexkategorie mit Komplexhomomorphismen bis auf Homotopie als Abbildungen. + +\begin{definition}[K-injektiv bzw. K-projektiv] + Ein Komplex $\com{X} \in \K$ heißt K-injektiv (bzw. K-projektiv), wenn für alle $\com{S} \in \K$, der Komplex + $\com{\mathrm{Hom}}(\com{S}, \com{X})$ (bzw. $\com{\mathrm{Hom}}(\com{X}, \com{S})$) exakt ist. +\end{definition} + +\begin{bem} + Mit \ref{hom-compl-cohomgroups} ist $\com{X} $ genau dann K-injektiv (bzw. K-projektiv), wenn + $\forall \com{S} \in K$ exakt: $\text{Mor}_{\mathcal{K}}(\com{S}[i] , \com{X}) = 0$ + (bzw. $\text{Mor}_{\mathcal{K}}(\com{X}, \com{S}[i]) = 0)$ $\forall i \in \Z$. Da Verschieben Exaktheit erhält + folgt also + \[ + \com{X} \text{ K-projektiv} \iff \text{Mor}_{\mathcal{K}}(\com{X} , \com{S} ) = 0 \quad \forall \com{S} \text{ exakt} + \] + \[ + \com{X} \text{ K-injektiv} \iff \text{Mor}_{\mathcal{K}}(\com{S} , \com{X} ) = 0 \quad \forall \com{S} \text{ exakt} + .\] +\end{bem} + +\subsection{Elementare Eigenschaften von K-injektiven und K-projektiven Komplexen} + +\begin{bem} + Ein exakter K-projektiver oder K-injektiver Komplex $\com{X}$ ist zusammenziehbar, d.h. ist nullhomotop. + \begin{proof} + Betrachte $\mathrm{id}_{\com{X}} \in \mathrm{Hom}^{0}(\com{X}, \com{X}) = \mathrm{Mor}_{\K}(\com{X} , \com{X}) = \mathrm{H}^0 \com{\mathrm{Hom}}(\com{X}, \com{X}) = 0$. Also ist $\mathrm{id}_{\com{X} } = 0$ und damit + $\com{X} = 0$ in $\K$. + \end{proof} +\end{bem} + +\begin{satz} + Sei $\com{X} \in \K$ mit $X^{i} = 0$ $\forall i \neq 0$. Dann ist $\com{X} $ K-injektiv (bzw. K-projektiv) genau + dann wenn $A^{0}$ injektiv (bzw. projektiv) in $\mathcal{A}$ ist. +\end{satz} + +\begin{proof} + ,,$\implies$'': Sei $\com{X} $ K-projektiv und $\com{S} = 0 \to M \to N \to P \to 0$ kurze exakte Folge in $\mathcal{A}$. Sei + $f\colon X^{0} \to P$. Das induziert einen Komplexhomomorphismus $\com{X} \to \com{S}$ +\[\begin{tikzcd} + 0 \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & X^0 \arrow{d}{f} \arrow[dashed]{dl}{k} \arrow{r} + & 0 \arrow{d} \arrow[dashed]{dl}\\ + M \arrow{r} & N \arrow{r}{v} & P \arrow{r} & 0 +\end{tikzcd}\] +Nach Voraussetzung ist dieser nullhomotop, d.h. es existiert $k \colon X^{0} \to N$, s.d. $f = vk$. Also ist +$\text{Hom}(X^{0}, N) \to \text{Hom}(X^{0}, P)$ surjektiv und damit $X^{0}$ projektiv. + +,,$\impliedby$'': Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ +Komplexhomomorphismus. Dann betrachte +\[ +diag +.\] +Da $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} k^{0}$. +\end{proof} + +\begin{satz}[] + \begin{enumerate}[(i)] + \item $\com{X} \in \mathcal{K}$ ist K-projektiv (bzw. K-injektiv) genau dann wenn $\com{X}[1]$ dies ist. + \item Falls zwei Punkte eines ausgezeichneten Dreiecks in $\mathcal{K}$ K-projektiv (bzw. K-injektiv) sind, + dann auch der dritte. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(i)] + \item Das folgt, daraus dass für $\com{X}, \com{S} \in \mathcal{K}$ gilt: $\com{S} $ exakt $\iff \com{S} [-1]$ + exakt und + \[ + \text{Mor}_{\mathcal{K}}(\com{X}, \com{S}[-1]) = \text{Mor}_{\mathcal{K}}(\com{X}[1], \com{S}) + .\] + \item Sei $(\com{X}, \com{Y}, \com{Z}, u, v, w)$ ein ausgezeichnetes Dreieck mit $\com{X}, \com{Y} $ K-projektiv + und $\com{S} $ exakt. Nach Anwenden von $\text{Mor}_{\mathcal{K}}(-, \com{S})$ + \ref{hom-cohom-func} ist dann + \[ + \underbrace{\text{Mor}_{\mathcal{K}}(\com{X}[1], \com{S})}_{= 0} + \to \text{Mor}_{\mathcal{K}}(\com{Z}, \com{S} ) + \to + \underbrace{\text{Mor}_{\mathcal{K}}(\com{Y}, \com{S} )}_{= 0} + \] exakt und die äußeren Terme $0$ nach Voraussetzung und (i). Also folgt + $\text{Mor}_{\mathcal{K}}(\com{Z}, \com{S}) = 0$, also $\com{Z} $ K-projektiv. Der allgemeine Fall folgt nun + mit \ref{TR2}. + \end{enumerate} +\end{proof} + +\begin{satz} + Sei $\com{P} \in \mathcal{K}$. Dann sind äquivalent + \begin{enumerate}[(i)] + \item $\com{P} $ K-projektiv + \item Für $\com{X} \to \com{Y} $ Quasiisomorphismus in $\mathcal{K}$ ist der natürliche Homomorphismus + \[ + \text{Mor}_{\mathcal{K}}(\com{P} , \com{X} ) \to \text{Mor}_{\mathcal{K}}(\com{P} , \com{Y} ) + \] ein Isomorphismus. + \item Für $\com{S} \in \mathcal{K}$ ist der natürliche Homomorphismus + \[ + \text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{D}}(\com{P} , \com{S} ) + \] ein Isomorphismus. + \end{enumerate} +\end{satz} + +\begin{proof} + (i)$\implies$(ii): Sei $f\colon \com{X} \to \com{Y} $ ein Quasiisomorphismus. Dann ist + \[ + \begin{tikzcd} + \com{X} \arrow{r}{f} & \com{Y} \arrow{r} & \com{C_f} \arrow{r} & \com{X}[1] + \end{tikzcd} + \] ein ausgezeichnetes Dreieck und $\com{C_f}$ ist nach \ref{mapping-cone-exact-for-qis} exakt. Anwenden von + $\text{Mor}_{\mathcal{K}}(\com{P} , -) $ liefert mit \ref{hom-cohom-func} eine exakte Folge: + \[ + \begin{tikzcd} + \underbrace{\text{Mor}_{\mathcal{K}}(\com{P} , \com{C_f}[-1]) \arrow{r}}_{= 0} & + \text{Mor}_{\mathcal{K}}(\com{P} , \com{X} ) \arrow{r} & + \text{Mor}_{\mathcal{K}}(\com{P} , \com{Y} ) \arrow{r} & + \underbrace{\text{Mor}_{\mathcal{K}}(\com{P}, \com{C_f} ) }_{= 0} + \end{tikzcd} + .\] Die äußeren Terme sind 0, da $\com{P} $ K-projektiv, also folgt der behauptete Isomorphismus. + + (ii)$\implies$(iii): Injektivität: Sei $f\colon \com{P} \to \com{S} $, s.d. $\text{id}^{-1}f = 0$. Nach + \ref{derived-cat-morphism-null} existiert ein $t\colon \com{S} \to \com{T} $ Quasiisomorphismus, s.d. $tf= 0$. + Nach (ii) ist $t_{*}\colon \text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{K}}(\com{P} , \com{T} ) $ + injektiv, also folgt $f = 0$. Surjektivität: Sei $a \in \text{Mor}_{\mathcal{D}}(\com{P} , \com{S} ) $. Dann + ist $a$ ein Diagramm in $\mathcal{K}$ + \[ + \begin{tikzcd} + & \com{M} & \\ + \com{P} \arrow{ur}{f} & & \arrow{ul}{s} \com{S} + \end{tikzcd} + \] mit $s$ Quasiisomorphismus. Nach (ii) ist $s_{*}\colon \text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{K}}(\com{P} , \com{M} ) $ surjektiv, also existiert ein $g\colon \com{P} \to \com{S} $, s.d. $sg = f$. Also + kommutiert + \[ + \begin{tikzcd} + & \com{S} \arrow{d}{s} & \\ + \com{P} \arrow{dr}{f} \arrow{ur}{g} & \com{M} & \arrow{l}{s} \arrow{ul}{\text{id}} \com{S} \arrow{dl}{s}\\ + & \com{M} \arrow{u}{\text{id}} & \\ + \end{tikzcd} + .\] Damit folgt $a = g\text{id}^{-1}$. + + (iii)$\implies$(i): Sei $\com{S} $ exakt. Dann ist $\com{S} \to 0$ ein Quasiisomorphismus, also + $\com{S} \stackrel{\sim }{=} 0$ in $\mathcal{D}$, also + \[ + \text{Mor}_{\mathcal{K}}(\com{X} , \com{S} ) + \stackrel{\text{(ii)}}{=} \text{Mor}_{\mathcal{D}}(\com{X} , \com{S} ) + = \text{Mor}_{\mathcal{D}}(\com{X} , \com{0} ) = 0 + .\] +\end{proof} + +\begin{satz} + Sei $\com{X} \in \mathcal{K}$. Dann sind äquivalent + \begin{enumerate}[(i)] + \item $\com{X} $ K-projektiv. + \item Für alle Diagramme in $\mathcal{K}$ + \[ + \begin{tikzcd} + & \com{M} \arrow{d}{s} \\ + \com{X} \arrow{r}{f} & \com{N}\\ + \end{tikzcd} + \] mit $s$ Quasiisomorphismus, existiert genau ein $g\colon \com{X} \to \com{M} $, s.d. + $sg= f$ in $\mathcal{K}$. + \item Für alle Quasiisomorphismen $u\colon \com{S} \to \com{X} $ in $\mathcal{K}$ existiert ein + $v\colon \com{A} \to \com{S} $, s.d. $uv = \text{id}_{\com{X} }$ in $\mathcal{K}$. + \end{enumerate} +\end{satz} + +\newpage +\section{Adjunktion von abgeleitetem Hom und Tensorprodukt} \end{document}