From 775955dcd2b2f7d07bb2edc55e83b6f80c611f03 Mon Sep 17 00:00:00 2001 From: flavis Date: Mon, 2 May 2022 18:53:03 +0200 Subject: [PATCH] final version --- sose2022/bachelorarbeit/arbeit.cls | 2 +- sose2022/bachelorarbeit/arbeit.pdf | Bin 662609 -> 481831 bytes sose2022/bachelorarbeit/arbeit.tex | 480 +++++++++++------ sose2022/bachelorarbeit/vortrag.pdf | Bin 258562 -> 168325 bytes sose2022/bachelorarbeit/vortrag.tex | 508 +++++++++++------- sose2022/metaethik/2022-04-22-Note-09-52.xopp | Bin 151993 -> 254845 bytes 6 files changed, 634 insertions(+), 356 deletions(-) diff --git a/sose2022/bachelorarbeit/arbeit.cls b/sose2022/bachelorarbeit/arbeit.cls index 9aba5c6..dccc7f2 100644 --- a/sose2022/bachelorarbeit/arbeit.cls +++ b/sose2022/bachelorarbeit/arbeit.cls @@ -201,7 +201,7 @@ } % replace all relations with align characters (&) and add the needed padding \regex_replace_all:nnN - { 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So erhalten wir ein System $(\tau^{\ge -n} \com{M})_{n \in \N}$ der abgeschnittenen Komplexe von $\com{M}$. -Induktiv konstruieren wir dann ein $\mathcal{J}$-spezielles inverses System +Induktiv erhalten wir dann ein $\mathcal{J}$-spezielles inverses System $(\com{I}_n)_{n \in \N}$ und ein inverses System von Quasiisomorphismen $f_n\colon \tau^{\ge -n} \com{M} \to \com{I}_n$, indem wir in jedem Schritt für $n \in \N$ aus $\tau^{\ge -n} \com{M}$ und $\com{I}_{n-1}$ einen nach unten beschränkten Komplex konstruieren, der klassisch durch einen K-injektiven aufgelöst werden kann. -So erhalten wir im Limes einen K-injektiven Komplex $\com{I}$ und einen Komplexhomomorphismus +Im Limes erhalten wir also einen K-injektiven Komplex $\com{I}$ und einen Komplexhomomorphismus $f\colon \com{M} \to \com{I}$. Da für $\mathcal{A} = A\text{-Mod}$ der inverse Limes jedoch nicht exakt ist, ist @@ -404,7 +417,9 @@ Funktor $\text{R}F$ von der derivierten Kategorie $\mathcal{D}(\mathcal{A})$ von $\mathcal{A}$ in die derivierte Kategorie $\mathcal{D}(\mathcal{B})$ von $\mathcal{B}$ konstruieren. Die derivierte Kategorie $\mathcal{D}(\mathcal{A})$ erhalten wir durch Lokalisierung, analog zum gleichnamigen Vorgang in der kommutativen Algebra: Wir starten mit der Kategorie -$\mathcal{K}(\mathcal{A})$, der Homotopiekategorie von $\mathcal{A}$, wobei die Morphismen +$\mathcal{K}(\mathcal{A})$, deren Objekte +Komplexe in $\mathcal{A}$ und deren +Morphismen Homotopieäquivalenzklassen von Komplexhomomorphismen sind. Daraus erhalten wir dann $\mathcal{D}(\mathcal{A})$ indem alle Quasiisomorphismen in $\mathcal{K}(\mathcal{A})$, das heißt Komplexhomomorphismen, die Isomorphismen auf den @@ -427,6 +442,14 @@ anderen Struktur, die einer triangulierten Kategorie. \item einer Klasse von Sextupeln $(X, Y, Z, u, v, w)$, den ausgezeichneten Dreiecken von $\mathcal{T}$, wobei $X, Y, Z \in \mathcal{T}$ und $u\colon X \to Y$, $v\colon Y \to Z$, $w\colon Z \to T(X)$ Morphismen sind. \end{enumerate} + Wenn $(X, Y, Z, u, v, w)$ ein ausgezeichnetes Dreieck ist, schreiben wir + auch: + \[ + \begin{tikzcd} + & Z \arrow[swap]{dl}{w} & \\ + X \arrow[swap]{rr}{u} & & Y \arrow[swap]{ul}{v}. + \end{tikzcd} + \] Ein Morphismus von ausgezeichneten Dreiecken $(X, Y, Z, u, v, w) \to (X', Y', Z', u', v', w')$ ist ein kommutatives Diagramm \[ @@ -441,21 +464,39 @@ anderen Struktur, die einer triangulierten Kategorie. isomorphe Sextupel $(X, Y, Z, u, v, w)$ wie oben, ist ein ausgezeichnetes Dreieck. Jeder Morphismus $u\colon X \to Y$ kann in ein ausgezeichnetes Dreieck $(X, Y, Z, u, v, w)$ eingebettet werden und das Sextupel $(X, X, 0, \text{id}_X, 0, 0)$ ist ein ausgezeichnetes Dreieck für alle $X \in \mathcal{T}$. - \item $(X, Y, Z, u, v, w)$ ist ein ausgezeichnetes Dreieck genau dann wenn $(Y, Z, T(X), v, w, -T(u))$ ein + \item $(X, Y, Z, u, v, w)$ ist genau dann ein ausgezeichnetes Dreieck, + wenn $(Y, Z, T(X), v, w, -T(u))$ ein ausgezeichnetes Dreieck ist. \item Für zwei ausgezeichnete Dreiecke $(X, Y, Z, u, v, w)$ und $(X', Y', Z', u', v', w')$, und Morphismen $f\colon X \to X'$, $g\colon Y \to Y'$, die mit $u$ und $u'$ kommutieren, existiert ein nicht notwendig eindeutiger Morphismus $h\colon Z \to Z'$, sodass $(f, g, h)$ ein Morphismus von ausgezeichneten Dreiecken ist. + \item Seien $(X, Y, Z', u, j, k)$, $(Y, Z, X', v, t, i)$ + und $(X, Z, Y', vu, r, s)$ ausgezeichnete Dreiecke. Dann existieren + Morphismen $f\colon Z' \to Y'$ und $g\colon Y' \to X'$, sodass + $(Z', Y', X', f, g, T(j) i)$ ein ausgezeichnetes Dreieck ist und die zwei gestrichelten + Dreiecke in + \[ + \begin{tikzcd} + & Y' \arrow[dashed]{ddl} \arrow[dashed]{dr}{g} & \\ + Z' \arrow[dashed]{ur}{f} \arrow[dashed]{d} & + & \arrow{ll}{T(j)i} + \arrow[swap]{ddl}{i} X' \\ + X \arrow[swap]{dr}{u} \arrow{rr}{vu} & & Z \arrow[dashed]{uul} + \arrow[dashed,swap]{u} \\ + & Y \arrow[swap]{uul}{j} \arrow[swap]{ur}{v} & + \end{tikzcd} + \] + kommutieren. \end{enumerate} \label{TR2} \end{definition} -\begin{bem} - Streng genommen ist die hier definierte Triangulierte Kategorie nur eine Pretriangulierte Kategorie. Für - eine vollständig triangulierte Kategorie fehlt noch das Oktaheder Axiom - (TR4 in \cite{hartshorne}), das wir im Folgenden jedoch nicht benötigen. -\end{bem} +%\begin{bem} +% Streng genommen ist die hier definierte Triangulierte Kategorie nur eine Pretriangulierte Kategorie. Für +% eine vollständig triangulierte Kategorie fehlt noch das Oktaheder Axiom +% (TR4 in \cite{hartshorne}), das wir im Folgenden jedoch nicht benötigen. +%\end{bem} \begin{definition}[Triangulierte Unterkategorie] Sei $\mathcal{T}$ eine triangulierte Kategorie mit Verschiebefunktor $T$. @@ -464,7 +505,7 @@ anderen Struktur, die einer triangulierten Kategorie. \begin{enumerate}[(i)] \item für jedes Objekt $X \in \mathcal{T}$ ist $X \in \mathcal{R}$, genau dann wenn $T(X) \in \mathcal{R}$ ist, und - \item falls zwei Punkte eines ausgezeichneten Dreiecks in $\mathcal{T}$ in $\mathcal{R}$ liegen, so auch + \item falls zwei Punkte eines ausgezeichneten Dreiecks von $\mathcal{T}$ in $\mathcal{R}$ liegen, so auch der dritte. \end{enumerate} \label{def:triangulated-subcategory} @@ -499,19 +540,27 @@ anderen Struktur, die einer triangulierten Kategorie. siehe Proposition 1.1 in \cite{hartshorne}. \end{proof} -\subsection{Homotopiekategorie} +\subsection{$\mathcal{K}(\mathcal{C})$ als triangulierte Kategorie} -Das kanonische Beispiel für eine triangulierte Kategorie ist die Homotopiekategorie einer additiven Kategorie $\mathcal{C}$. +Für eine additive Kategorie $\mathcal{C}$ sei im Folgenden $\mathcal{K}om(\mathcal{C})$ +die Komplexkategorie von $\mathcal{C}$. Weiter sei $\mathcal{K}(\mathcal{C})$ die Kategorie mit den +selben Objekten wie $\mathcal{K}om(\mathcal{C})$ und mit Homotopieäquivalenzklassen +von Komplexhomomorphismen als Morphismen. +Wir geben im Folgenden eine triangulierte Struktur auf $\mathcal{K}(\mathcal{C})$. -\begin{definition}[Homotopiekategorie] - Sei $\mathcal{C}$ eine additive Kategorie. Dann ist die \emph{Homotopiekategorie} - $\mathcal{K}(\mathcal{C})$ von $\mathcal{C}$, die Kategorie, deren Objekte - Komplexe mit Objekten aus $\mathcal{C}$ sind und - deren Morphismen Homotopieäquivalenzklassen von Komplexhomomorphismen sind. -\end{definition} +%Um eine triangulierte Struktur +%Das kanonische Beispiel für eine triangulierte Kategorie ist die Kategorie $\mathcal{K}(\mathcal{C})$ +%für eine additive Kategorie $\mathcal{C}$. +% +%\begin{definition} +% Sei $\mathcal{C}$ eine additive Kategorie. Dann ist +% $\mathcal{K}(\mathcal{C})$, die Kategorie, deren Objekte +% Komplexe mit Objekten aus $\mathcal{C}$ sind und +% deren Morphismen Homotopieäquivalenzklassen von Komplexhomomorphismen sind. +%\end{definition} %\begin{bem} -In der Homotopiekategorie haben wir einen natürlichen Kategorienautomorphismus +In $\mathcal{K}(\mathcal{C})$ haben wir einen natürlichen Kategorienautomorphismus $T\colon \mathcal{K}(\mathcal{C}) \to \mathcal{K}(\mathcal{C})$, der durch Verschieben nach links gegeben ist, das heißt für $\com{X} \in \mathcal{K}(\mathcal{C})$ durch @@ -529,12 +578,13 @@ heißt für $\com{X} \in \mathcal{K}(\mathcal{C})$ durch \end{bem} Um die ausgezeichneten Dreiecke in $\mathcal{K}(\mathcal{C})$ zu erklären, benötigen wir -den Abbildungskegel eines Morphismus in $\mathcal{K}(\mathcal{C})$: +den Abbildungskegel eines Komplexhomomorphismus: \begin{definition}[Abbildungskegel] - Seien $\com{X}, \com{Y} \in \mathcal{K}$ zwei Komplexe und $f\colon \com{X} \to \com{Y}$ ein + Seien $\com{X}, \com{Y} \in \mathcal{K}om(\mathcal{C})$ zwei Komplexe und + $f\colon \com{X} \to \com{Y}$ ein Komplexhomomorphismus. Dann sei der \emph{Abbildungskegel} - $\com{C}_f \in \mathcal{K}(\mathcal{C})$ definiert durch + $\com{C}_f \in \mathcal{K}om(\mathcal{C})$ definiert durch \[ C_f^{n} = X^{n+1} \oplus Y^{n} \] mit Differential @@ -557,8 +607,20 @@ den Abbildungskegel eines Morphismus in $\mathcal{K}(\mathcal{C})$: \end{enumerate} \end{bem} -\begin{satz}[Homotopiekategorie ist trianguliert] - Sei $\mathcal{C}$ eine additive Kategorie. Dann ist die Homotopiekategorie $\mathcal{K}(\mathcal{C})$ +\begin{bem}[Abbildungskegel in $\mathcal{K}(\mathcal{C})$] + Seien $\com{X}, \com{Y} \in \mathcal{K}om(\mathcal{C})$ und $f, g \colon \com{X} \to \com{X} $ + homotope Komplexhomomorphismen. Dann sind + $\com{C}_f$ und $\com{C}_g$ homotopieäquivalent, aber die Wahl der Homotopieäquivalenz + ist nicht kanonisch. Also sind $\com{C}_f$ und $\com{C}_g$ nur unkanonisch isomorph + in $\mathcal{K}(\mathcal{C})$. + Wenn wir + im Folgenden von $\com{C}_f \in \mathcal{K}(\mathcal{C})$ für ein + $f\colon \com{X} \to \com{Y} $ in $\mathcal{K}(\mathcal{C})$ sprechen, ist damit + ein beliebiger Vertreter der Isomorphieklasse gemeint. +\end{bem} + +\begin{satz}[$\mathcal{K}(\mathcal{C})$ ist trianguliert] + Sei $\mathcal{C}$ eine additive Kategorie. Dann ist $\mathcal{K}(\mathcal{C})$ mit den folgenden Daten trianguliert: \begin{enumerate}[(a)] \item Der Verschiebefunktor $T\colon \mathcal{K}(\mathcal{C}) \to \mathcal{K}(\mathcal{C})$ wie in \ref{eq:shift-functor}. @@ -579,7 +641,7 @@ den Abbildungskegel eines Morphismus in $\mathcal{K}(\mathcal{C})$: siehe Kapitel 1, §2 in \cite{hartshorne}. \end{proof} -Sei nun $\mathcal{A}$ eine abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die Homotopiekategorie. +Sei nun $\mathcal{A}$ eine abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$. \begin{lemma} Der Funktor $H\colon \mathcal{K} \to \mathcal{A}b$ der einen Komplex $\com{A}$ @@ -596,7 +658,7 @@ Daraus erhalten wir insbesondere folgendes Kriterium, ob ein Komplexhomomorphism \begin{lemma}[] Seien $\com{A}, \com{B} \in \mathcal{K}$ und $f\colon \com{A} \to \com{B} $ ein Komplexhomomorphismus. - Dann ist $f$ ein Quasiisomorphismus, genau dann wenn $\com{C}_f$ exakt ist. + Dann ist $f$ genau dann ein Quasiisomorphismus, wenn $\com{C}_f$ exakt ist. \label{mapping-cone-exact-for-qis} \end{lemma} @@ -661,9 +723,9 @@ Daraus erhalten wir insbesondere folgendes Kriterium, ob ein Komplexhomomorphism \subsection{Lokalisierung von Kategorien} -Wie anfangs erwähnt, ist die derivierte Kategorie $\mathcal{D}(\mathcal{A})$ einer +Wie anfangs erwähnt ist die derivierte Kategorie $\mathcal{D}(\mathcal{A})$ einer abelschen Kategorie $\mathcal{A}$ -eine Lokaliserung der Homotopiekategorie $\mathcal{K}(\mathcal{A})$. +eine Lokaliserung von $\mathcal{K}(\mathcal{A})$. Diese funktioniert analog zum gleichnamigen Prozess in der kommutativen Algebra, was uns zu folgendem Begriff führt: @@ -689,7 +751,7 @@ uns zu folgendem Begriff führt: \item Für $f, g \colon X \to Y$ Morphismen in $\mathcal{C}$, sind die folgenden Bedingungen äquivalent: \begin{enumerate}[(i)] \item Es existiert ein $s\colon Y \to Y'$ in $\mathcal{S}$, sodass $sf = sg$. - \item Es existiert ein $t\colon X \to X'$ in $\mathcal{S}$, sodass $ft = gt$. + \item Es existiert ein $t\colon X' \to X$ in $\mathcal{S}$, sodass $ft = gt$. \end{enumerate} \end{enumerate} \label{def:mult-system} @@ -697,7 +759,7 @@ uns zu folgendem Begriff führt: \begin{definition}[Lokalisierung] Seien $\mathcal{C}$ eine Kategorie und $\mathcal{S}$ eine Klasse von Morphismen in $\mathcal{C}$. Dann - ist die \emph{Lokalisierung} von $\mathcal{C}$ in Bezug auf $\mathcal{S}$ eine Kategorie $\mathcal{C}_{\mathcal{S}}$ + ist eine \emph{Lokalisierung} von $\mathcal{C}$ in Bezug auf $\mathcal{S}$ eine Kategorie $\mathcal{C}_{\mathcal{S}}$ zusammen mit einem Funktor $Q\colon \mathcal{C} \to \mathcal{C}_{\mathcal{S}}$, sodass \begin{enumerate}[(a)] \item $Q(s)$ ein Isomorphismus ist für alle $s \in \mathcal{S}$ und @@ -707,6 +769,11 @@ uns zu folgendem Begriff führt: \label{def:localisation} \end{definition} +\begin{bem} + Falls in der Situation von \ref{def:localisation} $\mathcal{C}_{\mathcal{S}}$ existiert, dann + ist diese Kategorie bis auf kanonische Kategorienäquivalenz eindeutig. +\end{bem} + \begin{definition} Sei $\mathcal{C}$ eine Kategorie und $\mathcal{S}$ ein multiplikatives System in $\mathcal{C}$. Dann definiere die Kategorie $\mathcal{C}_{\mathcal{S}}$ durch @@ -714,13 +781,13 @@ uns zu folgendem Begriff führt: \item $ob(\mathcal{C}) = ob(\mathcal{C}_{\mathcal{S}})$. \item Für $X, Y \in \mathcal{C}$ setze $\operatorname{Mor}_{\mathcal{C}_{\mathcal{S}}}(X, Y) = \{ (f, Z, s) \mid f \colon Z \to Y, - s \colon Z \to X \text{ mit } s \in \mathcal{S}\} / \sim $ wobei + s \colon Z \to X \text{ mit } s \in \mathcal{S}\} / \sim $, wobei $(f, Z, s) \sim (f', Z', s')$ genau dann wenn ein kommutatives Diagramm \[ \begin{tikzcd} - & Z \arrow{dl}{s} \arrow{dr}{f} & \\ + & Z \arrow[swap]{dl}{s} \arrow{dr}{f} & \\ X & W \arrow{l}{t} \arrow{u}{g} \arrow{d}{h} & Y \\ - & \arrow{ul}{s'} Z' \arrow{ur}{f'} & + & \arrow{ul}{s'} Z' \arrow[swap]{ur}{f'} & \end{tikzcd} \] mit $t \in \mathcal{S}$ in $\mathcal{C}$ existiert. \item Für $(f, U, s) \in \operatorname{Mor}_{\mathcal{C}_{\mathcal{S}}}(X, Y)$, @@ -770,7 +837,7 @@ uns zu folgendem Begriff führt: Wir nehmen im Folgenden stillschweigend an, dass $\mathcal{C}_{\mathcal{S}}$ eine (echte) Kategorie ist. \item Die Lokalisierung $\mathcal{C}_{\mathcal{S}}$ einer Kategorie $\mathcal{C}$ - kann auch dual, dass heißt durch Umdrehen aller Pfeile, + kann auch dual, das heißt durch Umdrehen aller Pfeile in \ref{constr:localisation}, konstruiert werden. Wenn im Folgenden der Kontext klar ist, dann schreiben wir auch einfach $X$ für $Q(X)$ für ein Objekt $X \in \mathcal{C}$. Ebenso schreiben wir für $X, Y \in \mathcal{C}$ auch $s^{-1}f$ bzw. in der dualen Konstruktion $fs^{-1}$ für die Äquivalenzklasse des Tripels @@ -814,8 +881,8 @@ an $\mathcal{S}$: \subsection{Derivierte Kategorie} -Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die -Homotopiekategorie. Wir bezeichnen im Folgenden die Klasse +Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und +$\mathcal{K} = \mathcal{K}(\mathcal{A})$. Wir bezeichnen im Folgenden die Klasse der Quasiisomorphismen in $\mathcal{K}$ als $\mathcal{Q}is$. \begin{lemma}[$\mathcal{Q}is$ ist multiplikativ] @@ -860,7 +927,7 @@ das folgende Lemma: \begin{proof} Wegen \hyperref[def:mult-system]{FR3} und \ref{lemma:qis-mult} genügt es die Äquivalenz von - (i) und (ii) zu zeigen. Es ist $\text{id}^{-1}f = 0$, genau dann wenn + (i) und (ii) zu zeigen. Es ist genau dann $\text{id}^{-1}f = 0$, wenn ein kommutatives Diagram \[ \begin{tikzcd} @@ -868,7 +935,7 @@ das folgende Lemma: \com{X} & \arrow[dashed]{l}{t} \com{X'} \arrow[dashed]{d} \arrow[dashed]{u} & \com{Y} \\ & \com{X} \arrow{ul}{\text{id}} \arrow{ur}{0} & \end{tikzcd} - \] mit $t$ Quasiisomorphismus existiert. Das zeigt die Behauptung. + \] mit einem Quasiisomorphismus $t$ existiert. Das zeigt die Behauptung. \end{proof} Wir möchten nun Ableitungen von Funktoren im Kontext von @@ -876,7 +943,7 @@ derivierten Kategorien betrachten. Seien dafür $\mathcal{A}$ und $\mathcal{B}$ abelsche Kategorien und $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein triangulierter (kovarianter) Funktor. -Das ist zum Beispiel der Fall, wenn $F$ induziert ist von einem additiven Funktor +Das ist zum Beispiel der Fall wenn $F$ induziert ist von einem additiven Funktor $F\colon \mathcal{A} \to \mathcal{B}$. Falls $F$ exakt ist werden Quasiisomorphismen auf Quasiisomorphismen geschickt und $F$ induziert daher @@ -889,7 +956,7 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien. Der \emph{rechts abgeleitete Funktor} von $F$ ist ein triangulierter Funktor \[ - \text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{A}) + \text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B}) \] zusammen mit einer natürlichen Transformation \[ \xi\colon Q_{\mathcal{B}} \circ F \to \text{R}F \circ Q_{\mathcal{A}} @@ -917,7 +984,7 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: \begin{bem} \begin{enumerate}[(a)] - \item Falls $\text{R}F$ existiert, ist dieser bis auf eindeutige natürliche Transformationen eindeutig. + \item Falls $\text{R}F$ existiert, ist dieser bis auf eindeutige natürliche Äquivalenz eindeutig. \item Wir schreiben $R^{i}F$ für $H^{i}(\text{R}F)$. Wenn $F$ induziert ist von einem links-exakten Funktor $F\colon \mathcal{A} \to \mathcal{B}$ und $\mathcal{A}$ genügend @@ -939,16 +1006,16 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein triangulierter Funktor. Angenommen es existiert eine triangulierte Unterkategorie $\mathcal{L} \subset \mathcal{K}(\mathcal{A})$, sodass \begin{enumerate}[(i)] - \item Jeder Komplex in $\mathcal{K}(\mathcal{A})$ besitzt einen Quasiisomorphismus in einen - Komplex aus $\mathcal{L}$. - \item $F|_{\mathcal{L}}$ ist exakt. + \item für jeden Komplex in $\mathcal{K}(\mathcal{A})$ ein Quasiisomorphismus in einen + Komplex aus $\mathcal{L}$ existiert, und + \item $F|_{\mathcal{L}}$ Exaktheit von Komplexen erhält. \end{enumerate} Dann existiert der Funktor $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$ und für die natürliche Transformation $\xi\colon Q_{\mathcal{B}} \circ F \to \text{R}F \circ Q_{\mathcal{A}}$ gilt, dass für alle $\com{I} \in \mathcal{L}$ die Abbildung \[ \xi(\com{I})\colon Q_{\mathcal{B}}(F(\com{I})) \to \text{R}F(Q_{\mathcal{A}}(\com{I})) - \] ein Isomorphismus in $\mathcal{D}(A)$ ist. + \] ein Isomorphismus in $\mathcal{D}(\mathcal{B})$ ist. \label{satz:existence-derived-functors} \end{satz} @@ -1030,7 +1097,7 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: .\] Wegen $d_{\com{Y}[n]} = (-1)^{n}d_{\com{Y}}$, induziert $(f^i)_{i \in \Z}$ also genau dann einen Komplexhomomorphismus $\com{X} \to \com{Y}[n]$, wenn $(f^{i})_{i \in \Z} \in \operatorname{ker } d^{n}$. - Weiter ist $(f^{i})_{i \in \Z} \in \operatorname{im } d^{n-1}$, genau dann wenn eine Familie + Weiter ist genau dann $(f^{i})_{i \in \Z} \in \operatorname{im } d^{n-1}$, wenn eine Familie $(k^{i})_{i \in \Z}\in \prod_{i \in \Z}^{} \operatorname{Hom}(X^{i}, Y^{i+n-1})$ existiert, sodass %\[ % f^{i} = d_{\com{Y}}^{i+n-1} k^{i} - (-1)^{n-1} k^{i+1} d_{\com{X} }^{i} @@ -1069,7 +1136,8 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \begin{proof} Nach der Definition von ausgezeichneten Dreiecken in $\mathcal{K}$ genügt es nachzurechnen, dass - für $f\colon \com{X} \to \com{Y} $ in $\mathcal{K}$ und $\com{S} \in \mathcal{K}$ gilt: + für $f\colon \com{X} \to \com{Y} $ in $\mathcal{K}om(\mathcal{A})$ + und $\com{S} \in \mathcal{K}om(\mathcal{A})$ gilt: \[ \com{C}_f \otimes_A \com{S} = \com{C}_{f \otimes \text{id}_{\com{S}}} .\] Das rechnet man gradweise nach und zeigt, dass die Differentiale übereinstimmen. @@ -1108,11 +1176,13 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \label{sec:resolutions} -Sei $\com{Y} \in \mathcal{K}$. +Sei in diesem Abschnitt $\mathcal{A}$ eine abelsche Kategorie und +$\mathcal{K} = \mathcal{K}(\mathcal{A})$. +Sei weiter $\com{Y} \in \mathcal{K}$. Um die Bedingungen von \ref{satz:existence-derived-functors} für $\com{\operatorname{Hom}}(\com{Y}, -)$ (bzw. $\com{\operatorname{Hom}}(-, \com{Y})$) zu erfüllen, benötigen wir -eine Unterkategorie $\mathcal{L}$ von Komplexen in $\mathcal{K}$, sodass +eine triangulierte Unterkategorie $\mathcal{L}$ von Komplexen in $\mathcal{K}$, sodass \begin{enumerate}[(i)] \item für jeden Komplex $\com{X} \in \mathcal{K}$ ein Quasiisomorphismus @@ -1150,7 +1220,7 @@ Das Ziel dieses Abschnitts ist das folgende Resultat: \label{satz:r-mod-existence-k-proj-and-k-inj-resolutions} \end{satz} -Die Vorgehensweise orientiert sich dabei an \cite{spaltenstein}. +Die Vorgehensweise orientiert sich dabei an den Abschnitten 1 bis 3 von \cite{spaltenstein}. \subsection{Elementare Eigenschaften von K-injektiven und K-projektiven Komplexen} @@ -1192,8 +1262,8 @@ Komplexen entwickelt. einen beliebigen Komplex $\com{X} \in \mathcal{K}$, der Komplex $\com{\operatorname{Hom}}(\com{X}, \com{I})$ exakt ist, denn \[ - H^{i}(\com{\operatorname{Hom}}(\com{X}, \com{I})) = \operatorname{Mor}_{\mathcal{K}}(\com{\operatorname{Hom}}(\com{X}, \com{I}[i])) - \stackrel{\ref{lemma:k-inj-exact-contractible}}{=} \operatorname{Mor}_{\mathcal{K}}(\com{\operatorname{Hom}}(\com{X}, 0)) = 0 + H^{i}(\com{\operatorname{Hom}}(\com{X}, \com{I})) = \operatorname{Mor}_{\mathcal{K}}(\com{X}, \com{I}[i]) + \stackrel{\ref{lemma:k-inj-exact-contractible}}{=} \operatorname{Mor}_{\mathcal{K}}(\com{X}, 0) = 0 .\] Analog gilt die duale Version für exakte K-projektive Komplexe. \label{satz:hom-exact-for-k-inj} %auch $\com{\operatorname{Hom}}(\com{X}, \com{I})$ exakt ist für alle $\com{M} \in \mathcal{K}$. @@ -1222,9 +1292,9 @@ $X^{0} = X$, zusammenhängt. Die folgende Aussage stellt diesen Zusammenhang her M \arrow{r} & N \arrow[twoheadrightarrow]{r}{v} & P \arrow{r} & 0 \end{tikzcd}\] Nach Voraussetzung ist dieser nullhomotop, das heißt es existiert ein $k \colon X^{0} \to N$, sodass $f = vk$. Also ist -$v_{*}\colon \operatorname{Hom}(X^{0}, N) \to \operatorname{Hom}(X^{0}, P)$ surjektiv und damit $X^{0}$ projektiv. +$v_{*}\colon \operatorname{Hom}_{\mathcal{A}}(X^{0}, N) \to \operatorname{Hom}_{\mathcal{A}}(X^{0}, P)$ surjektiv und damit $X^{0}$ projektiv. -($\Leftarrow$) Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ +($\Leftarrow$) Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ ein Komplexhomomorphismus. Dann betrachte \[ \begin{tikzcd} @@ -1236,7 +1306,8 @@ Komplexhomomorphismus. Dann betrachte \end{tikzcd} .\] Da $d^{0}f^{0} = 0$, faktorisiert $f^{0}$ über $\operatorname{ker } d^{0} = \operatorname{im }d^{-1}$. Weil -$X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d^{-1} k^{0}$. +$X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d^{-1} k^{0}$ +und damit ist $f$ nullhomotop. \end{proof} \begin{satz}[] @@ -1322,7 +1393,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d \[ \begin{tikzcd} & \com{S} \arrow{d}{s} & \\ - \com{P} \arrow{dr}{f} \arrow{ur}{g} & \com{Y} & \arrow{l}{s} \arrow{ul}{\text{id}} \com{S} \arrow{dl}{s}\\ + \com{P} \arrow{dr}{f} \arrow{ur}{g} & \com{Y} & \arrow{l}{s} \arrow[swap]{ul}{\text{id}} \com{S} \arrow{dl}{s}\\ & \com{Y} \arrow{u}{\text{id}} & \\ \end{tikzcd} .\] Damit folgt $a = g\text{id}^{-1}$. @@ -1346,7 +1417,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d & \com{X} \arrow{d}{s} \\ \com{P} \arrow{r}{f} & \com{Y}\\ \end{tikzcd} - \] mit $s$ Quasiisomorphismus, existiert genau ein $g\colon \com{P} \to \com{X} $, sodass + \] mit einem Quasiisomorphismus $s$, existiert genau ein $g\colon \com{P} \to \com{X} $, sodass $sg= f$ in $\mathcal{K}$. \item Für alle Quasiisomorphismen $s\colon \com{S} \to \com{P} $ in $\mathcal{K}$ existiert ein $g\colon \com{P} \to \com{S} $, sodass $sg = \text{id}_{\com{P} }$ in $\mathcal{K}$. @@ -1358,7 +1429,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d \[ \begin{tikzcd} & \com{X} \arrow{d}{\text{id}^{-1}s} \\ - \com{P} \arrow[dashed]{ur}{g} \arrow{r}{\text{id}^{-1}f} & \com{Y} + \com{P} \arrow[dashed]{ur}{g} \arrow[swap]{r}{\text{id}^{-1}f} & \com{Y} \end{tikzcd} .\] Da $s$ ein Quasiisomorphismus ist, induziert $s$ einen Isomorphismus in $\mathcal{D}$. Also existiert genau ein @@ -1390,7 +1461,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d & \com{Q} \arrow{dr}{f} \arrow{dl}{s} & \\ \com{P} & & \com{S} \end{tikzcd} - \] in $\mathcal{K}$ mit $s$ Quasiisomorphismus. Nach (iii) existiert ein $t\colon \com{P} \to \com{Q}$ mit + \] in $\mathcal{K}$ mit einem Quasiisomorphismus $s$. Nach (iii) existiert ein $t\colon \com{P} \to \com{Q}$ mit $st = \text{id}_{\com{P} }$. Dann ist \[ \begin{tikzcd} @@ -1398,7 +1469,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d \com{P} & \com{P} \arrow{l}{\text{id}} \arrow{u}{t} \arrow{d}{\text{id}} & \com{S} \\ & \com{P} \arrow{ul}{\text{id}} \arrow{ur}{ft} & \\ \end{tikzcd} - \] ein kommutatives Diagramm in $\mathcal{K}$, also folgt $s^{-1}f = \text{id}^{-1}(ft)$ in $\mathcal{D}$. + \] ein kommutatives Diagramm in $\mathcal{K}$, also folgt $a = s^{-1}f = \text{id}^{-1}(ft)$ in $\mathcal{D}$. \end{proof} Durch Umdrehen aller Pfeile erhalten wir analog: @@ -1417,8 +1488,8 @@ Durch Umdrehen aller Pfeile erhalten wir analog: \com{Y} \arrow{r}{f} \arrow{d}{s} & \com{I} \\ \com{X} \end{tikzcd} - \] mit $s$ Quasiisomorphismus, existiert genau ein $g\colon \com{X} \to \com{I} $, sodass das Diagramm - kommutiert. + \] mit $s$ Quasiisomorphismus, existiert genau ein $g\colon \com{X} \to \com{I} $, + sodass $gs = f$ in $\mathcal{K}$. \item Für jeden Quasiisomorphismus $s\colon \com{I} \to \com{S} $ in $\mathcal{K}$ existiert ein $g\colon \com{S} \to \com{I} $, sodass $gs = \text{id}_{\com{I} }$ in $\mathcal{K}$. \end{enumerate} @@ -1439,12 +1510,19 @@ Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen \item Falls $n = 1$, dann ist $\com{I}_n = 0$. \item Falls $n > 1$, dann sei $\com{C}_n$ der Kern der natürlichen Abbildung $\com{I} _n \to \com{I}_{n-1}$. Dann ist $\com{C}_n \in \mathcal{J}$ und - die kurze exakte Folge + die kurze Folge \[ - 0 \to \com{C}_n \to \com{I}_n \to \com{I} _{n-1} \to 0 - \] zerfällt stufenweise. + \begin{tikzcd} + 0 \arrow{r} & \com{C}_n \arrow{r} + & \com{I}_n \arrow{r} + & \com{I}_{n-1} \arrow{r} + & 0 + \end{tikzcd} + %0 \to \com{C}_n \to \com{I}_n \to \com{I} _{n-1} \to 0 + \] ist exakt und zerfällt stufenweise. \end{enumerate} - \item Die Klasse $\mathcal{J}$ heißt abgeschlossen unter speziellen inversen Limites, falls jedes + \item Die Klasse $\mathcal{J}$ heißt \emph{abgeschlossen unter speziellen inversen Limites}, + falls jedes $\mathcal{J}$-spezielle inverse System in $\mathcal{K}$ einen Limes in $\mathcal{J}$ besitzt und jeder Komplex in $\mathcal{K}$, der isomorph zu einem Komplex in $\mathcal{J}$ ist, bereits in $\mathcal{J}$ ist. \end{enumerate} @@ -1484,9 +1562,9 @@ Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen \cdots\arrow{r} & A^{2} \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \\ & \vdots & \vdots & \vdots & \vdots \\ \end{tikzcd} - \] denn für $n > 1$ ist $\com{\operatorname{ker } p}_n$ = $[\cdots \to 0 \to \underbrace{A^{n-2}}_{\in \mathcal{J}_0} \to 0 \to \cdots ]$. Nach + \] denn für $n > 1$ ist $\operatorname{ker } p_n$ = $[\cdots \to 0 \to \underbrace{A^{n-2}}_{\in \mathcal{J}_0} \to 0 \to \cdots ]$. Nach Voraussetzung ist also $\operatorname{ker } p_n$ in $\mathcal{J}$ und die kurze exakte Folge - $0 \to \com{\operatorname{ker } p}_n \to \com{S}_n \to \com{S}_{n-1} \to 0$ zerfällt gradweise. Also folgt + $0 \to \operatorname{ker } p_n \to \com{S}_n \to \com{S}_{n-1} \to 0$ zerfällt gradweise. Also folgt $\com{A} = \lim \com{S}_n \in \mathcal{J}$. \end{proof} @@ -1517,7 +1595,7 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. \end{definition} \begin{bsp} - Spezielle inverse Systeme erfüllen (R). + Spezielle inverse Systeme in $\mathcal{A}b$ erfüllen (R). \end{bsp} \begin{lemma} @@ -1667,9 +1745,10 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. \end{proof} \begin{satz} - Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(B)$ abgeschlossen + Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(\mathcal{B})$ abgeschlossen unter speziellen inversen Limites. Angenommen inverse Limites existieren in $\mathcal{A}$ und sei - $F\colon \mathcal{K}(A) \to \mathcal{K}(B)$ ein kovarianter Funktor, der mit inversen Limites vertauscht und + $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein kovarianter Funktor, + der mit inversen Limites vertauscht und gradweise zerfallende kurze exakte Folgen erhält. Dann ist $F^{-1}(\mathcal{J}) \subset \mathcal{K}(\mathcal{A})$ abgeschlossen unter speziellen inversen Limites. @@ -1687,13 +1766,13 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. \item Für $n > 1$ ist nach Voraussetzung \[ \begin{tikzcd} - 0 \arrow{r} & \operatorname{ker } \com{p}_n \arrow{r} & \com{S}_n \arrow{r}{p_n} & \com{S}_{n-1} \arrow{r} & 0 + 0 \arrow{r} & \operatorname{ker } p_n \arrow{r} & \com{S}_n \arrow{r}{p_n} & \com{S}_{n-1} \arrow{r} & 0 \end{tikzcd} \] - exakt, zerfällt gradweise und $\operatorname{ker } \com{p}_n$ ist in $F^{-1}(\mathcal{J})$. Nach Voraussetzung ist damit + exakt, zerfällt gradweise und $\operatorname{ker } p_n$ ist in $F^{-1}(\mathcal{J})$. Nach Voraussetzung ist damit \[ \begin{tikzcd} - 0 \arrow{r} & F(\operatorname{ker } \com{p}_n) \arrow{r} & F(\com{S}_n) \arrow{r}{F(p_n)} & F(\com{S}_{n-1}) \arrow{r} & 0 + 0 \arrow{r} & F(\operatorname{ker } p_n) \arrow{r} & F(\com{S}_n) \arrow{r}{F(p_n)} & F(\com{S}_{n-1}) \arrow{r} & 0 \end{tikzcd} \] exakt und zerfällt gradweise. Aus der Exaktheit folgt damit auch $\operatorname{ker } F(p_n) = F(\operatorname{ker } p_n)$, also $\operatorname{ker } F(p_n) \in \mathcal{J}$. @@ -1714,7 +1793,7 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. Sei $\mathcal{E}$ die Klasse der exakten Komplexe und für $\com{T} \in \mathcal{I}$ sei $\mathcal{H}_{\com{T}}$ die Klasse der Komplexe $\com{A} $, sodass $\com{\operatorname{Hom}}(\com{T}, \com{A})$ exakt ist. Dann - ist $\mathcal{H}_{\com{T}} = \com{\operatorname{Hom}}(\com{T}, -)^{-1}(\mathcal{E})$. + ist $\mathcal{H}_{\com{T}} = \com{\operatorname{Hom}}(\com{T}, -)^{-1}(\mathcal{E})$ und $\mathcal{E}$ mit $\com{\operatorname{Hom}}(\com{T}, -)$ erfüllt die Voraussetzungen von \ref{satz:complete-inv-system-functor}, denn: @@ -1742,14 +1821,21 @@ Für die Klasse der K-projektiven Komplexe betrachten wir die duale Version von \item Falls $n = 1$, dann ist $\com{P}_n = 0$. \item Falls $n > 1$, dann sei $\com{C}_n$ der Kokern der natürlichen Abbildung $\com{P}_{n-1} \to \com{P}_{n}$. Dann ist $\com{C}_n \in \mathcal{P}$ und - die kurze exakte Folge + die kurze Folge \[ - 0 \to \com{P}_{n-1} \to \com{P}_{n} \to \com{C}_n \to 0 - \] zerfällt stufenweise. + \begin{tikzcd} + 0 \arrow{r} & \com{P}_{n-1} \arrow{r} + & \com{P}_{n} \arrow{r} + & \com{C}_n \arrow{r} + & 0 + \end{tikzcd} + %0 \to \com{P}_{n-1} \to \com{P}_{n} \to \com{C}_n \to 0 + \] ist exakt und zerfällt stufenweise. \end{enumerate} - \item Die Klasse $\mathcal{P}$ heißt abgeschlossen unter speziellen direkten Kolimites, falls jedes + \item Die Klasse $\mathcal{P}$ heißt \emph{abgeschlossen unter speziellen direkten Kolimites}, + falls jedes $\mathcal{P}$-spezielle direkte System in $\mathcal{K}$ einen Kolimes in $\mathcal{P}$ besitzt und jeder - Komplex in $\mathcal{K}$, der isomorph ist zu einem Komplex in $\mathcal{P}$, bereits in $\mathcal{P}$ ist. + Komplex in $\mathcal{K}$, der isomorph zu einem Komplex in $\mathcal{P}$ ist, bereits in $\mathcal{P}$ ist. \end{enumerate} \end{definition} @@ -1768,9 +1854,11 @@ und insbesondere die folgenden Ergebnisse: %\end{proof} \begin{satz} - Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(B)$ abgeschlossen - unter speziellen inversen Limites. Angenommen direkte Kolimites existieren in $\mathcal{A}$ und sei - $F\colon \mathcal{K}(A) \to \mathcal{K}(B)$ ein kontravarianter Funktor, der direkte Kolimites in + Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(\mathcal{B})$ abgeschlossen + unter speziellen inversen Limites. Angenommen direkte Kolimites existieren in $\mathcal{A}$ + und sei + $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ + ein kontravarianter Funktor, der direkte Kolimites in inverse Limites überführt und gradweise zerfallende kurze exakte Folgen erhält. Dann ist $F^{-1}(\mathcal{J}) \subset \mathcal{K}(\mathcal{A})$ abgeschlossen unter speziellen direkten Kolimites. @@ -1793,7 +1881,7 @@ und insbesondere die folgenden Ergebnisse: Angenommen inverse (bzw. direkte) Limites existieren in $\mathcal{K}$ und sei $\mathcal{G}$ eine Klasse von Komplexen in $\mathcal{K}$. Dann sei $\leftfinal{\mathcal{G}}$ (bzw. $\rightfinal{\mathcal{G}})$ die kleinste Klasse von Komplexen in $\mathcal{K}$, die abgeschlossen - unter speziellen inversen Limites (bzw. direkten Kolimites) ist und $\mathcal{G}$ enthält. + unter speziellen inversen Limites (bzw. speziellen direkten Kolimites) ist und $\mathcal{G}$ enthält. \end{definition} \subsection{Existenz von K-projektiven und K-injektiven Auflösungen} @@ -1808,6 +1896,29 @@ zu beweisen. Dazu verallgemeinern wir zunächst die Begriffe K-injektive und K-p ein Quasiisomorphismus $\com{X} \to \com{J} $ mit $\com{J} \in \mathcal{J}$. \end{definition} +\begin{bem}[Notation] + Für einen Komplex $\com{X} \in \mathcal{K}$ und $n \in \N$ schreiben wir + $\tau_{\le n} \com{X}$ (bzw. $\tau^{\ge -n} \com{X}$) für den + Komplex + \[ + \begin{tikzcd} + \cdots \arrow{r} & X^{n-2} \arrow{r} & X^{n-1} \arrow{r} & \operatorname{ker } d^n \arrow{r} & 0 \arrow{r} & \cdots + \end{tikzcd} + \] bzw. + \[ + \begin{tikzcd} + \cdots \arrow{r} & 0 \arrow{r} & \operatorname{coker }d^{-n-1} \arrow{r} & X^{-n+1} \arrow{r} & X^{-n+2} \arrow{r} & \cdots + \end{tikzcd} + .\] Für $i \in \Z$ ist dann $H^{i}(\tau_{\le n} \com{X}) = \begin{cases} + H^{i}(\com{X}) & i \le n \\ + 0 & i > n + \end{cases}$ + bzw. $H^{i}(\tau^{\ge -n}\com{X}) = \begin{cases} + 0 & i < -n \\ + H^{i}(\com{X}) & i \ge -n + \end{cases}$. +\end{bem} + \subsubsection{Linksauflösungen} Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. @@ -1839,17 +1950,17 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. $s\colon \com{P} \to \tau_{\le n}\com{A}$. Durch Komposition mit dem natürlichen Komplexhomomorphismus $\tau_{\le n}\com{A} \to \com{A}$ erhalten wir ein $f\colon \com{P} \to \com{A} $. Für $i > n$ ist nun - $H^{i}(\com{P}) = H^{i}(\tau_{\le n}\com{A}) = 0 $. Für $i \le n$ ist + $H^{i}(\com{P}) \simeq H^{i}(\tau_{\le n}\com{A}) = 0 $. Für $i \le n$ ist $H^{i}(\tau_{\le n}\com{A}) = H^{i}(\com{A})$, also induziert $f$ den gewünschten Isomorphismus. (ii)$\implies$(i): Sei $\com{A} \in \mathcal{K}$ nach oben beschränkt. Ohne Einschränkung ist $A^{i} = 0$ für alle $i > 0$. Wähle $n= 0$ in (ii). Dann existiert ein $f\colon \com{P} \to \com{A}$ mit - $\com{P} \in \mathcal{P}$, $H^{i}(\com{P}) = 0$ und $f$ induziert + $\com{P} \in \mathcal{P}$, $H^{i}(\com{P}) = 0$ für $i > 0$ und $f$ induziert %$H^{i}(\com{P}) \xrightarrow[\sim]{0 = H^{i}(f)} 0 = H^{i}(\com{A})$ für $i > 0$ und Isomorphismen $H^{i}(\com{P})\xrightarrow[\sim]{H^{i}(f)} H^{i}(\com{A})$ für $i \le 0$. - Da $H^{i}(\com{P}) = 0 = H^{i}(\com{A})$ für $i < 0$ ist also $f$ ein Quasiisomorphismus. + Da $H^{i}(\com{P}) = 0 = H^{i}(\com{A})$ für $i > 0$, ist also $f$ ein Quasiisomorphismus. \end{proof} \begin{bem}[] @@ -1871,11 +1982,12 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. Sei $\mathcal{P}$ die Klasse der nach oben beschränkten Komplexe $\com{P} \in \mathcal{K}$ mit $P^{i}$ projektiv für alle $i \in \Z$. Falls $\mathcal{A}$ genügend viele Projektive hat, erfüllt - $\mathcal{P}$ die äquivalenten Bedingungen aus \ref{lemma:class-compl-cond}. + $\mathcal{P}$ die äquivalenten Bedingungen aus \ref{lemma:class-compl-cond} (siehe beispielsweise + Lemma 4.6 in \cite{hartshorne}). Ein solches $\com{P}$ ist K-projektiv, denn: Für alle eingradigen Komplexe $\com{Q}$ mit $Q^{i}$ projektiv für alle $i \in \Z$ ist $\com{Q} $ nach \ref{satz:single-degree-compl-k-proj} K-projektiv. Da - nach \ref{kor:k-proj-closed} die Klasse der K-projektiven + nach \ref{kor:k-proj-closed} die Klasse der K-Projektiven abgeschlossen unter speziellen direkten Kolimites ist, folgt mit dem Dual von \ref{lemma:bounded-compl-in-complete-class}, dass $\com{P} $ K-projektiv ist. @@ -2039,7 +2151,7 @@ Im Folgenden nehmen wir an, dass $\mathcal{P}$ die äquivalenten Bedingungen von Sei $p_{n-1}\colon \com{P}_{n-1} = \com{P} \to \com{P}_n$ die natürliche Abbildung. Dann ist $p_{n-1}$ gradweise gegeben durch die natürliche Inklusion $P^{i} \to Q^{i+1} \oplus P^{i}$. Also folgt - $\operatorname{coker } p_{n-1} = \com{Q}[1] \in \mathcal{P}$ und wir haben gradweise + $\operatorname{coker } p_{n-1} = \com{Q}[1] \in \mathcal{P}$ und wir haben zerfallende exakte Folgen: \[ \begin{tikzcd} @@ -2048,7 +2160,10 @@ Im Folgenden nehmen wir an, dass $\mathcal{P}$ die äquivalenten Bedingungen von \end{tikzcd} .\] Also ist $(\com{P}_n)_{n \in \N}$ ein $\mathcal{P}$-spezielles direktes System. - Außerdem ist nach Definition von $h$: $f_n p_{n-1} = h p_{n-1} = f = a_{n-1} f_{n-1}$, + Außerdem ist nach Definition von $h$: + \[ + f_n p_{n-1} = h p_{n-1} = f = a_{n-1} f_{n-1} + ,\] also kommutiert \[ \begin{tikzcd} @@ -2089,7 +2204,7 @@ Daraus folgt nun sofort: \begin{korollar}[] Angenommen direkte Kolimites existieren in $\mathcal{A}$ und - $\colim$ ist exakt. Falls $\mathcal{A}$ außerdem genug Projektive hat, + $\colim$ ist exakt. Falls $\mathcal{A}$ außerdem genügend viele Projektive hat, besitzt jeder Komplex in $\mathcal{K}$ eine K-projektive Auflösung. \label{satz:existence-k-proj-resolution} \end{korollar} @@ -2114,12 +2229,12 @@ $\mathcal{I}$-Rechtsauflösung.} %\end{enumerate} \begin{bsp} - Falls $\mathcal{A}$ genügend Injektive hat, können wir, dual zu Beispiel + Falls $\mathcal{A}$ genügend viele Injektive hat, können wir, dual zu Beispiel \ref{bsp:bounded-above-projectives}, $\mathcal{I}$ als die Klasse der nach unten beschränkten Komplexe mit in $\mathcal{A}$ injektiven Objekten wählen. \end{bsp} -Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von +Durch Umdrehen der Pfeile erhalten wir die dualen Aussagen von \ref{lemma:constr-dir-system} und \ref{satz:existence-left-resolutions}: \begin{lemma}[] @@ -2236,7 +2351,7 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \[ H^{m}(\com{I}) \longrightarrow H^{m}(\com{I}_N) \] ein Isomorphismus. Erneute Betrachtung des linken Vierecks von \eqref{eq:diag-hi-in} für $i=m$ und - $n=N$ liefert nun, dass $H^{i}(f)$ ein Isomorphismus ist. + $n=N$ liefert nun, dass $H^{m}(f)$ ein Isomorphismus ist. \end{proof} \begin{bem} @@ -2249,7 +2364,9 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \label{sec:application} -Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (links-)Moduln. +Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (Links-)Moduln und +weiterhin $\mathcal{K} = \mathcal{K}(\mathcal{A})$. Die +Argumentation folgt im Wesentlichen den Anfängen der Abschnitte 5 und 6 von \cite{spaltenstein}. \subsection{Abgeleitete $\com{\operatorname{Hom}}$ Funktoren} @@ -2283,7 +2400,7 @@ Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (links-)Mo %\end{satz} \begin{satz} - Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\text{R}\com{\operatorname{Hom}}(\com{M}, \com{N})$ wohldefiniert + Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\operatorname{R}\com{\operatorname{Hom}}(\com{M}, \com{N})$ wohldefiniert und kann mithilfe einer K-projektiven Auflösung von $\com{M} $ oder einer K-injektiven Auflösung von $\com{N} $ berechnet werden. \label{satz:derived-hom} @@ -2314,6 +2431,31 @@ Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (links-)Mo .\end{align*} \end{proof} +\noindent Analog zu \ref{hom-compl-cohomgroups} erhalten wir + +\begin{lemma}[] + Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist + \[ + H^{0}(\operatorname{R}\com{\operatorname{Hom}}(\com{M}, \com{N})) \simeq + \operatorname{Mor}_{\mathcal{D}}(\com{M}, \com{N}) + .\] + \label{rhom-compl-cohomgroups} +\end{lemma} + +\begin{proof} + Nach \ref{satz:existence-k-inj-resolution} existiert ein Quasiisomorphismus + $f\colon \com{N} \to \com{I}$, wobei $\com{I}$ K-injektiv ist. Also induziert + $f$ einen Isomorphismus in $\mathcal{D}$ und es folgt + \begin{salign*} + H^{0}(\operatorname{R}\com{\operatorname{Hom}}(\com{M}, \com{N})) + &\simeq H^{0}(\operatorname{R}\com{\operatorname{Hom}}(\com{M}, \com{I})) \\ + &\simeq H^{0} (\com{\operatorname{Hom}}(\com{M}, \com{I})) \\ + &\stackrel{\ref{hom-compl-cohomgroups}}{=} \operatorname{Mor}_{\mathcal{K}}(\com{M}, \com{I}) \\ + &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} \operatorname{Mor}_{\mathcal{D}}(\com{M}, \com{I})\\ + &\simeq \operatorname{Mor}_{\mathcal{D}}(\com{M}, \com{N}) + .\end{salign*} +\end{proof} + \subsection{Abgeleitetes Tensorprodukt} Sei von nun an $R = A$ ein kommutativer Ring. Um das Tensorprodukt abzuleiten, benötigen wir noch @@ -2337,13 +2479,13 @@ eine weitere Klasse von Komplexen in $\mathcal{K}$: $n \in \Z$: \[ (\com{M} \otimes_A \com{S})^{n} = \bigoplus_{i+j=n} M^{i} \otimes_A S^{j} - = M^{0} \otimes S^{n} = (M^{0} \otimes \com{S} )^{n} + = M^{0} \otimes_A S^{n} = (M^{0} \otimes_A \com{S} )^{n} \] und für $m \in M^{0}$, $s \in S^{n}$: \[ - d_{\com{M} \otimes_A \com{S} }^{n}(m \otimes s) - = \underbrace{d_{M}^{0}(m)}_{= 0} \otimes_A s + (-1)^{0} m \otimes_A d_{S}(s) - = m \otimes_A d_S(s) - = d_{M^{0} \otimes_A \com{S} } + d_{\com{M} \otimes_A \com{S}}^{n}(m \otimes s) + = \underbrace{d_{M}^{0}(m)}_{= 0} \otimes s + (-1)^{0} m \otimes d^{n}_{S}(s) + = m \otimes d^{n}_S(s) + = d^{n}_{M^{0} \otimes_A \com{S} } .\] Also ist $\com{M} \otimes_A \com{S} = M^{0} \otimes_A \com{S}$. Damit folgt die Behauptung aus den Definitionen. \end{proof} @@ -2391,7 +2533,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe (ii)$\implies$(i): Sei $\com{S} $ exakt. Wegen \ref{lemma:0.10} genügt es zu zeigen, dass für jeden K-injektiven Komplex $\com{I} \in \mathcal{K}$, - $\com{\operatorname{Hom}}(\com{S} \otimes_A \com{M}, I) $ exakt ist. Dazu + $\com{\operatorname{Hom}}(\com{S} \otimes_A \com{M}, \com{I}) $ exakt ist. Dazu sei $\com{I}$ ein K-injektiver Komplex. Dann ist \[ \com{\operatorname{Hom}}(\com{S} \otimes_A \com{M}, \com{I}) @@ -2403,7 +2545,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \begin{enumerate}[(a)] \item Falls $\com{M}, \com{N} \in \mathcal{K}$ K-flach sind, dann ist auch $\com{M} \otimes_A \com{N} $ K-flach. - \item $\com{M} \in \mathcal{K}$ ist K-flach genau dann wenn $\com{M}[1]$ + \item $\com{M} \in \mathcal{K}$ ist genau dann K-flach, wenn $\com{M}[1]$ K-flach ist. \item Falls zwei Punkte eines ausgezeichneten Dreiecks in $\mathcal{K}$ K-flach sind, @@ -2419,7 +2561,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \item Seien $\com{M}, \com{N} \in \mathcal{K}$ K-flach und $\com{S} $ exakt. Dann ist \[ - (\com{M} \otimes_A \com{N}) \otimes \com{S} = + (\com{M} \otimes_A \com{N}) \otimes_A \com{S} = \com{M} \otimes_A (\com{N} \otimes_A \com{S}) \] und die rechte Seite ist exakt. \item Seien $\com{S}, \com{M} \in \mathcal{K}$. @@ -2432,7 +2574,8 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe .\] Da Verschieben Exaktheit erhält, folgt daraus die Äquivalenz. \item Sei $(\com{M}, \com{N}, \com{P}, u, v, w)$ ein ausgezeichnetes Dreieck in $\mathcal{K}$ mit $\com{M}$ und $\com{N} $ K-flach. Sei weiter - $\com{S} $ exakt. Da $- \otimes_A \com{S} $ nach \ref{satz:tor-is-triangulated} + $\com{S} \in \mathcal{K}$ exakt. + Da $- \otimes_A \com{S} $ nach \ref{satz:tor-is-triangulated} ein triangulierter Funktor ist, erhalten wir das ausgezeichnete Dreieck $(\com{M} \otimes_A \com{S}, \com{N} \otimes_A \com{S}, \com{P} \otimes_A \com{S}, u \otimes \text{id}_{\com{S} }, v \otimes \text{id}_{\com{S} }, w \otimes \text{id}_{\com{S} })$ und damit für $i \in \Z$ die exakte Folge @@ -2445,7 +2588,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe .\] Da die äußeren Terme nach Voraussetzung $0$ sind, folgt die Exaktheit von $\com{P} \otimes_A \com{S} $ und damit, dass $\com{P} $ K-flach ist. - Der allgemeine Fall folgt nun mit \ref{TR2}. + Der allgemeine Fall folgt nun mit \hyperref[TR2]{TR2}. \end{enumerate} \end{proof} @@ -2457,11 +2600,13 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \end{satz} \begin{proof} - Sei $\com{M} $ K-projektiv und $\com{S} $ exakt. Sei weiter $\com{I} $ K-injektiv. Dann folgt + Sei $\com{M} $ K-projektiv und $\com{S} $ exakt. Sei weiter $\com{I} \in \mathcal{K}$ + K-injektiv. Dann folgt \[ \com{\operatorname{Hom}} (\com{M} \otimes_A \com{S}, \com{I}) \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{S}, \com{I} )) - .\] Es ist $\com{\operatorname{Hom}}(\com{S}, \com{I})$ exakt, da $\com{I} $ K-injektiv und damit + .\] Es ist $\com{\operatorname{Hom}}(\com{S}, \com{I})$ exakt, da $\com{I} $ K-injektiv ist. + Damit ist die rechte Seite auch exakt, da $\com{M} $ K-projektiv ist. Also folgt die Behauptung mit \ref{lemma:0.10}. Das Insbesondere folgt nun aus \ref{satz:existence-k-proj-resolution}. @@ -2479,7 +2624,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe $\com{P} \to \com{N} $. Da $\com{M} $ K-flach ist, erhält $\com{M} \otimes_A -$ nach \ref{kor:exactness-preserver-preserves-qis} Quasiisomorphismen. Damit folgt \begin{equation} - H^{i}(\com{M} \otimes_A \com{N}) = H^{i}(\com{M} \otimes_A \com{P}) + H^{i}(\com{M} \otimes_A \com{N}) \simeq H^{i}(\com{M} \otimes_A \com{P}) \label{eq:cohom-groups-1} .\end{equation} Da $\com{P} $ K-flach ist, folgt mit der Exaktheit von $\com{M} $, dass @@ -2489,7 +2634,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe Damit erhalten wir: \begin{satz} - Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\com{M} \otimes_A^{\text{L}} \com{N}$ wohldefiniert und + Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\com{M} \otimes_A^{\operatorname{L}} \com{N}$ wohldefiniert und kann mithilfe einer K-flachen Auflösung einer der Faktoren berechnet werden. \label{satz:derived-tor} \end{satz} @@ -2506,7 +2651,7 @@ Damit erhalten wir: \item Nach \ref{satz:tor-exact-for-k-flat} erhält $\com{N} \otimes_A -|_{\mathcal{L}}$ Exaktheit von Komplexen. \end{enumerate} - Also existiert $\com{N} \otimes_A^{\text{L}} -$. Analog zeigt man die Existenz von + Also existiert $\com{N} \otimes_A^{\operatorname{L}} -$. Analog zeigt man die Existenz von $- \otimes_A^{L} \com{N}$ und wie im Beweis von \ref{satz:derived-hom}, dass beide Ableitungen übereinstimmen. \end{proof} @@ -2519,8 +2664,8 @@ Jetzt können wir alles zusammentragen und erhalten: Seien $\com{M}, \com{N}, \com{P} \in \mathcal{D}$. Dann existiert ein natürlicher Isomorphismus, der funktoriell in allen Variablen ist: \[ - \text{R}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) - = \text{R}\com{\operatorname{Hom}}(\com{M}, \text{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) + \operatorname{R}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\operatorname{L}} \com{N}, \com{P}) + = \operatorname{R}\com{\operatorname{Hom}}(\com{M}, \operatorname{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) .\] \label{satz:adjunction-rhom-rtor} \end{satz} @@ -2533,50 +2678,55 @@ Jetzt können wir alles zusammentragen und erhalten: Dann folgt \begin{align*} - \text{R}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) - &= \text{R}\com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ + \operatorname{R}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\operatorname{L}} \com{N}, \com{P}) + &= \operatorname{R}\com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ &= \com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P} ) \\ &\stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ - &= \text{R}\com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ - &= \text{R}\com{\operatorname{Hom}}(\com{M}, \text{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) + &= \operatorname{R}\com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + &= \operatorname{R}\com{\operatorname{Hom}}(\com{M}, \operatorname{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) .\end{align*} Das vorletzte Gleichheitszeichen gilt, weil nach \ref{satz:homs-of-k-flat-are-k-inj} $\com{\operatorname{Hom}}(\com{N}, \com{P})$ K-injektiv ist. \end{proof} -\begin{korollar}[] +\begin{korollar} Seien $\com{M}, \com{N}, \com{P} \in \mathcal{D}$. Dann existiert ein natürlicher Isomorphismus, der funktoriell in allen Variablen ist: \[ - \operatorname{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) - = \operatorname{Mor}_{\mathcal{D}}(\com{M}, \text{R}\com{\operatorname{Hom}}(\com{N} , \com{P} )) + \operatorname{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\operatorname{L}} \com{N}, \com{P}) + = \operatorname{Mor}_{\mathcal{D}}(\com{M}, \operatorname{R}\com{\operatorname{Hom}}(\com{N} , \com{P} )) .\] Insbesondere gilt folgende Funktoradjunktion in $\mathcal{D}$: \[ - - \otimes_A^{\text{L}} \com{N} \dashv \text{R}\com{\operatorname{Hom}}(\com{N}, -) + - \otimes_A^{\operatorname{L}} \com{N} \dashv \operatorname{R}\com{\operatorname{Hom}}(\com{N}, -) .\] \end{korollar} \begin{proof} - Wir können wieder annehmen, dass $\com{N}$ K-flach und $\com{P} $ K-injektiv ist. - Dann betrachte: - \begin{salign*} - \operatorname{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) - &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} - \operatorname{Mor}_{\mathcal{K}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ - &\stackrel{\ref{hom-compl-cohomgroups}}{=} - H^{0}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ - &= H^{0}\com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ - &\stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} - H^{0}\com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ - &\stackrel{\ref{hom-compl-cohomgroups}}{=} - \operatorname{Mor}_{\mathcal{K}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ - &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} - \operatorname{Mor}_{\mathcal{D}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ - &= \operatorname{Mor}_{\mathcal{D}}(\com{M}, \text{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) - .\end{salign*} - Das vorletzte Gleichheitszeichen gilt erneut, weil nach \ref{satz:homs-of-k-flat-are-k-inj} - $\com{\operatorname{Hom}}(\com{N}, \com{P})$ K-injektiv ist. + %$H^{0}$ angewendet auf den Isomorphismus aus \ref{satz:adjunction-rhom-rtor} zeigt + %mit \ref{rhom-compl-cohomgroups} die Behauptung. + Man wende \ref{rhom-compl-cohomgroups} auf den Isomorphismus + aus \ref{satz:adjunction-rhom-rtor} an. + + %Wir können wieder annehmen, dass $\com{N}$ K-flach und $\com{P} $ K-injektiv ist. + %Dann betrachte: + %\begin{salign*} + % \operatorname{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) + % &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} + % \operatorname{Mor}_{\mathcal{K}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ + % &\stackrel{\ref{hom-compl-cohomgroups}}{=} + % H^{0}\com{\operatorname{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ + % &= H^{0}\com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ + % &\stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} + % H^{0}\com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + % &\stackrel{\ref{hom-compl-cohomgroups}}{=} + % \operatorname{Mor}_{\mathcal{K}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + % &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} + % \operatorname{Mor}_{\mathcal{D}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + % &= \operatorname{Mor}_{\mathcal{D}}(\com{M}, \text{R}\com{\operatorname{Hom}}(\com{N}, \com{P})) + %.\end{salign*} + %Das vorletzte Gleichheitszeichen gilt erneut, weil nach \ref{satz:homs-of-k-flat-are-k-inj} + %$\com{\operatorname{Hom}}(\com{N}, \com{P})$ K-injektiv ist. \end{proof} \bibliographystyle{alpha} diff --git a/sose2022/bachelorarbeit/vortrag.pdf b/sose2022/bachelorarbeit/vortrag.pdf index b667298b8f7626c51f999cd1489b362d9fc196d8..9b08bc7d0625f01e8cfb1522ff721a64a2031564 100644 GIT binary patch literal 168325 zcma%iQ*>lq)NPE4ZQFJ_w%M_5TNS5c+qOHlt&W|Jt&Vm3>w6#W82`&(Pp3wWvCmw4 z%{kXObL^r}5|dzHVdMZ%%qkNRr&f-@s^aQspBifW+ 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\end{enumerate} +\end{definition} + +Wir zeigen nun zunächst, dass die Klasse $\mathcal{I}$ der K-Injektiven abgeschlossen unter +speziellen inversen Systemen ist, das heißt, dass der inverse Limes +$\mathcal{I}$-spezieller inverser Systeme wieder in $\mathcal{I}$ liegt. + +Dazu beobachtet man + +\begin{lemma} + Die Klasse der exakten Komplexe ist abgeschlossen unter speziellen inversen Limites. +\end{lemma} + +\begin{proof} + Diagrammjagd. +\end{proof} + +\begin{lemma} + Sei $\mathcal{B}$ eine weitere abelsche Kategorie, + $\mathcal{J} \subset \mathcal{K}(\mathcal{B})$ abgeschlossen unter speziellen inversen Limites. + Angenommen inverse Limites existieren in $\mathcal{A}$ und sei + $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein kovarianter + Funktor der mit inversen Limites vertauscht und gradweise zerfallende kurze exakte Folgen erhält. + + Dann ist $F^{-1}(\mathcal{J}) \subset \mathcal{K}(\mathcal{A})$ abgeschlossen unter speziellen + inversen Limites. +\end{lemma} + +\begin{proof} + Das liegt daran, dass $F$ $F^{-1}(\mathcal{J})$-spezielle inverse Systeme in + $\mathcal{J}$-spezielle inverse Systeme überführt. +\end{proof} + +\begin{korollar}[] +Die Klasse $\mathcal{I}$ der K-injektiven Komplexe +ist abgeschlossen unter speziellen inversen Limites. +\end{korollar} + +\begin{proof} + Sei $\mathcal{E}$ die Klasse der exakten Komplexe in $\mathcal{A}$, + dann ist für jeden exakten Komplex $\com{T} $ in $\mathcal{A}$ die Klasse + $\com{\operatorname{Hom}}(\com{T}, -)^{-1}(\mathcal{E})$ abgeschlossen + unter speziellen inversen Limites. Mit + \[ + \mathcal{I} = \bigcap_{\com{T} \in \mathcal{E}} + \com{\operatorname{Hom}} (\com{T}, -)^{-1}(\mathcal{E}) + \] folgt jetzt die Behauptung. +\end{proof} + +Sei nun $\com{M} \in \mathcal{K}$ ein beliebiger Komplex. +Wir wollen ein $\mathcal{I}$-spezielles inverses System konstruieren, dessen Limes eine Auflösung +von $\com{M} $ liefert. + +\begin{definition}[Abschneiden] + Für $n \in \N$ betrachten wir den Komplex $\tau^{\ge -n} \com{M} $ + \[ + \begin{tikzcd} + 0 \arrow{r} & \arrow{r} \operatorname{coker } d_{M}^{-n-1} + & \arrow{r} M^{-n+1} + & \arrow{r} M^{-n+2} + & \cdots + \end{tikzcd} + .\] +\end{definition} + +\begin{bem}[] + Dann ist für $i \ge -n$ + \[ + H^{i}(\tau^{\ge -n}\com{M}) = H^{i}(\com{M}) + .\] +\end{bem} + +\begin{satz}[] + Es existiert ein $\mathcal{I}$-spezielles inverses System $(\com{I}_n)_{n \in \N}$ + mit einem inversen System von Quasiisomorphismen $f_n\colon \tau^{\ge -n} \com{M} \to \com{I}_n$. +\end{satz} + +\begin{proof} + Wir gehen induktiv vor: Setze $I_1 = 0$ und $f_1 = 0$. Sei $\com{M}_n = \tau^{\ge -n} \com{M}$. + Dann ist $\com{M}_2$ nach unten beschränkt, also existiert ein K-injektiver Komplex + $\com{I}_2$ und ein Quasiisomorphismus $f_2 \colon \com{M}_2 \to \com{I}_2$. + + Sei nun $n \ge 3$, $\com{I}_{n-1}$ und $f_{n-1}$ bereits konstruiert. + Aus $\com{I}_{n-1}$ und $\com{M}_n$ konstruiert man einen neuen nach unten beschränkten + Komplex, der wieder durch einen K-injektiven aufgelöst werden kann. Durch geeignete Modifikation + erhält man dann $\com{I}_n$ und $f_n$. +\end{proof} + +Wir nehmen nun an, dass in $\mathcal{A}$ inverse Limites existieren. + +\begin{bem}[] + \begin{enumerate}[(a)] + \item Sei $\com{I} $ der Limes des Systems + $(\com{I}_n)_{n \in \N}$ und $f\colon \com{M} \to \com{I}$ der Limes des Systems + $(f_n)_{n \in \N}$. + \item Dann sind zwar die $f_n$ Quasiisomorphismen, aber + $f$ a priori kein Quasiisomorphismus, da der inverse Limes im Allgemeinen, + insbesondere für $\mathcal{A} = A\text{-Mod}$, nicht exakt ist. + \item Abhilfe: Mittag Leffler + Diagrammjagd. Also + $f$ für $\mathcal{A} = A\text{-Mod}$ ein Quasiisomorphismus. +\end{enumerate} +\end{bem} + +\begin{korollar}[] +Existenz von $R\com{\operatorname{Hom}}(\com{N}, -)$. +\end{korollar} + +\begin{bem}[Umdrehen der Pfeile] + Dual dazu definieren wir K-projektive Komplexe und Auflösungen und zeigen, dass jeder + Komplex in $A$-Mod ebenfalls eine K-projektive Auflösung hat. +\end{bem} + +\begin{korollar}[] +Existenz von $R\com{\operatorname{Hom}}(-, \com{N})$. +\end{korollar} + +\begin{definition}[K-flach] + Analog zu K-injektiv für $- \otimes_A \com{N}$. +\end{definition} + +\begin{satz}[] + K-projektiv $\implies$ K-flach +\end{satz} + +\begin{korollar}[] + Es existieren K-flache Auflösungen. +\end{korollar} + +\begin{korollar} + Abgeleitetes Tensorprodukt existiert. +\end{korollar} + +Die K-flachen Komplexe hängen noch auf wichtige Weise mit den K-injektiven zusammen: +\begin{satz}[] + Sei $\com{M} \in \mathcal{K}$. Dann ist $\com{M} $ genau dann K-flach, wenn + $\com{\operatorname{Hom}}(\com{M}, \com{I})$ K-injektiv ist für jeden K-Injektiven + $\com{I} \in \mathcal{K}$. +\end{satz} + + +\begin{korollar} + Seien $\com{M}, \com{N}, \com{P} \in \mathcal{D}(\mathcal{A})$. Dann existiert + ein natürlicher Isomorphismus + \[ + R\com{\operatorname{Hom}}(\com{M} \otimes_A^{L} \com{N}, \com{P}) + = R\com{\operatorname{Hom}}(\com{M}, R\com{\operatorname{Hom}}(\com{N}, \com{P})) + .\] +\end{korollar} + +\begin{proof} + Da in $\mathcal{D}(\mathcal{A})$ Quasiisomorphismen Isomorphismen induzieren und + wir die Existenz der verschiedenen Auflösungen kennen, können wir ohne Einschränkung annehmen, + dass $\com{P} $ K-injektiv und $\com{N} $ K-flach ist. Dann ist + \begin{align*} + R\com{\operatorname{Hom}}(\com{M} \otimes_A^{L} \com{N}, \com{P}) + &= \com{\operatorname{Hom}}(\com{M} \otimes_A^{L} \com{N}, \com{P}) \\ + &= \com{\operatorname{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ + &= \com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + &= \com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + &= R\com{\operatorname{Hom}}(\com{M}, \com{\operatorname{Hom}}(\com{N}, \com{P})) \\ + &= R\com{\operatorname{Hom}}(\com{M}, R\com{\operatorname{Hom}}(\com{N}, \com{P})) + .\end{align*} +\end{proof} + +Die eigentliche Adjunktion bekommen wir durch Anwenden von $H^{0}$ \end{document} diff --git a/sose2022/metaethik/2022-04-22-Note-09-52.xopp b/sose2022/metaethik/2022-04-22-Note-09-52.xopp index 6fd506a4035feefcf14e56007a4f2b5663086ba1..a689145eade3cb6fbd3311dbf0e6477712ec2926 100644 GIT binary patch delta 253324 zcmV(u9o%YQ%rnl_9^OZsp8r*YeT 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