From 81453e5c4e12860bb87adf3874ae3243925ce610 Mon Sep 17 00:00:00 2001 From: flavis Date: Sun, 3 Apr 2022 18:21:36 +0200 Subject: [PATCH] first revision --- sose2022/bachelorarbeit/arbeit.cls | 6 +- sose2022/bachelorarbeit/arbeit.pdf | Bin 387716 -> 395092 bytes sose2022/bachelorarbeit/arbeit.tex | 645 +++++++++++++++++------------ 3 files changed, 389 insertions(+), 262 deletions(-) diff --git a/sose2022/bachelorarbeit/arbeit.cls b/sose2022/bachelorarbeit/arbeit.cls index 6fa8437..fcfcd0f 100644 --- a/sose2022/bachelorarbeit/arbeit.cls +++ b/sose2022/bachelorarbeit/arbeit.cls @@ -45,8 +45,8 @@ } % PARAGRAPH no indent but skip -\setlength{\parskip}{3mm} -\setlength{\parindent}{0mm} +%\setlength{\parskip}{3mm} +%\setlength{\parindent}{0mm} \newtheorem{satz}{Satz}[section] \newtheorem{lemma}[satz]{Lemma} @@ -78,7 +78,7 @@ % HEADERS -\pagestyle{fancy} +\pagestyle{headings} \newcommand{\incfig}[1]{% \def\svgwidth{\columnwidth} diff --git a/sose2022/bachelorarbeit/arbeit.pdf b/sose2022/bachelorarbeit/arbeit.pdf index e6e67efb0bfe6b43c863b0789b7ab951c7f32678..87cfcc1f25828eab55675b5a27e75b539263f828 100644 GIT binary patch delta 236888 zcmZ5nV{{l@*NttfQDa+;Z8x?W+nU%-8rwD-+qP}9vETOj{(b*u)|$CEcb|Rs*|R2S z0ee0g2ZdZwRGglffgO%~dVXLTjx%{@5*#D>`T>j<2##r!?-xW6d-V<;D+ptKXdi>D zl_Vw$D*B4iJ!$qe$&g)&j6WNJn%ewrwx!b*JBtuDq!f+G&uq!+rR(g>kNW zYK+@gYUq%B-NlRA52j;LXFNk(giwoort-&|r*8Hiu8BP>E%l1i(3GQ;2pq{A_USs2 zqje~m^^y(pm?P!YG)SkOPhKzdG9o@9(U#=e-(cv;4&dPE5Nxd6Y{@$-;M4%YO57U9 zAD>QPk2zIMCV4Q~tcq)<_vu(ZbL=`BC$6XRl{xHq*dG_tF*krpeDrfwWM^zM4 zR;34`HkU4^Qmwn@$~OehSE)y5Nl@lG?fG(Dw}1WA+O`66&2Y*EDFon{+QKQE7D20g zWzs|GGa;VlLuRJvq$pGd-7c5Kj+3kc?_NqRWs;Vr2NS*DbsU^%&};0{?Y*MymS8!l zc}&^s7$9I=Z5h|@t{On zn&Q`>fwgQ149=VuL6$j?7cpLpUA{beMtAHT2>1_53yrxX=jv*r^|%Y}cj1n9hlM

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$\mathcal{A}, \mathcal{B}$ abelsche Kategorien und sei $F\colon \mathcal{A} \to \mathcal{B}$ ein 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 -mit den klassischen Rechtsableitungen von $F$ bei $X$ übereinstimmen, falls -$F$ linksexakt ist. +$X \in \mathcal{A}$ einen Komplex $\text{R}F(X)$ zu definieren, dessen Kohomologiegruppen, +falls $F$ linksexakt ist, +mit den klassischen Rechtsableitungen von $F$ bei $X$ übereinstimmen. Allgemeiner definieren wir $\text{R}F(\com{X})$ für jeden Komplex $\com{X}$ in $\mathcal{A}$. Genauer gesagt konstruieren wir einen @@ -39,7 +39,7 @@ $\mathcal{K}(\mathcal{A})$, der Komplexkategorie von $\mathcal{A}$, wobei die Mo 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 -Kohomologiegruppen induzieren, in $\mathcal{D}(\mathcal{A})$ zu Isomorphismen erklären. +Kohomologiegruppen induzieren, in $\mathcal{D}(\mathcal{A})$ zu Isomorphismen erklärt werden. Dieser Lokalisierungsprozess wird im Folgenden skizziert. Die Vorgehensweise orientiert sich an Kapitel 1 von \cite{hartshorne}. @@ -76,7 +76,7 @@ anderen Struktur, die einer triangulierten Kategorie. 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 $h\colon Z \to Z'$, so dass $(f, g, h)$ ein Morphismus + ein nicht notwendig eindeutiger Morphismus $h\colon Z \to Z'$, sodass $(f, g, h)$ ein Morphismus von ausgezeichneten Dreiecken ist. \end{enumerate} \label{TR2} @@ -232,8 +232,8 @@ Daraus erhalten wir insbesondere folgendes Kriterium, ob ein Komplexhomomorphism \end{lemma} \begin{proof} - Es ist $(\com{A}, \com{B}, \com{C}_f, f, i, p)$ ein ausgezeichnetes Dreieck in $\mathcal{K}$ mit - $i\colon \com{B} \to \com{C}_f$ und $p\colon \com{C}_f \to \com{A}[1]$ die natürlichen + Es ist $(\com{A}, \com{B}, \com{C}_f, f, \iota, \rho)$ ein ausgezeichnetes Dreieck in $\mathcal{K}$ mit + $\iota\colon \com{B} \to \com{C}_f$ und $\rho\colon \com{C}_f \to \com{A}[1]$ die natürlichen Morphismen. Also erhalten wir mit \ref{lemma:cohom-is-cohom-functor} für $i \in \Z$ eine exakte Folge \[ @@ -248,6 +248,48 @@ Daraus erhalten wir insbesondere folgendes Kriterium, ob ein Komplexhomomorphism .\] Die Exaktheit liefert nun die Äquivalenz. \end{proof} +\begin{bem} + Falls in der Situation von \ref{mapping-cone-exact-for-qis}, + $H^{i}(f)$ ein Isomorphismus ist für $i \gg 0$ (bzw. $i \ll 0$), folgt aus der exakten Folge im Beweis, + dass $H^{i}(\com{C}_f) = 0$ für $i \gg 0$ (bzw. $i \ll 0$). + \label{bem:mapping-cone-h-bounded} +\end{bem} + +\begin{korollar} + Sei $\mathcal{B}$ eine weitere abelsche Kategorie und + $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ + ein triangulierter Funktor. + Dann erhält $F$ genau dann Exaktheit von Komplexen, wenn $F$ Quasiisomorphismen erhält. + \label{kor:exactness-preserver-preserves-qis} +\end{korollar} + +\begin{proof} + ($\Rightarrow$) Seien $\com{X}, \com{Y} \in \mathcal{K}(\mathcal{A})$ + und $f\colon \com{X} \to \com{Y}$ 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 in $\mathcal{K}(\mathcal{A})$, wobei $\com{C}_f$ + nach \ref{mapping-cone-exact-for-qis} exakt ist. Nach Voraussetzung ist + \[ + \begin{tikzcd} + F(\com{X}) \arrow{r}{F(f)} & F(\com{Y}) \arrow{r} & F(\com{C}_f) \arrow{r} & \com{X}[1] + \end{tikzcd} + \] ebenfalls ein ausgezeichnetes Dreieck und wir erhalten für $i \in \Z$ die exakte Folge: + \[ + \begin{tikzcd} + H^{i-1}(F(\com{C}_f)) \arrow{r} & H^{i}(F(\com{X})) \arrow{r} & H^{i}(F(\com{Y})) \arrow{r} & + H^{i}(F(\com{C}_f)) + \end{tikzcd} + .\] Die äußeren Terme sind $0$, weil $F$ Exaktheit erhält. Aus der Exaktheit der Folge, folgt damit + der gewünschte Isomorphismus. + + ($\Leftarrow$) Sei $\com{X} \in \mathcal{K}(\mathcal{A})$ exakt. Dann ist $\com{X} \to \com{0}$ + ein Quasiisomorphismus, also auch $F(\com{X}) \to F(\com{0} )$. Da $F$ additiv ist, folgt $F(\com{0}) = \com{0}$ und damit + die Behauptung. +\end{proof} + \subsection{Lokalisierung von Kategorien} Wie anfangs erwähnt, ist die derivierte Kategorie $\mathcal{D}(\mathcal{A})$ einer @@ -348,9 +390,12 @@ uns zu folgendem Begriff führt: Allgemeinen auch $\text{Mor}_{\mathcal{C}_{\mathcal{S}}}(X, Y)$ für $X, Y \in \mathcal{C}_{\mathcal{S}}$ keine Menge. Das - heißt streng genommen ist $\mathcal{C}_{\mathcal{S}}$ nur - eine große Kategorie. Auf diese mengentheoretischen Probleme gehen wir - im Folgenden jedoch nicht ein. + heißt streng genommen ist $\mathcal{C}_{\mathcal{S}}$ im Allgemeinen nur + eine große Kategorie. In unseren Anwendungsfällen kann man jedoch zeigen, dass $\mathcal{C}_{\mathcal{S}}$ + eine (echte) Kategorie ist. Für Details + siehe Bemerkung 10.3.6 in \cite{set-theoretic}. + 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, konstruiert werden. Wenn im Folgenden der Kontext klar ist, dann schreiben wir auch einfach @@ -373,6 +418,7 @@ an $\mathcal{S}$: von Morphismen. Wir nennen $\mathcal{S}$ kompatibel mit der Triangulation, wenn die folgenden Axiome erfüllt sind: \begin{enumerate}[(FR1), leftmargin=14mm] + \setcounter{enumi}{3} \item $s \in \mathcal{S} \iff T(s) \in \mathcal{S}$. \item \hyperref[TR2]{TR3} unter der zusätzlichen Annahme, dass $f, g \in \mathcal{S}$ und der zusätzlichen Forderung, dass $h \in \mathcal{S}$ ist. @@ -486,8 +532,13 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: \[ \eta\colon \text{R}F \to G \] von Funktoren von $\mathcal{D}(\mathcal{A})$ nach $\mathcal{D}(\mathcal{B})$, sodass + das folgende Diagramm kommutiert: \[ - \zeta = (\eta \circ Q_{\mathcal{A}}) \circ \xi + \begin{tikzcd} + Q_{\mathcal{B}} \circ F \arrow{r}{\xi} \arrow{dr}{\zeta} + & \text{R}F \circ Q_{\mathcal{A}} \arrow[dashed]{d}{\eta} \\ + & G \circ Q_{\mathcal{A}} + \end{tikzcd} .\] \end{definition} @@ -498,8 +549,7 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: links-exakten Funktor $F\colon \mathcal{A} \to \mathcal{B}$ und $\mathcal{A}$ genügend Injektive hat, sind das genau die klassischen rechts-abgeleiteten Funktoren von $F$. \item Es gibt den analogen Begriff der Linksableitung $\text{L}F$ (kovarianter) - Funktoren, bei dem sich die Pfeile umdrehen. - % TODO: präzisieren!!! + Funktoren, bei dem sich die Pfeile der natürlichen Transformationen umdrehen. \end{enumerate} \label{bem:derived-functors} \end{bem} @@ -507,17 +557,17 @@ $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$: \begin{satz} Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien und $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein triangulierter Funktor. Angenommen es - existiert eine triangulierte Unterkategorie $\mathcal{L} \subset \mathcal{K}(\mathcal{A})$, s.d. + 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. \end{enumerate} Dann existiert der Funktor $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$ und - eine natürliche Transformation $\xi\colon Q_{\mathcal{B}} \circ F \to \text{R}F \circ Q_{\mathcal{A}}$, sodass + 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 \[ - \xi(\com{I})\colon Q_{\mathcal{B}}(F(\com{I}) \to \text{R}F(Q_{\mathcal{A}}(\com{I})) + \xi(\com{I})\colon Q_{\mathcal{B}}(F(\com{I})) \to \text{R}F(Q_{\mathcal{A}}(\com{I})) \] ein Isomorphismus in $\mathcal{D}(A)$ ist. \label{satz:existence-derived-functors} \end{satz} @@ -545,18 +595,20 @@ wird uns erlauben das klassische Adjunktionsresultat \] für einen $A$-Modul $M$ auf die abgeleiteten Funktoren zu übertragen. \begin{definition} - Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe in einer beliebigen abelschen Kategorie $\mathcal{A}$. - Dann sei $\com{\text{Hom}}(\com{A}, \com{B}) \in \mathcal{K}(\mathcal{A}b)$ definiert durch + Seien $\com{X}, \com{Y} \in \mathcal{K}$ zwei Komplexe in einer beliebigen abelschen Kategorie $\mathcal{A}$. + Dann sei $\com{\text{Hom}}(\com{X}, \com{Y}) \in \mathcal{K}(\mathcal{A}b)$ definiert durch \[ - \text{Hom}^{n}(\com{A}, \com{B}) = \prod_{i \in \Z} \text{Hom}(A^{i}, B^{i+n}) + \text{Hom}^{n}(\com{X}, \com{Y}) = \prod_{i \in \Z} \text{Hom}(X^{i}, Y^{i+n}) \] mit Differentialen \[ - d^{n}(f) = d_{\com{B} } f - (-1)^{n} f d_{\com{A}} - \] für $f \in \text{Hom}^{n}(\com{A}, \com{B})$. + d^{n}(f) = d_{\com{Y} } f - (-1)^{n} f d_{\com{X}} + \] für $f \in \text{Hom}^{n}(\com{X}, \com{Y})$. + \label{def:hom-compl} \end{definition} \begin{definition} - Seien $\com{M}, \com{N} \in \mathcal{K}(A\text{-Mod})$. Dann sei + Sei $A$ ein kommutativer Ring und + seien $\com{M}, \com{N} \in \mathcal{K}(A\text{-Mod})$. Dann sei $\com{M} \otimes_A \com{N} \in \mathcal{K}(A\text{-Mod})$ definiert durch \[ (\com{M} \otimes_A \com{N})^{n} = \bigoplus_{i \in \Z} M^{i} \otimes_A N^{n-i} @@ -564,7 +616,22 @@ wird uns erlauben das klassische Adjunktionsresultat \[ d^{n}(m \otimes n) = d_{\com{M} }(m) \otimes n + (-1)^{i} m \otimes d_{\com{N} }(n) \] für $m \in M^{i}, n \in N^{n-i}$. + \label{def:tor-compl} \end{definition} +\begin{bem} + \begin{enumerate}[(a)] + \item Die Konstruktionen in \ref{def:hom-compl} und \ref{def:tor-compl} sind + funktoriell in beiden Variablen und induzieren daher entsprechende Funktoren. + \item + Wie für das klassiche Tensorprodukt von $A$-Moduln, existieren für + $\com{M}, \com{N}, \com{P} \in \mathcal{K}(A\text{-Mod})$ + natürliche Isomorphismen + \[ + \com{M} \otimes_A \com{N} = \com{N} \otimes_A \com{M} \text{ und } + (\com{M} \otimes_A \com{N}) \otimes_A \com{P} = \com{M} \otimes_A (\com{N} \otimes_A \com{P}) + .\] + \end{enumerate} +\end{bem} Die Kohomologiegruppen von $\com{\text{Hom}}(\com{X}, \com{Y})$ für Komplexe $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: @@ -609,7 +676,7 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \end{lemma} \begin{proof} - Das gilt gradweise nach Definition von $\lim$ und man verifiert, dass die gradweisen Homomorphismen, + Das gilt gradweise nach Definition von $\lim$ und man verifiert, dass die gradweisen Homomorphismen Komplexhomomorphismen bilden. \end{proof} @@ -620,6 +687,14 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \label{satz:tor-is-triangulated} \end{lemma} +\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: + \[ + \com{C}_f \otimes_A \com{S} = \com{C}_{f \otimes \text{id}_{\com{S}}} + .\] Das rechnet man gradweise nach und zeigt, dass die Differentiale übereinstimmen. +\end{proof} + \begin{satz}[Adjunktion der Hom- und Tensorproduktkomplexe] Seien $\com{M}, \com{N}, \com{P} \in \mathcal{K}(A\text{-Mod})$. Dann existiert ein natürlicher Isomorphismus, der funktoriell in allen Variablen ist: @@ -631,10 +706,12 @@ $\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: \end{satz} \begin{proof} + Das gilt gradweise, weil für beliebige $A$-Moduln $M, N, P$ ein natürlicher Isomorphismus + \[ + \text{Hom}_{A}(M \otimes_A N, P) \to \text{Hom}_{A}(M, \text{Hom}_{A}(N, P)) + \] existiert. Man verifiziert, dass die gradweisen Isomorphismen Komplexhomomorphismen bilden. \end{proof} -% TODO: Bedingung (I) an Indexmengen - % TODO: verstehe ich nicht aber habe alternative %\begin{lemma}[] % Sei $\com{A} \in \mathcal{K}$ und $\mathcal{G}$ eine Klasse von Objekten @@ -668,7 +745,7 @@ Dazu definieren wir: \begin{definition}[K-injektiv] Ein Komplex $\com{I} \in \mathcal{K}$ heißt K-injektiv, wenn der Funktor - $\com{\text{Hom}}(-, \com{I})$ Exaktheit erhält. Eine K-injektive + $\com{\text{Hom}}(-, \com{I})$ Exaktheit von Komplexen erhält. Eine K-injektive Auflösung eines Komplexes $\com{X} \in \mathcal{K}$ ist ein Quasiisomorphismus $\com{X} \to \com{I} $ mit $\com{I} \in \mathcal{K}$ K-injektiv. @@ -676,7 +753,7 @@ Dazu definieren wir: \begin{definition}[K-projektiv] Ein Komplex $\com{P} \in \mathcal{K}$ heißt K-projektiv, wenn der Funktor - $\com{\text{Hom}}(\com{P}, -)$ Exaktheit erhält. Eine K-projektive + $\com{\text{Hom}}(\com{P}, -)$ Exaktheit von Komplexen erhält. Eine K-projektive Auflösung eines Komplexes $\com{X} \in \mathcal{K}$ ist ein Quasiisomorphismus $\com{P} \to \com{X} $ mit $\com{P} \in \mathcal{K}$ K-projektiv. @@ -717,7 +794,8 @@ Komplexen entwickelt. \begin{lemma} Ein exakter K-projektiver oder K-injektiver Komplex $\com{X} \in \mathcal{K}$ - ist zusammenziehbar, d.h. ist nullhomotop. + ist zusammenziehbar, das heißt ist nullhomotop, also in $\mathcal{K}$ isomorph zum Nullkomplex. + \label{lemma:k-inj-exact-contractible} \end{lemma} \begin{proof} @@ -726,6 +804,19 @@ Komplexen entwickelt. $\com{X} \stackrel{\sim }{=} 0$ in $\K$. \end{proof} +\begin{bem} + Aus \ref{lemma:k-inj-exact-contractible} folgt, dass für einen exakten und K-injektiven Komplex + $\com{I} \in \mathcal{K}$ und + einen beliebigen Komplex $\com{X} \in \mathcal{K}$, + der Komplex $\com{\text{Hom}}(\com{X}, \com{I})$ exakt ist, denn + \[ + H^{i}(\com{\text{Hom}}(\com{X}, \com{I})) = \text{Mor}_{\mathcal{K}}(\com{\text{Hom}}(\com{X}, \com{I}[i])) + \stackrel{\ref{lemma:k-inj-exact-contractible}}{=} \text{Mor}_{\mathcal{K}}(\com{\text{Hom}}(\com{X}, 0)) = 0 + .\] Analog gilt die duale Version für exakte K-projektive Komplexe. + \label{satz:hom-exact-for-k-inj} + %auch $\com{\text{Hom}}(\com{X}, \com{I})$ exakt ist für alle $\com{M} \in \mathcal{K}$. +\end{bem} + Für ein Objekt $X \in \mathcal{A}$ stellt sich die Frage, ob Injektivität (bzw. Projektivität) von $X$ in $\mathcal{A}$ mit K-injektivität (bzw. K-projektivität) des Komplexes $\com{X}$, wobei $X^{i} = 0$ für alle $i \neq 0$ und $X^{0} = X$, zusammenhängt. Die folgende Aussage stellt diesen Zusammenhang her: @@ -740,7 +831,8 @@ $X^{0} = X$, zusammenhängt. Die folgende Aussage stellt diesen Zusammenhang her \begin{proof} Wir zeigen nur den K-projektiven Fall. Der K-injektive folgt dann durch Umdrehen aller Pfeile. - ,,$\implies$'': Sei $\com{X} $ K-projektiv und $\com{S} = [0 \to M \to N \to P \to 0]$ eine kurze exakte Folge in $\mathcal{A}$. Sei + + ($\Leftarrow$) Sei $\com{X} $ K-projektiv und $\com{S} = [0 \to M \to N \to P \to 0]$ eine 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} @@ -750,7 +842,7 @@ $X^{0} = X$, zusammenhängt. Die folgende Aussage stellt diesen Zusammenhang her Nach Voraussetzung ist dieser nullhomotop, das heißt es existiert ein $k \colon X^{0} \to N$, sodass $f = vk$. Also ist $v_{*}\colon \text{Hom}(X^{0}, N) \to \text{Hom}(X^{0}, P)$ surjektiv und damit $X^{0}$ projektiv. -,,$\impliedby$'': Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ +($\Rightarrow$) Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ Komplexhomomorphismus. Dann betrachte \[ \begin{tikzcd} @@ -832,7 +924,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d \end{tikzcd} .\] Die äußeren Terme sind 0, da $\com{P} $ K-projektiv ist, 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 + (ii)$\implies$(iii): Injektivität: Sei $f\colon \com{P} \to \com{S} $, sodass $\text{id}^{-1}f = 0$. Nach \ref{derived-cat-morphism-null} existiert ein Quasiisomorphismus $t\colon \com{S} \to \com{T} $ , sodass $tf= 0$. 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 @@ -842,7 +934,7 @@ $X^0$ projektiv ist, existiert $k^{0}\colon X^{0} \to S^{-1}$, sodass $f^{0} = d & \com{Y} & \\ \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{Y} ) $ surjektiv, also existiert ein $g\colon \com{P} \to \com{S} $, s.d. $sg = f$. Also + \] mit $s$ Quasiisomorphismus. Nach (ii) ist $s_{*}\colon \text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{K}}(\com{P} , \com{Y} ) $ surjektiv, also existiert ein $g\colon \com{P} \to \com{S} $, sodass $sg = f$. Also kommutiert \[ \begin{tikzcd} @@ -987,7 +1079,7 @@ Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen \begin{lemma} Sei $\mathcal{J}_0$ eine Klasse von Objekten von $\mathcal{A}$. Sei weiter $\mathcal{J}$ eine unter speziellen inversen Limites abgeschlossene Klasse - von Objekten in $\mathcal{K}$, so dass jeder Komplex $\com{A} \in \mathcal{K}$ mit nur einem nicht-null + von Objekten in $\mathcal{K}$, sodass jeder Komplex $\com{A} \in \mathcal{K}$ mit nur einem nicht-null Term und mit $A^{i} \in \mathcal{J}_0$ für $i \in \Z$, in $\mathcal{J}$ enthalten ist. Dann ist jeder nach unten beschränkte Komplex $\com{A} \in \mathcal{K}$ mit $A^{i} \in \mathcal{J}_0$ für $i \in \Z$ in $\mathcal{J}$ enthalten. @@ -996,9 +1088,10 @@ Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen \end{lemma} \begin{proof} - Sei $\com{A} \in \mathcal{K}^{+}$ mit $A^{i} \in \mathcal{J}_0$ für alle $i \in \Z$. Ohne Einschränkung + Sei $\com{A} \in \mathcal{K}$ nach unten beschränkt mit + $A^{i} \in \mathcal{J}_0$ für alle $i \in \Z$. Ohne Einschränkung sei $A^{i} = 0$ für alle $i < 0$. Dann sind die Spalten des nachstehenden Diagramms ein - $\mathcal{J}$-spezielles inverses System $(\com{S}_n)_{n \ge 0}$ mit + $\mathcal{J}$-spezielles inverses System $(\com{S}_n)_{n \in \N}$ mit Übergangsabbildungen $p_n$, \[ \begin{tikzcd} @@ -1008,7 +1101,7 @@ Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen \cdots\arrow{r} & A^{2} \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \\ & \vdots & \vdots & \vdots & \vdots \\ \end{tikzcd} - \] denn für $n > 0$ ist $\com{\text{ker } p}_n$ = $[\cdots \to 0 \to \underbrace{A^{n-1}}_{\in \mathcal{J}_0} \to 0 \to \cdots ]$. Nach + \] denn für $n > 1$ ist $\com{\text{ker } p}_n$ = $[\cdots \to 0 \to \underbrace{A^{n-2}}_{\in \mathcal{J}_0} \to 0 \to \cdots ]$. Nach Voraussetzung ist also $\text{ker } p_n$ in $\mathcal{J}$ und die kurze exakte Folge $0 \to \com{\text{ker } p}_n \to \com{S}_n \to \com{S}_{n-1} \to 0$ zerfällt gradweise. Also folgt $\com{A} = \lim \com{S}_n \in \mathcal{J}$. @@ -1066,7 +1159,7 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. der Übergangsabbildungen $A_n \to A_{n-1}$, $B_n \to B_{n-1}$, $C_n \to C_{n-1}$ und $D_n \to D_{n-1}$. - Sei weiter $N \in \N$, s.d. für alle $n > N$ die Folge + Sei weiter $N \in \N$, sodass für alle $n > N$ die Folge \[ \begin{tikzcd} A_n' \arrow{r} & B_n' \arrow{r} & C_n' \arrow{r} & D_n' @@ -1146,7 +1239,7 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. h_{N+1}(z) = h_{N+1}(g_{N+1}(z)) = 0 .\] Da die untere Zeile exakt ist, existiert nun - ein $\tilde{y} \in \text{ker } p_B$, s.d. $g_{N+1}(\tilde{y}) = z$. Also ist + ein $\tilde{y} \in \text{ker } p_B$, sodass $g_{N+1}(\tilde{y}) = z$. Also ist $y - \tilde{y} \in \text{ker } g_{N+1}$ und \[ p_B(y - \tilde{y}) = p_B(y) = b @@ -1211,13 +1304,13 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. \item Für $n > 1$ ist nach Voraussetzung \[ \begin{tikzcd} - 0 \arrow{r} & \text{ker } p_n \arrow{r} & \com{S}_n \arrow{r}{p_n} & \com{S}_{n-1} \arrow{r} & 0 + 0 \arrow{r} & \text{ker } \com{p}_n \arrow{r} & \com{S}_n \arrow{r}{p_n} & \com{S}_{n-1} \arrow{r} & 0 \end{tikzcd} \] - exakt, zerfällt gradweise und $\text{ker } p_n$ ist in $F^{-1}(\mathcal{J})$. Nach Voraussetzung ist damit + exakt, zerfällt gradweise und $\text{ker } \com{p}_n$ ist in $F^{-1}(\mathcal{J})$. Nach Voraussetzung ist damit \[ \begin{tikzcd} - 0 \arrow{r} & F(\text{ker } p_n) \arrow{r} & F(\com{S}_n) \arrow{r}{F(p_n)} & F(\com{S}_{n-1}) \arrow{r} & 0 + 0 \arrow{r} & F(\text{ker } \com{p}_n) \arrow{r} & F(\com{S}_n) \arrow{r}{F(p_n)} & F(\com{S}_{n-1}) \arrow{r} & 0 \end{tikzcd} \] exakt und zerfällt gradweise. Aus der Exaktheit folgt damit auch $\text{ker } F(p_n) = F(\text{ker } p_n)$, also $\text{ker } F(p_n) \in \mathcal{J}$. @@ -1225,7 +1318,7 @@ technisches Hilfswerkzeug für inverse Systeme in $\mathcal{A}b$. Also ist $F(\lim \com{S}_n) = \lim F(\com{S}_n) \in \mathcal{J}$ und damit $\lim \com{S}_n \in F^{-1}(\mathcal{J})$. \end{proof} -\begin{korollar}[] +\begin{korollar} Sei $\mathcal{I}$ eine Klasse von Komplexen und angenommen in $\mathcal{A}$ existieren inverse Limites. Dann ist die Klasse aller Komplexe $\com{A} \in \mathcal{K}$, sodass @@ -1316,7 +1409,7 @@ 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 nennen wir $\underset{\leftarrow}{\mathcal{G}}$ (bzw. $\underset{\rightarrow}{\mathcal{G}})$ die kleinste Klasse von Komplexen in $\mathcal{K}$, die abgeschlossen - unter speziellen inversen (bzw. direkten) Limites ist und $\mathcal{G}$ enthält. + unter speziellen inversen Limites (bzw. direkten Colimites) ist und $\mathcal{G}$ enthält. \end{definition} \subsection{Existenz von K-projektiven und K-injektiven Auflösungen} @@ -1333,59 +1426,68 @@ zu beweisen. Dazu verallgemeinern wir zunächst die Begriffe K-injektive und K-p \subsubsection{Linksauflösungen} -Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedingung genügt: -\begin{enumerate}[(L1)] - \item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine - Linksauflösung $\com{P} \to \com{A} $ durch einen nach oben beschränkten - Komplex $\com{P} \in \mathcal{P}$. -\end{enumerate} +Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. +%\begin{enumerate}[(L1)] +% \item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine +% $\mathcal{P}$-Linksauflösung. +%\end{enumerate} -% TODO: beschränktheit notwendig -%\begin{lemma}[] -% Die folgenden Bedingungen an $\mathcal{P}$ sind äquivalent: -% \begin{enumerate}[(i)] -% %\item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ -% % hat eine $\mathcal{P}$-Linksauflösung. -% \item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine -% Linksauflösung $\com{P} \to \com{A} $ durch einen nach oben beschränkten -% Komplex $\com{P} \in \mathcal{P}$. -% \item Für alle $\com{A} \in \mathcal{K} $ und $n \in \Z$, existiert ein $\com{P} \in \mathcal{P}$ mit -% $H^{i}(\com{P}) = 0$ für $i > n$ und ein Komplexhomomorphismus $f\colon \com{P} \to \com{A} $, der -% einen Isomorphismus $H^{i}(\com{P}) \to H^{i}(\com{A})$ induziert für $i \le n$. -% \end{enumerate} -% \label{lemma:class-compl-cond} -%\end{lemma} +\begin{lemma}[] + Die folgenden Bedingungen an $\mathcal{P}$ sind äquivalent: + \begin{enumerate}[(i)] + \item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ + hat eine $\mathcal{P}$-Linksauflösung. + %\item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine + % Linksauflösung $\com{P} \to \com{A} $ durch einen nach oben beschränkten + % Komplex $\com{P} \in \mathcal{P}$. + \item Für alle $\com{A} \in \mathcal{K} $ und $n \in \Z$, existiert ein $\com{P} \in \mathcal{P}$ mit + $H^{i}(\com{P}) = 0$ für $i > n$ und ein Komplexhomomorphismus $f\colon \com{P} \to \com{A} $, der + einen Isomorphismus $H^{i}(\com{P}) \to H^{i}(\com{A})$ induziert für $i \le n$. + \end{enumerate} + \label{lemma:class-compl-cond} +\end{lemma} -%\begin{proof} -% (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 (iii). Dann existiert ein $f\colon \com{P'} \to \com{A} $ mit -% $\com{P'} \in \mathcal{P}$ und $f$ induziert -% $H^{i}(\com{P}) \xrightarrow[\sim]{0 = H^{i}(f)} 0 = H^{i}(\com{A})$ für $i > 0$ und -% $H^{i}(\com{P})\xrightarrow[\sim]{H^{i}(f)} H^{i}(\com{A})$ für $i \le 0$. Also ist -% $f$ ein Quasiisomorphismus. -% -% (i) $\implies$ (ii): -% Sei $\com{A} \in \mathcal{K}$ und $n \in \Z$. -% Dann ist $\tau_{\le n} \com{A}$ nach oben beschränkt, also existiert -% ein $\com{P} \in \mathcal{P}$ und ein Quasiisomorphismus -% $s\colon \com{P} \to \tau_{\le n}\com{A}$. 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 -% $H^{i}(\tau_{\le n}\com{A}) = H^{i}(\com{A})$, also induziert $f$ den gewünschten Isomorphismus. -%\end{proof} +\begin{proof} + (i)$\implies$(ii): + Sei $\com{A} \in \mathcal{K}$ und $n \in \Z$. + Dann ist $\tau_{\le n} \com{A}$ nach oben beschränkt, also existiert + ein $\com{P} \in \mathcal{P}$ und ein Quasiisomorphismus + $s\colon \com{P} \to \tau_{\le n}\com{A}$. 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 + $H^{i}(\tau_{\le n}\com{A}) = H^{i}(\com{A})$, 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 (iii). Dann existiert ein $f\colon \com{P} \to \com{A} $ mit + $\com{P} \in \mathcal{P}$ und $f$ induziert + $H^{i}(\com{P}) \xrightarrow[\sim]{0 = H^{i}(f)} 0 = H^{i}(\com{A})$ für $i > 0$ und + $H^{i}(\com{P})\xrightarrow[\sim]{H^{i}(f)} H^{i}(\com{A})$ für $i \le 0$. Also ist + $f$ ein Quasiisomorphismus. +\end{proof} + +\begin{bem}[] + Sei $\com{A} \in \mathcal{K}$. Angenommen $\mathcal{P}$ genügt den äquivalenten Bedingungen + von \ref{lemma:class-compl-cond} und es existiert ein $n \in \Z$, sodass + für alle $i > n$, $H^{i}(\com{A}) = 0$ ist. Dann hat $\com{A} $ eine + $\mathcal{P}$-Linksauflösung, denn nach \ref{lemma:class-compl-cond} (ii) existiert dann + ein $\com{P} \in \mathcal{P}$ und ein Komplexhomomorphismus $f\colon \com{P} \to \com{A}$, sodass + $H^{i}(\com{P}) = 0 = H^{i}(\com{A})$ für $i > n$ und + $H^{i}(f)$ ein Isomorphismus ist für $i \le n$. Also ist $f$ ein Quasiisomorphismus. + \label{bem:p-left-resolutions-for-h-bounded} +\end{bem} \begin{bsp} - Falls $\mathcal{A}$ genügend Projektive hat, können wir $\mathcal{P}$ als - die Klasse - der nach oben beschränkten Komplexe $\com{P} \in \mathcal{K}$ mit $P^{i}$ - projektiv für alle $i \in \Z$ wählen. - %Sei $\mathcal{P}$ die Klasse + %Falls $\mathcal{A}$ genügend Projektive hat, können wir $\mathcal{P}$ als + %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 Projektive hat, erfüllt - %$\mathcal{P}$ die äquivalenten Bedingungen aus \ref{lemma:class-compl-cond}. + %projektiv für alle $i \in \Z$ wählen. + 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 Projektive hat, erfüllt + $\mathcal{P}$ die äquivalenten Bedingungen aus \ref{lemma:class-compl-cond}. 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 @@ -1408,12 +1510,11 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \begin{proof} Wir gehen induktiv vor. Setze $\com{P}_{1} = 0$ und $f_{1} = 0$. - Nach (L1) existiert ein Quasiisomorphismus - $f_2 \colon \com{P}_2 \to \tau_{\le 2}\com{A}$, wobei $\com{P}_2 \in \mathcal{P}$ - nach oben beschränkt ist. + Nach den äquivalenten Bedingungen von \ref{lemma:class-compl-cond} existiert ein Quasiisomorphismus + $f_2 \colon \com{P}_2 \to \tau_{\le 2}\com{A}$ mit $\com{P}_2 \in \mathcal{P}$. Sei nun $n \ge 3$ und seien $\com{P}_{1}, \ldots, \com{P}_{n-1}$ und $f_{1}, \ldots, f_{n-1}$ - konstruiert mit $\com{P}_i$ nach oben beschränkt. Dann + konstruiert wie im Lemma. Dann setze $\com{P} = \com{P}_{n-1}$ und $\com{B} = \tau_{\le n} \com{A} $. Es sei $a_{n-1}\colon \tau_{\le n-1}\com{A} \to \tau_{\le n} \com{A} $ der natürliche Komplexhomomorphismus und $f = a_{n-1}f_{n-1}\colon \com{P} \to \com{B}$. Es gilt dann @@ -1421,11 +1522,12 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi f d_P = d_B f \label{eq:f-comp-hom} \end{equation} - Da $\com{B}$ und $\com{P} $ nach oben beschränkt sind und - $C_f^{i} = P^{i+1} \oplus B^{i}$, ist $\com{C}_f$ auch nach oben beschränkt. - Also existiert mit (L1) ein Quasiisomorphismus - $g\colon \com{Q} \to \com{C}_f[-1]$ mit $\com{Q}[1] \in \mathcal{P}$ und - $\com{Q} $ nach oben beschränkt. Da + Da $\com{B}$ nach oben beschränkt ist + und $H^{i}(\com{P}) \stackrel{\sim }{=} H^{i}(\tau_{\le n-1}\com{A}) = 0$ für $i \gg 0$, folgt + nach \ref{bem:mapping-cone-h-bounded} + $H^{i}(\com{C}_f) = 0$ für $i \gg 0$. + Also existiert nach \ref{bem:p-left-resolutions-for-h-bounded} ein Quasiisomorphismus + $g\colon \com{Q} \to \com{C}_f[-1]$ mit $\com{Q}[1] \in \mathcal{P}$. Da $C_f^{i}[-1] = P^{i} \oplus B^{i-1}$ für $i \in \Z$, ist $g$ gradweise gegeben durch Morphismen $g'_i\colon Q^{i} \to P^{i} $ und $g''_i\colon Q^{i} \to B^{i-1}$ in $\mathcal{A}$. @@ -1442,14 +1544,15 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \end{equation} In Matrixnotation ist \begin{align*} - d_{C_f} &= \begin{pmatrix} d_{P}[1] & 0 \\ f[1] & d_{B} \end{pmatrix} + d_{C_f} = \begin{pmatrix} d_{P[1]} & 0 \\ f[1] & d_{B} \end{pmatrix} + = \begin{pmatrix} - d_{P} & 0 \\ f & d_{B} \end{pmatrix} \intertext{Also folgt} - d_{C_f}[-1] &= - d_{C_f} = \begin{pmatrix} -d_{P}[1] & 0 \\ -f[1] & -d_{B} \end{pmatrix} + d_{C_f}[-1] = - d_{C_f} = \begin{pmatrix} d_{P} & 0 \\ -f & -d_{B} \end{pmatrix} .\end{align*} Auswerten der Kommutativität von \eqref{eq:1} in beiden Summanden liefert nun \begin{align} d_P g' &= g' d_Q \label{eq:g'-comp-hom} \\ - g''d_Q &= -fg' - d_Bg'' \label{eq:g''} + -fg' - d_Bg'' &= g''d_Q \label{eq:g''} .\end{align} Aus \eqref{eq:g'-comp-hom} folgt, dass $g'\colon \com{Q} \to \com{P} $ ein Komplexhomomorphismus ist. Setze nun @@ -1457,7 +1560,7 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \[ h(x,y) = g''[1](x) + f(y) .\] - Betrachte nun für $i \in \Z$ das folgende Diagramm: + Betrachte für $i \in \Z$ das folgende Diagramm: \[ \begin{tikzcd} \cdots \arrow{r} & Q^{i+1} \oplus P^{i} \arrow{r}{d_{C_{-g'}}} \arrow{d}{h} & Q^{i+2} \oplus P^{i+1} \arrow{r} \arrow{d}{h} @@ -1466,10 +1569,10 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \end{tikzcd} .\] In Matrixnotation ist \begin{salign*} - h d_{C_{-g'}} &= \begin{pmatrix} g''[1] & f \end{pmatrix} - \begin{pmatrix} d_Q[1] & 0 \\ -g'[1] & d_P \end{pmatrix} \\ + h d_{C_{-g'}} &= \begin{pmatrix} g'' & f \end{pmatrix} + \begin{pmatrix} -d_Q & 0 \\ -g' & d_P \end{pmatrix} \\ &= \begin{pmatrix} - g''[1] d_Q[1] - f g'[1] & f d_P + -g'' d_Q - f g' & f d_P \end{pmatrix} \\ &\stackrel{\eqref{eq:g''}}{=} \begin{pmatrix} @@ -1481,29 +1584,45 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \end{pmatrix} \\ &= d_B h .\end{salign*} + %\begin{salign*} + % h d_{C_{-g'}} &= \begin{pmatrix} g''[1] & f \end{pmatrix} + % \begin{pmatrix} d_Q[1] & 0 \\ -g'[1] & d_P \end{pmatrix} \\ + % &= \begin{pmatrix} + % g''[1] d_Q[1] - f g'[1] & f d_P + % \end{pmatrix} \\ + % &\stackrel{\eqref{eq:g''}}{=} + % \begin{pmatrix} + % d_B g'' & f d_P + % \end{pmatrix} \\ + % &\stackrel{\eqref{eq:f-comp-hom}}{=} + % \begin{pmatrix} + % d_B g'' & d_B f + % \end{pmatrix} \\ + % &= d_B h + %.\end{salign*} Also ist $h$ ein Komplexhomomorphismus. Bleibt zu zeigen, dass $h$ ein Quasiisomorphismus ist. Dafür genügt es nach \ref{mapping-cone-exact-for-qis} zu zeigen, dass $\com{C}_h$ exakt ist. Behauptung: $\com{C}_h = \com{C}_{-g}[1]$. Es ist gradweise für $ i \in \Z$ \[ - C_h^{i} = C_{-g}^{i+1} \oplus B^{i} = (Q^{i+2} \oplus P^{i+1}) \oplus B^{i} + C_h^{i} = C_{-g'}^{i+1} \oplus B^{i} = (Q^{i+2} \oplus P^{i+1}) \oplus B^{i} = Q^{i+2} \oplus (P^{i+1} \oplus B^{i}) = Q^{i+2} \oplus C_f^i = C_{-g}^{i}[1] .\] Für die Differentiale gilt, wieder in Matrixnotation: \begin{align*} d_{C_h} = \begin{pmatrix} - d_{C_{-g'}}[1] & 0 \\ - h[1] & d_B \end{pmatrix}[1] + d_{C_{-g'}[1]} & 0 \\ + h[1] & d_B \end{pmatrix} = \begin{pmatrix} - \begin{pmatrix} d_Q[1] & 0 \\ - -g'[1] & d_P - \end{pmatrix}[1] & 0 \\ - \begin{pmatrix} g''[1] & f \end{pmatrix}[1] & d_B + -\begin{pmatrix} -d_Q & 0 \\ + -g' & d_P + \end{pmatrix} & 0 \\ + \begin{pmatrix} g'' & f \end{pmatrix} & d_B \end{pmatrix} = \begin{pmatrix} - -d_Q & 0 & 0 \\ + d_Q & 0 & 0 \\ g' & -d_P & 0 \\ g'' & f & d_B \end{pmatrix} @@ -1512,27 +1631,24 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \begin{align*} d_{C_{-g}[1]} = \begin{pmatrix} - d_Q[1] & 0 \\ - -g & d_{C_f[-1]} + d_{Q[1]} & 0 \\ + -g[1] & d_{C_f[-1]} \end{pmatrix} [1] - = \begin{pmatrix} d_Q[1] & 0 \\ - \begin{pmatrix} -g' \\ -g'' \end{pmatrix}[1] - & \begin{pmatrix} d_P[1] & 0 \\ f[1] & d_{B} \end{pmatrix}[-1] - \end{pmatrix}[1] + = -\begin{pmatrix} -d_Q & 0 \\ + \begin{pmatrix} -g' \\ -g'' \end{pmatrix} + & -\begin{pmatrix} -d_P & 0 \\ f & d_{B} \end{pmatrix} + \end{pmatrix} = \begin{pmatrix} - - d_Q & 0 & 0 \\ + d_Q & 0 & 0 \\ g' & -d_P & 0 \\ g'' & f & d_B \end{pmatrix} .\end{align*} Also folgt die Behauptung. Da $g$ und demnach $-g$ ein Quasiisomorphismus ist und Verschieben Exaktheit erhält, - folgt damit mit \ref{mapping-cone-exact-for-qis} die Exaktheit von $\com{C}_{-g}[1]$. - - Setze nun $\com{P}_n \coloneqq \com{C}_{-g'}$ und $f_n \coloneqq h$. Da nach - Voraussetzung $\com{P} $ und $\com{Q} $ nach oben beschränkt sind, ist auch - $\com{C}_{-g'}$ nach oben beschränkt. + folgt damit mit \ref{mapping-cone-exact-for-qis} die Exaktheit von $\com{C}_h = \com{C}_{-g}[1]$. + Setze nun $\com{P}_n \coloneqq \com{C}_{-g'}$ und $f_n \coloneqq h$. 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 @@ -1556,7 +1672,7 @@ Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$, die folgender Bedi \] und $(f_n)_{n \in \N}$ ist ein direktes System. \end{proof} -Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: +Daraus folgt nun sofort: \begin{satz} Angenommen direkte Colimites existieren in $\mathcal{A}$ und @@ -1568,7 +1684,7 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: \end{satz} \begin{proof} - Sei $\com{A} \in \mathcal{K}$ und $(\com{P}_n)_{n \ge -1}$, $(f_n)_{n \ge -1}$ wie + Sei $\com{A} \in \mathcal{K}$ und $(\com{P}_n)_{n \in \N}$, $(f_n)_{n \in \N}$ wie in \ref{lemma:constr-dir-system}. Da direkte Colimites in $\mathcal{A}$ existieren und sich diese in $\mathcal{K}$ gradweise bilden, existieren direkte Colimites in $\mathcal{K}$. Nach der Definition von $\underset{\rightarrow}{\mathcal{P}}$ ist dann @@ -1578,7 +1694,7 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: \[ f\coloneqq \colim f_n \colon \com{P} \longrightarrow \colim \tau_{\le n}\com{A} = \com{A} - .\] Da $\colim$ exakt, folgt für $i \in \Z$: + .\] Da $\colim$ exakt ist, folgt für $i \in \Z$: \[ H^{i}(f) = H^{i}(\colim f_n) = \colim \underbrace{H^{i}(f_n)}_{\text{Isomorphismus}} .\] Also ist $f$ ein Quasiisomorphismus. @@ -1586,8 +1702,8 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: \begin{korollar}[] Angenommen direkte Colimites existieren in $\mathcal{A}$ und - $\colim$ ist exakt. - Dann hat jeder Komplex in $\mathcal{K}$ eine K-projektive Linksauflösung. + $\colim$ ist exakt. Falls $\mathcal{A}$ außerdem genug Projektive hat, + besitzt jeder Komplex in $\mathcal{K}$ eine K-projektive Auflösung. \label{satz:existence-k-proj-resolution} \end{korollar} @@ -1600,16 +1716,19 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: Sei $\mathcal{I}$ eine Klasse von Komplexen in $\mathcal{K}$. Im Folgenden nehmen wir an, dass $\mathcal{I}$ die folgende Eigenschaft erfüllt: +\vspace{2mm} -\begin{enumerate}[(1)] - \item Jeder nach unten beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine - Auflösung $\com{A} \to \com{I} $ nach rechts mit $\com{I} \in \mathcal{I}$ und - $\com{I}$ nach unten beschränkt. -\end{enumerate} +\noindent\hspace{9mm} \emph{Jeder nach unten beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine +$\mathcal{I}$-Rechtsauflösung.} + +%\begin{enumerate}[(1)] +% \item \emph{Jeder nach unten beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine +% $\mathcal{I}$-Rechtsauflösung.} +%\end{enumerate} \begin{bsp} - Falls $\mathcal{A}$ genügend Injektive hat, können wir dual zu Beispiel - \ref{bsp:bounded-above-projectives} $\mathcal{I}$ als die Klasse + Falls $\mathcal{A}$ genügend 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} @@ -1618,9 +1737,9 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \begin{lemma}[] Sei $\com{A} \in \mathcal{K}$. Dann existiert ein $\mathcal{I}$-spezielles - inverses System $(\com{I}_n)_{n \ge -1}$ und ein inverses System von + inverses System $(\com{I}_n)_{n \in \N}$ und ein inverses System von Kettenhomomorphismen $f_n\colon \tau^{\ge-n}\com{A} \to \com{I}_n$, sodass - $f_n$ ein Quasiisomorphismus ist. + $f_n$ ein Quasiisomorphismus ist für $n \ge 2$. \label{lemma:constr-inv-system} \end{lemma} @@ -1628,7 +1747,7 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von Angenommen inverse Limites existieren in $\mathcal{A}$ und $\lim$ ist exakt. Dann hat jeder Komplex in $\mathcal{K}$ eine - $\underset{\rightarrow}{\mathcal{P}}$-Rechtsauflösung. + $\underset{\leftarrow}{\mathcal{I}}$-Rechtsauflösung. \label{satz:existence-right-resolutions} \end{satz} @@ -1640,22 +1759,22 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von Diese Voraussetzung wird jedoch nur verwendet, um zu zeigen, dass $f = \lim f_n\colon \com{A} \to \lim \com{I}_n$ ein Quasiisomorphismus ist. Wir können uns der speziellen Struktur des inversen Systems - $(\com{I}_n)_{n \ge -1}$ bedienen, um für $\mathcal{A} = R\text{-Mod}$ zu zeigen, + $(\com{I}_n)_{n \in \N}$ bedienen, um für $\mathcal{A} = R\text{-Mod}$ zu zeigen, dass $f$ dennoch ein Quasiisomorphismus ist. \end{bem} \begin{satz}[] Sei $R$ ein Ring und $\mathcal{A}$ die Kategorie der R-links-Moduln. Dann - hat jeder Komplex in $\mathcal{K}$ eine K-injektive Rechtsauflösung. + hat jeder Komplex in $\mathcal{K}$ eine K-injektive Auflösung. \label{satz:existence-k-inj-resolution} \end{satz} \begin{proof} - Seien $(\com{I}_n)_{n \ge -1}$ und $(f_n)_{n \ge -1}$ wie in + Seien $(\com{I}_n)_{n \in \N}$ und $(f_n)_{n \in \N}$ wie in \ref{lemma:constr-inv-system}. Seien $\com{I} = \lim \com{I}_n$ und $f = \lim f_n$. Es genügt zu zeigen, dass $f$ ein Quasiisomorphismus ist. - Sei $i \in \Z$ beliebig. Für $n > 1$ haben wir folgendes kommutative Diagramm: + Sei $i \in \Z$ beliebig. Für $n \ge 3$ haben wir folgendes kommutative Diagramm: \[ \begin{tikzcd} \com{I} \arrow{r} \arrow{d}{f} & \com{I}_n \arrow{r}{p_n} \arrow{d}{f_n} & \com{I_{n-1}} \arrow{d}{f_{n-1}} \\ @@ -1671,11 +1790,11 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \label{eq:diag-hi-in} .\end{equation} Die rechten beiden vertikalen Pfeile sind Isomorphismen, da $f_k$ ein - Quasiisomorphismus ist für alle $k \ge -1$. + Quasiisomorphismus ist für alle $k \ge 2$. Sei nun $n \ge -i+1$. Dann ist $i \ge -n + 1 \ge -n$, also ist $H^{i}(\com{A}) = H^{i}(\tau^{\ge -n}\com{A}) = H^{i}(\tau^{\ge -(n-1)} \com{A})$. Also - sind die Abbildungen in der unteren Zeile in \eqref{eq:diag-hi-in} Isomorphismen und + sind die Morphismen in der unteren Zeile von \eqref{eq:diag-hi-in} Isomorphismen und damit ist $H^{i}(p_n)\colon H^{i}(\com{I}_n) \to H^{i}(\com{I}_{n-1})$ ein Isomorphismus. @@ -1686,20 +1805,20 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von 0 \arrow{r} & \text{ker } p_n \arrow{r} & \com{I}_n \arrow{r}{p_n} & \com{I}_{n-1} \arrow{r} & 0 \end{tikzcd} - .\] Das liefert eine lange exakte Kohomologiefolge: + .\] Das liefert für $j \in \Z$ eine lange exakte Kohomologiefolge: \begin{equation} \begin{tikzcd} - H^{i-1}(\com{I}_{n}) \arrow{r}{H^{i-1}(p_n)} & H^{i-1}(\com{I}_{n-1}) \arrow{r} - & H^{i}(\text{ker } p_n) \arrow{r} - & H^{i}(\com{I}_n) \arrow{r}{H^{i}(p_n)} - & H^{i}(\com{I}_{n-1}) + H^{j-1}(\com{I}_{n}) \arrow{r}{H^{j-1}(p_n)} & H^{j-1}(\com{I}_{n-1}) \arrow{r} + & H^{j}(\text{ker } p_n) \arrow{r} + & H^{j}(\com{I}_n) \arrow{r}{H^{j}(p_n)} + & H^{j}(\com{I}_{n-1}) \end{tikzcd} \label{eq:long-ex-hi-in} \end{equation} - Anwenden des obigen Arguments auf $i-1$ liefert für - $n \ge -(i-1) + 1 = -i+2 \ge -i+1$ Isomorphismen $H^{i}(p_n)$ und $H^{i-1}(p_n)$. + Anwenden des obigen Arguments für $i = j$ und $i = j-1$ liefert für + $n \ge -(j-1) + 1 \ge -j+1$ Isomorphismen $H^{j}(p_n)$ und $H^{j-1}(p_n)$. Aufgrund der Exaktheit von \eqref{eq:long-ex-hi-in} folgt, dann dass - $H^{i}(\text{ker } p_n) = 0$ für alle $n \ge -i+2$. + $H^{j}(\text{ker } p_n) = 0$ für alle $n \ge -(j-1)+1 = -j+2$. Sei nun $m \in \Z$ beliebig. Dann setze $N \coloneqq -m + 1$. Dann ist für alle $n > N$: @@ -1718,36 +1837,103 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von für $n > N$ exakt. Das System \begin{equation*} \begin{tikzcd} - (I_n^{m-1})_{n\ge -1} \arrow{r} & - (I_n^{m})_{n\ge -1} \arrow{r} & - (I_n^{m+1})_{n\ge -1} \arrow{r} & - (I_n^{m+2})_{n\ge -1} + (I_n^{m-1})_{n \in \N} \arrow{r} & + (I_n^{m})_{n \in \N} \arrow{r} & + (I_n^{m+1})_{n \in \N} \arrow{r} & + (I_n^{m+2})_{n \in \N} \end{tikzcd} \end{equation*} erfüllt damit die Bedingungen von \ref{0.11}. Also ist die natürliche Abbildung \[ H^{m}(\com{I}) \longrightarrow H^{m}(\com{I}_N) - \] ein Isomorphismus. Erneute Betrachtung von \eqref{eq:diag-hi-in} für $i=m$ und + \] 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. \end{proof} -\begin{bem}[] +\begin{bem} Damit ist \ref{satz:r-mod-existence-k-proj-and-k-inj-resolutions} bewiesen. \end{bem} \newpage -\section{Adjunktion von abgeleitetem Hom und Tensorprodukt} +\section{Ableitungen und Adjunktion} + +Sei $R$ ein Ring und im Folgenden $\mathcal{A}$ die Kategorie der $R$ (links-)Moduln. + +\subsection{Abgeleitete $\com{\text{Hom}}$ Funktoren} -Sei $A$ ein kommutativer Ring und im Folgenden $\mathcal{A}$ die Kategorie der $A$-Moduln. +%\begin{satz}[] +% Sei $\com{I} \in \mathcal{K}$ K-injektiv und exakt. Dann ist $\com{\text{Hom}}(\com{M} , \com{I} )$ exakt für alle +% $\com{M} \in \mathcal{K}$. +% +% \label{satz:hom-exact-for-k-inj} +%\end{satz} +% +%\begin{proof} +% Sei $\com{M} \in \mathcal{K}$ beliebig. Dann existiert nach +% \ref{satz:existence-k-proj-resolution} ein $\com{P} \in \mathcal{K}$ K-projektiv und ein Quasiisomorphismus +% $\com{P} \to \com{M}$. Da $\com{I} $ K-injektiv, ist $\com{\text{Hom}}(-, \com{I})$ ein exakter Funktor, also +% folgt +% \begin{equation} +% H^{i}(\com{\text{Hom}}(\com{M}, \com{I})) = \com{\text{Hom}}(H^{i}(\com{M}), \com{I}) +% = \com{\text{Hom}}(H^{i}(\com{P}), \com{I}) +% = H^{i}(\com{\text{Hom}}(\com{P} , \com{I} )) +% \label{eq:cohom-groups-2} +% .\end{equation} +% Da $\com{P} $ K-projektiv und $\com{I} $ exakt, folgt die Exaktheit der rechten Seite und damit die Behauptung. +%\end{proof} -\subsection{K-flache Komplexe} +%Umdrehen der Pfeile liefert -Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Komplexen. +%\begin{satz}[] +% Sei $\com{P} \in \mathcal{K}$ K-projektiv und exakt. Dann ist $\com{\text{Hom}}(\com{P} , \com{M} )$ exakt für alle +% $\com{M} \in \mathcal{K}$. +% \label{satz:hom-exact-for-k-proj} +%\end{satz} -\begin{definition}[K-flacher Komplex] - Ein Komplex $\com{M} \in \mathcal{K}$ heißt K-flach, wenn für jeden exakten Komplex - $\com{S} \in \mathcal{K}$ auch $\com{M} \otimes_A \com{S}$ exakt ist. +\begin{satz} + Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\text{R}\com{\text{Hom}}(\com{M}, \com{N})$ 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} +\end{satz} + +\begin{proof} + In der Notation von \ref{satz:existence-derived-functors} wähle $\mathcal{L}$ als die volle Unterkategorie der + K-injektiven Komplexe von $\mathcal{K}$. Dann ist für $\com{M} \in \mathcal{K}$ beliebig: + \begin{enumerate}[(i)] + \item $\mathcal{L}$ ist trianguliert nach \ref{satz:k-proj-triangulated}. + \item Für alle $\com{N} \in \mathcal{K}$ existiert wegen \ref{satz:existence-k-inj-resolution} ein Quasiisomorphismus $\com{N} \to \com{I} $ + mit $\com{I} \in \mathcal{L}$. + \item Nach \ref{satz:hom-exact-for-k-inj} erhält $\com{\text{Hom}}(\com{M}, -)|_{\mathcal{L}}$ + Exaktheit von Komplexen. + \end{enumerate} + Also existiert R$\com{\text{Hom}}(\com{M} , -)$. Analog berechnet sich R$\com{\text{Hom}}(-, \com{N})$ für + $\com{N} \in \mathcal{K}$ unter Wahl von $\mathcal{L}$ + als die volle Unterkategorie der K-projektiven Komplexe von $\mathcal{K}$. Beide Ableitungen stimmen überein, denn + für $\com{M}, \com{N} \in \mathcal{K}$ beliebig und $\com{P} \to \com{M} $ und $\com{N} \to \com{I} $ K-projektive + bzw. K-injektive Auflösungen gilt mit wiederholter Anwendung von \ref{satz:existence-derived-functors} und + wegen $\com{P} \stackrel{\sim }{=} \com{M} $ und $\com{N} \stackrel{\sim }{=} \com{I} $ in $\mathcal{D}$: + \begin{align*} + \text{R}\com{\text{Hom}}(\com{M} , - )(\com{N}) + &= \text{R}\com{\text{Hom}}(\com{P} , -)(\com{I}) \\ + &= \com{\text{Hom}}(\com{P}, \com{I}) \\ + &= \text{R}\com{\text{Hom}}(-, \com{I} )(\com{P}) \\ + &= \text{R}\com{\text{Hom}}(-, \com{N} )(\com{M}) + .\end{align*} +\end{proof} + +\subsection{Abgeleitetes Tensorprodukt} + +Sei von nun an $R = A$ ein kommutativer Ring. Um das Tensorprodukt abzuleiten, benötigen wir noch +eine weitere Klasse von Komplexen in $\mathcal{K}$: + +\begin{definition}[K-flach] + Ein Komplex $\com{M} \in \mathcal{K}$ heißt K-flach, wenn der Funktor + $\com{M} \otimes_A -$ Exaktheit von Komplexen erhält. Eine K-flache + Auflösung eines Komplexes $\com{N} \in \mathcal{K}$ ist + ein Quasiisomorphismus $\com{M} \to \com{N} $ mit + $\com{P} \in \mathcal{K}$ K-flach. \end{definition} \begin{satz} @@ -1759,7 +1945,7 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom Sei $\com{M} \in \mathcal{K}$ wie im Satz. Dann ist für $\com{S} \in \mathcal{K}$ und $n \in \Z$: \[ - (\com{M} \otimes_A \com{S})^{n} = \bigoplus_{i+j=n} M^{i} \otimes_A S^{J} + (\com{M} \otimes_A \com{S})^{n} = \bigoplus_{i+j=n} M^{i} \otimes_A S^{j} = M^{0} \otimes S^{n} = (M^{0} \otimes \com{S} )^{n} \] und für $m \in M^{0}$, $s \in S^{n}$: \[ @@ -1788,9 +1974,9 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \text{Mor}_{\mathcal{D}}(\com{A} , \com{I} ) \stackrel{\ref{satz:mork=mord-for-k-inj}}{=} \text{Mor}_{\mathcal{K}}(\com{A} , \com{I} ) = 0 .\] - Mit Yoneda folgt nun, dass $\com{A} \stackrel{\sim }{=} 0$ in $\mathcal{D}$. Da per Definition + Mit Yoneda folgt nun, dass $\com{A} \stackrel{\sim }{=} 0$ in $\mathcal{D}$. Da $H^{i}(-)\colon \mathcal{K} \to \mathcal{A}b$ über den kanonischen Funktor $Q\colon \mathcal{K} \to \mathcal{D}$ - faktorisiert, folgt $H^{i}(\com{A}) = 0$ für $i \in \Z$, also $\com{A}$ exakt. + faktorisiert, folgt $H^{i}(\com{A}) = 0$ für $i \in \Z$, also ist $\com{A}$ exakt. \end{proof} \begin{satz}[] @@ -1812,12 +1998,12 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe .\] Weil $\com{M} $ K-flach ist, folgt $\com{S} \otimes_A \com{M} $ exakt, also wegen $\com{I} $ K-injektiv, die Behauptung. - (ii)$\implies$(i): Sei $\com{S} $ azyklisch. $A$-Mod hat genügend Injektive, also genügt es wegen \ref{lemma:0.10} zu - zeigen, dass für jeden injektiven $A$-Modul $I$, $\com{\text{Hom}}(\com{S} \otimes_A \com{M}, I) $ exakt ist. Dazu - sei $I$ injektiver $A$-Modul und $\com{I} = [ \cdots \to 0 \to I \to 0 \cdots]$. - Dann ist nach \ref{satz:single-degree-compl-k-proj} $\com{I} $ K-injektiv und damit + (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{\text{Hom}}(\com{S} \otimes_A \com{M}, I) $ exakt ist. Dazu + sei $\com{I}$ ein K-injektiver Komplex. Dann ist \[ - \com{\text{Hom}}(\com{S} \otimes_A \com{M}, I) = \com{\text{Hom}}(\com{S} \otimes_A \com{M}, \com{I}) + \com{\text{Hom}}(\com{S} \otimes_A \com{M}, \com{I}) \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \com{\text{Hom}}(\underbrace{\com{S}}_{\text{exakt}}, \underbrace{\com{\text{Hom}}(\com{M}, \com{I})}_{\text{K-injektiv}}) \] exakt. \end{proof} @@ -1825,7 +2011,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \begin{satz}[] \begin{enumerate}[(a)] \item Falls $\com{M}, \com{N} \in \mathcal{K}$ K-flach sind, dann ist - auch $\com{A} \otimes_A \com{B} $ K-flach. + auch $\com{M} \otimes_A \com{N} $ K-flach. \item $\com{M} \in \mathcal{K}$ ist K-flach genau dann wenn $\com{M}[1]$ K-flach ist. \item Falls zwei Punkte eines ausgezeichneten Dreiecks in $\mathcal{K}$ K-flach @@ -1847,12 +2033,12 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \] und die rechte Seite ist exakt. \item Seien $\com{S}, \com{M} \in \mathcal{K}$. Dann sind $- \otimes_A \com{S} $ und $\com{M} \otimes_A -$ nach - \ref{satz:tor-is-triangulated} ein triangulierter Funktor, also folgt + \ref{satz:tor-is-triangulated} triangulierte Funktoren, also folgt \[ \com{M}[1] \otimes_A \com{S} = (\com{M} \otimes_A \com{S})[1] = \com{M} \otimes_A \com{S}[1] - .\] Da Verschieben Exaktheit erhält folgt daraus die Äquivalenz. + .\] 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} @@ -1875,6 +2061,7 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \begin{satz}[] Sei $\com{M} \in \mathcal{K}$ K-projektiv. Dann ist $\com{M} $ K-flach. + Insbesondere hat jeder Komplex $\com{N} \in \mathcal{K}$ eine K-flache Auflösung. \label{satz:k-proj-is-k-flat} \end{satz} @@ -1884,7 +2071,8 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \com{\text{Hom}} (\com{M} \otimes_A \com{S}, \com{I}) \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \com{\text{Hom}}(\com{M}, \com{\text{Hom}}(\com{S}, \com{I} ) .\] Es ist $\com{\text{Hom}}(\com{S}, \com{I})$ exakt, da $\com{I} $ K-injektiv und damit die rechte Seite, da - $\com{M} $ K-projektiv ist. Also folgt die Behauptung mit \ref{lemma:0.10}. + $\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}. \end{proof} \begin{satz}[] @@ -1895,103 +2083,40 @@ Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexe \begin{proof} Sei $\com{M} \in \mathcal{K}$ K-flach und exakt und sei $\com{N} \in \mathcal{K}$ beliebig. Dann existiert nach - \ref{satz:existence-k-proj-resolution} ein $\com{P} \in \mathcal{K}$ K-projektiv und ein Quasiisomorphismus - $\com{P} \to \com{N} $. Da $\com{M} $ K-flach, ist $\com{M} \otimes_A -$ ein exakter Funktor also folgt + \ref{satz:k-proj-is-k-flat} ein K-flacher Komplex $\com{P} \in \mathcal{K}$ und ein Quasiisomorphismus + $\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}) = \com{M} \otimes_A H^{i}(\com{N}) - = \com{M} \otimes_A H^{i}(\com{P}) - = H^{i}(\com{M} \otimes_A \com{P}) + H^{i}(\com{M} \otimes_A \com{N}) = H^{i}(\com{M} \otimes_A \com{P}) \label{eq:cohom-groups-1} .\end{equation} - Wegen \ref{satz:k-proj-is-k-flat} ist $\com{P} $ K-flach, also folgt mit der Exaktheit von $\com{M} $, dass + Da $\com{P} $ K-flach ist, folgt mit der Exaktheit von $\com{M} $, dass $\com{M} \otimes_A \com{P} $ exakt ist. Damit folgt die Behauptung aus \eqref{eq:cohom-groups-1}. \end{proof} -\begin{satz}[] - Sei $\com{I} \in \mathcal{K}$ K-injektiv und exakt. Dann ist $\com{\text{Hom}}(\com{M} , \com{I} )$ exakt für alle - $\com{M} \in \mathcal{K}$. - - \label{satz:hom-exact-for-k-inj} -\end{satz} - -\begin{proof} - Sei $\com{M} \in \mathcal{K}$ beliebig. Dann existiert nach - \ref{satz:existence-k-proj-resolution} ein $\com{P} \in \mathcal{K}$ K-projektiv und ein Quasiisomorphismus - $\com{P} \to \com{M}$. Da $\com{I} $ K-injektiv, ist $\com{\text{Hom}}(-, \com{I})$ ein exakter Funktor, also - folgt - \begin{equation} - H^{i}(\com{\text{Hom}}(\com{M}, \com{I})) = \com{\text{Hom}}(H^{i}(\com{M}), \com{I}) - = \com{\text{Hom}}(H^{i}(\com{P}), \com{I}) - = H^{i}(\com{\text{Hom}}(\com{P} , \com{I} )) - \label{eq:cohom-groups-2} - .\end{equation} - Da $\com{P} $ K-projektiv und $\com{I} $ exakt, folgt die Exaktheit der rechten Seite und damit die Behauptung. -\end{proof} - -Umdrehen der Pfeile liefert +Damit erhalten wir: -\begin{satz}[] - Sei $\com{P} \in \mathcal{K}$ K-projektiv und exakt. Dann ist $\com{\text{Hom}}(\com{P} , \com{M} )$ exakt für alle - $\com{M} \in \mathcal{K}$. - \label{satz:hom-exact-for-k-proj} -\end{satz} - -\subsection{Abgeleitete $\com{\text{Hom}}$ Funktoren} - -\begin{satz}[] - Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\text{R}\com{\text{Hom}}(\com{M}, \com{N})$ 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} -\end{satz} - -\begin{proof} - In der Notation von \ref{satz:existence-derived-functors} wähle $\mathcal{L}$ als die volle Unterkategorie der - K-injektiven Komplexe von $\mathcal{K}$. Dann ist für $\com{M} $ beliebig: - \begin{enumerate}[(i)] - \item $\mathcal{L}$ ist trianguliert nach \ref{satz:k-proj-triangulated}. - \item Für alle $\com{N} \in \mathcal{K}$ existiert ein Quasiisomorphismus $\com{N} \to \com{I} $ wegen \ref{satz:existence-k-inj-resolution} - mit $\com{I} \in \mathcal{L}$. - \item Nach \ref{satz:hom-exact-for-k-inj} ist $\com{\text{Hom}}(\com{M}, -)|_{\mathcal{L}}$ exakt. - \end{enumerate} - Also existiert R$\com{\text{Hom}}(\com{M} , -)$. Analog berechnet sich R$\com{\text{Hom}}(-, \com{N})$ für - $\com{N} \in \mathcal{K}$ unter Wahl von $\mathcal{L}$ - als die volle Unterkategorie der K-projektiven Komplexe von $\mathcal{K}$. Beide Ableitungen stimmen überein, denn - für $\com{M}, \com{N} \in \mathcal{K}$ beliebig und $\com{P} \to \com{M} $ und $\com{N} \to \com{I} $ K-projektive - bzw. K-injektive Auflösungen gilt mit wiederholter Anwendung von \ref{satz:existence-derived-functors} und - wegen $\com{P} \stackrel{\sim }{=} \com{M} $ und $\com{N} \stackrel{\sim }{=} \com{I} $ in $\mathcal{D}$: - \begin{align*} - \text{R}\com{\text{Hom}}(\com{M} , - )(\com{N}) - &= \text{R}\com{\text{Hom}}(\com{P} , -)(\com{I}) \\ - &= \com{\text{Hom}}(\com{P}, \com{I}) \\ - &= \text{R}\com{\text{Hom}}(-, \com{I} )(\com{P}) \\ - &= \text{R}\com{\text{Hom}}(-, \com{N} )(\com{M}) - .\end{align*} -\end{proof} - -\subsection{Abgeleitetes Tensorprodukt} - -\begin{satz}[] +\begin{satz} Seien $\com{M}, \com{N} \in \mathcal{D}$. Dann ist $\com{M} \otimes_A^{\text{L}} \com{N}$ wohldefiniert und kann mithilfe einer K-flachen Auflösung einer der Faktoren berechnet werden. \label{satz:derived-tor} \end{satz} \begin{proof} - Erneut in der Notation von \ref{satz:existence-derived-functors} wähle $\mathcal{L}$ + In der Notation (der Linksableitungsversion) von \ref{satz:existence-derived-functors} wähle $\mathcal{L}$ als die volle Unterkategorie der K-flachen Komplexe von $\mathcal{K}$. Dann ist für $\com{N}$ beliebig: \begin{enumerate}[(i)] \item $\mathcal{L}$ ist trianguliert nach \ref{satz:k-flat-triangulated}. - \item Für alle $\com{M} \in \mathcal{K}$ existiert nach - \ref{satz:existence-k-proj-resolution} und \ref{satz:k-proj-is-k-flat} + \item Für alle $\com{M} \in \mathcal{K}$ existiert nach \ref{satz:k-proj-is-k-flat} ein Quasiisomorphismus $\com{P} \to \com{M}$ mit $\com{P} \in \mathcal{L}$. - \item Nach \ref{satz:tor-exact-for-k-flat} ist - $\com{N} \otimes_A -|_{\mathcal{L}}$ exakt. + \item Nach \ref{satz:tor-exact-for-k-flat} erhält + $\com{N} \otimes_A -|_{\mathcal{L}}$ Exaktheit von Komplexen. \end{enumerate} - Also existiert $\com{N} \otimes_A^{\text{L}} -$ und analog für - $- \otimes_A^{L} \com{N}$. + Also existiert $\com{N} \otimes_A^{\text{L}} -$. 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} \subsection{Adjunktion} @@ -2010,7 +2135,7 @@ Jetzt können wir alles zusammentragen und erhalten: \begin{proof} Nach \ref{satz:derived-hom} und \ref{satz:derived-tor} sind alle Terme wohldefiniert - und wir können mit \ref{satz:existence-k-proj-resolution} und + und wir können mit \ref{satz:k-proj-is-k-flat} ohne Einschränkung annehmen, dass $\com{N} $ K-flach ist, und mit \ref{satz:existence-k-inj-resolution}, dass $\com{P} $ K-injektiv ist. @@ -2068,6 +2193,8 @@ Jetzt können wir alles zusammentragen und erhalten: Robin Hartshorne. Residues and duality. \emph{Lecture Notes in Math.}. 20, Springer-Verlag (1966) \bibitem{spaltenstein} N. Spaltenstein. Resolutions of unbounded complexes. \emph{Composito Mathematica 65}. (1988) +\bibitem{set-theoretic} +Charles A. Weibel. An introduction to homological algebra. \emph{Cambridge studies in advanced mathematics}. 38 (1988) \end{thebibliography} \end{document}