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z%!7Rl@>Yis?wA+^)rVG{zm%o!1#BaP<`VqV5^0v2UhSI|aE>|enRRf3JcK}y6Ohvo z=%tFaHusz^HNv&z!reJOaKZF;bAjqMQ#%Zrc>#fvURawh?KqEkPk+nk zf^~l~vP$aGYTc8f=x-LML1wB%4g-DpGeQm1ckL4^6rXyF6=c*2Zf3OEZffwQI7m+*{ysEc%FC~RI4T&7b@Mm|F zJq7RBZW;b)45n!`-rfvUQKlW!7`UXJg>_JV#6wT=!cJhGd-25XzLxvpu7E{+ z%E!5o5Mn)0t&AARBf8LQd`LX}5x|3YQ{(nLRi}{pYj n$ und ein Komplexhomomorphismus $f\colon \com{P} \to \com{A} $, der - einen Isomorphismus $H^{i}(\com{P}) \to H^{i}(\com{A})$ induziert für $i \le n$. +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} -\begin{proof} - (1) $\implies$ (2): 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. - - (2) $\implies$ (1): Sei $\com{A} \in \mathcal{K}^{-}$. Ohne Einschränkung ist $A^{i} = 0$ für alle $i > 0$. Dann - existiert für $n = 0$ 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} +% 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} -Im Folgenden nehmen wir an, dass $\mathcal{P}$ die Eigenschaft (1) erfüllt. +%\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{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. +\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 + %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 @@ -1367,35 +1399,36 @@ Im Folgenden nehmen wir an, dass $\mathcal{P}$ die Eigenschaft (1) erfüllt. \label{bsp:bounded-above-projectives} \end{bsp} -Unser Ziel ist nun für einen gegebenen Komplex $\com{A}$ eine Auflösung nach Links durch einen Komplex -aus $\mathcal{P}$ zu konstruieren. Dazu schneiden wir den Komplex nach oben ab und lösen schrittweise auf. Das -folgende Lemma führt diese Konstruktion aus. - \begin{lemma} - Sei $\com{A} \in \mathcal{K}$. Dann existiert ein $\mathcal{P}$-spezielles direktes System $(\com{P}_n)_{n\ge -1}$ und + Sei $\com{A} \in \mathcal{K}$. Dann existiert ein $\mathcal{P}$-spezielles direktes System $(\com{P}_n)_{n \in \N}$ und ein direktes System von Kettenhomomorphismen $f_n \colon \com{P}_n \to \tau_{\le n}\com{A}$, sodass - $f_n$ ein Quasiisomorphismus ist für alle $n \ge 0$. + $f_n$ ein Quasiisomorphismus ist für alle $n \ge 2$. \label{lemma:constr-dir-system} \end{lemma} \begin{proof} - Wir gehen induktiv vor. Setze $\com{P}_{-1} = 0$ und $f_{-1} = 0$. Nach (1) existiert ein Quasiisomorphismus - $f_0 \colon \com{P}_0 \to \tau_{\le 0}\com{A} $ mit $\com{P}_0 \in \mathcal{P}$. - - Sei nun $n \ge 1$ und seien $\com{P}_{-1}, \ldots, \com{P}_{n-1}$ und $f_{-1}, \ldots, f_{n-1}$ konstruiert mit $\com{P}_i \in \mathcal{K}^{-}$. Dann - setze $\com{P} = \com{P}_{n-1}$, $\com{B} = \tau_{\le n} \com{A} $. Es sei + 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. + + 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 + 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 \begin{equation} f d_P = d_B f \label{eq:f-comp-hom} \end{equation} - Da $\com{B} = \tau_{\le n}\com{A}$ und $\com{P} $ nach oben beschränkt sind und - $C_f^{i} = P^{i+1} \oplus B^{i}$, existiert mit (1) 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 gradweise - $C_f^{i}[-1] = P^{i} \oplus B^{i-1}$ ist $g = (g_i)_{i \in \Z}$ ist $g$ gradweise - gegeben durch $g'_i\colon Q^{i} \to P^{i} $ und $g''_i\colon Q^{i} \to B^{i-1}$. + 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 + $C_f^{i}[-1] = P^{i} \oplus B^{i-1}$ 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 $\mathcal{A}$. Betrachte für $i \in \Z$ das folgende kommutative Diagramm: \begin{equation} @@ -1413,7 +1446,7 @@ folgende Lemma führt diese Konstruktion aus. \intertext{Also folgt} d_{C_f}[-1] &= - d_{C_f} = \begin{pmatrix} -d_{P}[1] & 0 \\ -f[1] & -d_{B} \end{pmatrix} .\end{align*} - Auswerten von \eqref{eq:1} in beiden Summanden liefert nun + 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''} @@ -1448,7 +1481,7 @@ folgende Lemma führt diese Konstruktion aus. \end{pmatrix} \\ &= d_B h .\end{salign*} - Also ist $h$ Komplexhomomorphismus. Bleibt zu zeigen, dass $h$ ein Quasiisomorphismus + 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: $\com{C}_h = \com{C}_{-g}[1]$. @@ -1511,7 +1544,7 @@ folgende Lemma führt diese Konstruktion aus. & Q^{i+1} \arrow{r} & 0 \end{tikzcd} .\] - Also ist $(\com{P}_n)_{n \ge -1}$ ein $\mathcal{P}$-spezielles direktes System. + Also ist $(\com{P}_n)_{n \in \N}$ ein $\mathcal{P}$-spezielles direktes System. Außerdem ist nach Definition von $h$: $f_n p_{n-1} = h p_{n-1} = f = a_{n-1} f_{n-1}$, also kommutiert \[ @@ -1520,7 +1553,7 @@ folgende Lemma führt diese Konstruktion aus. & \com{P}_{n} \arrow{d}{f_n = h} \\ \tau_{\le n-1}\com{A} \arrow{r}{a_{n-1}} & \tau_{\le n}\com{A} \end{tikzcd} - \] und $(f_n)_{n \ge -1}$ ist ein direktes System. + \] und $(f_n)_{n \in \N}$ ist ein direktes System. \end{proof} Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: @@ -1528,7 +1561,6 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: \begin{satz} Angenommen direkte Colimites existieren in $\mathcal{A}$ und $\colim$ ist exakt. - Dann hat jeder Komplex in $\mathcal{K}$ eine $\underset{\rightarrow}{\mathcal{P}}$-Linksauflösung. @@ -1555,7 +1587,6 @@ 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. \label{satz:existence-k-proj-resolution} \end{korollar} @@ -1567,18 +1598,18 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: \subsubsection{Rechtsauflösungen} -Sei $\mathcal{J}$ eine Klasse von Komplexen in $\mathcal{K}$. Im Folgenden nehmen wir an, -dass $\mathcal{J}$ die folgende Eigenschaft erfüllt: +Sei $\mathcal{I}$ eine Klasse von Komplexen in $\mathcal{K}$. Im Folgenden nehmen wir an, +dass $\mathcal{I}$ die folgende Eigenschaft erfüllt: \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 $\com{I} \in \mathcal{J}$ und + Auflösung $\com{A} \to \com{I} $ nach rechts mit $\com{I} \in \mathcal{I}$ und $\com{I}$ nach unten beschränkt. \end{enumerate} \begin{bsp} Falls $\mathcal{A}$ genügend Injektive hat, können wir dual zu Beispiel - \ref{bsp:bounded-above-projectives} $\mathcal{J}$ als die Klasse + \ref{bsp:bounded-above-projectives} $\mathcal{I}$ als die Klasse der nach unten beschränkten Komplexe mit in $\mathcal{A}$ injektiven Objekten wählen. \end{bsp} @@ -1586,7 +1617,7 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \ref{lemma:constr-dir-system} und \ref{satz:existence-left-resolutions}: \begin{lemma}[] - Sei $\com{A} \in \mathcal{K}$. Dann existiert ein $\mathcal{J}$-spezielles + 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 Kettenhomomorphismen $f_n\colon \tau^{\ge-n}\com{A} \to \com{I}_n$, sodass $f_n$ ein Quasiisomorphismus ist. @@ -1596,7 +1627,6 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \begin{satz}[] Angenommen inverse Limites existieren in $\mathcal{A}$ und $\lim$ ist exakt. - Dann hat jeder Komplex in $\mathcal{K}$ eine $\underset{\rightarrow}{\mathcal{P}}$-Rechtsauflösung. @@ -1605,7 +1635,7 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \begin{bem} Leider findet \ref{satz:existence-right-resolutions} in $\mathcal{A} = R\text{-Mod}$ für - $R$ ein Ring keine Anwendung, da hier $\lim$ nicht exakt ist. + einen Ring $R$ keine Anwendung, da hier $\lim$ nicht exakt ist. 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. @@ -1701,6 +1731,10 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von $n=N$ liefert nun, dass $H^{i}(f)$ ein Isomorphismus ist. \end{proof} +\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}