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z2_RLqODf|4GMeBTPc9ZfTtlSef`~AEY++}q7zgf^V-$;laC{-GPn;EQLDEEl9wns5 zI%g>`H}BWz*ElmRSWvtwP8i}U-Iq4li&vujY_=V0EwNyAW70gHLr|)T#eJZE%Q%q! zw{VYw&~mL*`J3AabU%SUz^JNl&M^@6mYpvtO&3uqx%*XlIy?rlfWBH(y~VabOMA4a zwO+=<`-6#XVHd*H3-m^| ziRhfu$+_>9ES5DjwFPvee^y5&5)`-pv_mYbNKkzwRMhu488$Ttllj?v^9R#yn}g*e&Q`~#&iN_w9)%CqG8leb@^!XTFIspj zP4y0vb~7pA=*ou2n_aNmhBiM*qV+a2(5e1Nz66?vR>`fxn?%-Vl~+B zCkJ^nGMM)z~Ce#CBAZE0({o)K 1$, 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 + \[ + 0 \to \com{C}_n \to \com{I}_n \to \com{I} _{n-1} \to 0 + \] zerfällt stufenweise. + \end{enumerate} + \item Die Klasse $\mathcal{J}$ heißt 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 ist zu einem Komplex in $\mathcal{J}$, bereits in $\mathcal{J}$ ist. + \end{enumerate} + +\end{definition} + +Im Folgenden möchten wir zeigen, dass die Klasse der K-projektiven Komplexe +abgeschlossen unter speziellen inversen Limites ist. Dazu benötigen wir die folgenden Lemmata: + +% TODO: beispiel funktioniert nicht mit N als indexmenge, wird nicht benoetigt +%\item Sei $\mathcal{J}$ eine Klasse von Komplexen abgeschlossen unter speziellen inversen Limites mit +% $\com{A} \in \mathcal{J} \iff \com{A}[1] \in \mathcal{J}$. Dann ist für $\com{A}, \com{B} \in \mathcal{J}$ +% und $u \colon \com{A} \to \com{B} $, auch $\com{C}_u$ in $\mathcal{J}$. Denn +% \[ +% \com{C}_u \to \com{A}[1] \to 0 +% \] ist ein $\mathcal{J}$-spezielles inverses System mit Limes $\com{C}_u$. + +\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 + 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. + + \label{lemma:bounded-compl-in-complete-class} +\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 $A^{i} = 0$ für alle $i < 0$. Dann sind die Spalten des nachstehenden Diagramms ein + $\mathcal{J}$-spezielles inverses System $(\com{S}_n)_{n \ge 0}$ mit + Übergangsabbildungen $p_n$, + \[ + \begin{tikzcd} + \cdots \arrow{r} & 0 \arrow{r} \arrow{d} & 0 \arrow{r} \arrow{d} & 0 \arrow{r} \arrow{d} & 0 \arrow{d}\\ + \cdots\arrow{r} & A^{0} \arrow{d} \arrow{r} & A^{0} \arrow{d} \arrow{r} & A^{0} \arrow{d} \arrow{r} & 0 \arrow{d}\\ + \cdots\arrow{r} & A^{1} \arrow{d} \arrow{r} & A^{1} \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d}\\ + \cdots\arrow{r} & A^{2} \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & 0 \arrow{d} \\ + & \vdots & \vdots & \vdots & \vdots \\ + \end{tikzcd} + \] 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 + 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 + $\com{A} = \lim \com{S}_n \in \mathcal{J}$. +\end{proof} + +\begin{lemma} + Die Klasse der exakten Komplexe in $\mathcal{K}(\mathcal{A}b)$ ist abgeschlossen unter speziellen inversen Limites. + + \label{lemma:exact-comp-complete-inv} +\end{lemma} + +\begin{proof} + Sei $(\com{S}_n)_{n \in \N}$ ein spezielles inverses System in $\mathcal{K}(\mathcal{A}b)$. Für $i \in \Z$ + erfüllt $(S_n^{i})_{n \in \N}$ die Bedingung P aus \ref{0.11}. Sei also $i \in \Z$ beliebig. Dann erfüllt + \[ + (S_n^{i-1})_{n \in \N} \to (S_n^{i})_{n \in \N} \to (S_n^{i+1})_{n \in \N} \to (S_n^{i+2})_{n \in \N} + \] die Bedingungen von \ref{0.11}, da nach Voraussetzung für alle $n > 1$ $\text{ker}(\com{S}_n \to \com{S}_{n-1})$ + exakt ist. Also ist + \[ + \begin{tikzcd} + \lim S_n^{i-1} \arrow{r} \arrow{d}{=} & \lim S_n^{i} \arrow{d}{=} \arrow{r} & \lim S_n^{i+1} \arrow{d}{=}\\ + (\lim S_n)^{i-1} \arrow{r} & (\lim S_n)^{i} \arrow{r} & (\lim S_n)^{i+1} + \end{tikzcd} + \] exakt. Da $i \in \Z$ beliebig, folgt $\lim S_n$ exakt. +\end{proof} + +\begin{satz} + Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(B)$ 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 + gradweise zerfallende kurze exakte Folgen erhält. + + Dann ist $F^{-1}(\mathcal{J}) \subset \mathcal{K}(\mathcal{A})$ abgeschlossen unter speziellen inversen Limites. + + \label{satz:complete-inv-system-functor} +\end{satz} + +\begin{proof} + Sei $(\com{S}_n)_{n \in \N}$ ein $F^{-1}(\mathcal{J})$-spezielles inverses System. Dann ist + $(F(\com{S}_n))_{n \in \N}$ ein $\mathcal{J}$-spezielles System, denn + \begin{enumerate}[(i)] + \item $F(S_1) = F(1) = 0$, da $F$ mit inversen Limites vertauscht und die Null der Limes des leeren Diagramms + ist. + \item Für $n > 1$ ist nach Voraussetzung + \[ + \begin{tikzcd} + 0 \arrow{r} & \text{ker } p_n \arrow{r} & \com{S}_n \arrow{r}{p_n} & \com{S}_{n-1} \arrow{r} & 0 + \end{tikzcd} + \] + exakt, zerfällt gradweise und $\text{ker } p_n$ ist in $F^{-1}(\mathcal{J})$. Nach Voraussetzung ist damit + \[ + \begin{tikzcd} + 0 \arrow{r} & F(\text{ker } p_n) \arrow{r} & F(\com{S}_n) \arrow{r}{F(p_n)} & F(\com{S}_{n-1}) \arrow{r} & 0 + \end{tikzcd} + \] exakt und zerfällt gradweise. Aus der Exaktheit folgt damit auch + $\text{ker } F(p_n) = F(\text{ker } p_n)$, also $\text{ker } F(p_n) \in \mathcal{J}$. + \end{enumerate} + Also $F(\lim \com{S}_n) = \lim F(\com{S}_n) \in \mathcal{J}$ und damit $\lim \com{S}_n \in F^{-1}(\mathcal{J})$. +\end{proof} + +\begin{korollar}[] + 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 + $\com{\text{Hom}}(\com{T}, \com{A})$ exakt ist für alle $\com{T} \in \mathcal{I}$, abgeschlossen unter speziellen + inversen Limites. Insbesondere ist die Klasse der K-injektiven Komplexe abgeschlossen unter speziellen inversen + Limites. +\end{korollar} + +\begin{proof} + Sei $\mathcal{J}$ die Klasse der exakten Komplexe und für $\com{T} \in \mathcal{I}$ sei + $\mathcal{E}_{\com{T}}$ die Klasse der Komplexe $\com{A} $, sodass + $\com{\text{Hom}}(\com{T}, \com{A})$ exakt ist. Dann + ist $\mathcal{E}_{\com{T}} = \com{\text{Hom}}(\com{T}, -)^{-1}(\mathcal{J})$. $\com{\text{Hom}}(\com{T}, -)$ erfüllt + die Voraussetzungen von \ref{satz:complete-inv-system-functor}, denn: + + \begin{enumerate}[(i)] + \item Nach \ref{lemma:exact-comp-complete-inv} ist + $\mathcal{J}$ abgeschlossen unter speziellen inversen Limites. + \item Wegen \ref{satz:adjunction-hom-tor-comp} ist $\com{\text{Hom}} (\com{T}, -)$ rechtsadjungiert und + vertauscht daher mit Limites. Außerdem ist $\com{\text{Hom}}(\com{T}, -)$ gradweise additiv, erhält also + gradweise zerfallende Folgen. + \end{enumerate} + Also ist $\mathcal{E}_{\com{T}}$ und damit $\bigcap_{\com{T} \in \mathcal{I}} \mathcal{E}_{\com{T} }$ + abgeschlossen unter speziellen inversen Limites. + + Das Insbesondere folgt wenn $\mathcal{I} = \mathcal{J}$ gesetzt wird. +\end{proof} + +Wenn wir alle Pfeile umdrehen erhalten wir die folgende Definition und Ergebnisse: + +\begin{definition}[Spezielles direktes System] + Sei $\mathcal{P} \subset \mathcal{K}$ eine Klasse von Komplexen. + \begin{enumerate}[(a)] + \item Ein direktes System $(\com{P}_n)_{n \in \N}$ in $\mathcal{K}$ heißt $\mathcal{P}$-spezielles + direktes System, falls folgende Bedingungen erfüllt sind: + \begin{enumerate}[(i)] + \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 + \[ + 0 \to \com{P}_{n-1} \to \com{P}_{n} \to \com{C}_n \to 0 + \] zerfällt stufenweise. + \end{enumerate} + \item Die Klasse $\mathcal{P}$ heißt abgeschlossen unter speziellen direkten Colimites, falls jedes + $\mathcal{P}$-spezielle direkte System in $\mathcal{K}$ einen Colimes in $\mathcal{P}$ besitzt und jeder + Komplex in $\mathcal{K}$, der isomorph ist zu einem Komplex in $\mathcal{P}$, bereits in $\mathcal{P}$ ist. + \end{enumerate} +\end{definition} + +Durch Umdrehen aller Pfeile, erhalten wir auch eine duale Version von \ref{lemma:bounded-compl-in-complete-class}. +Ebenfalls analog gilt: + +% brauche ich nicht +%\begin{lemma} +% Die Klasse der exakten Komplexe in $\mathcal{K}(\mathcal{A}b)$ ist abgeschlossen unter speziellen direkten Colimites. +% +% \label{lemma:exact-comp-complete-inv} +%\end{lemma} +% +%\begin{proof} +% +%\end{proof} + +\begin{satz} + Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(B)$ abgeschlossen + unter speziellen inversen Colimites. Angenommen direkte Colimites existieren in $\mathcal{A}$ und sei + $F\colon \mathcal{K}(A) \to \mathcal{K}(B)$ ein kontravarianter Funktor, der direkte Colimites 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 Colimites. + + \label{satz:complete-inv-system-functor} +\end{satz} + +\begin{korollar}[] + Sei $\mathcal{I}$ eine Klasse von Komplexen und angenommen in $\mathcal{A}$ existieren direkte Colimites. + + Dann ist die Klasse aller Komplexe $\com{A} \in \mathcal{K}$, sodass + $\com{\text{Hom}}(\com{A}, \com{T})$ exakt ist für alle $\com{T} \in \mathcal{I}$, abgeschlossen unter speziellen + direkten Colimites. Insbesondere ist die Klasse der K-projektiven Komplexe abgeschlossen unter speziellen direkten + Colimites. + \label{kor:k-proj-closed} +\end{korollar} + + +\begin{definition}[] + 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. +\end{definition} + +\subsection{Existenz von K-projektiven und K-injektiven Auflösungen} + +Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. Die folgenden Bedingungen an $\mathcal{P}$ sind +äquivalent: + +\begin{enumerate}[(1)] + \item Jeder nach oben beschränkte Komplex $\com{A} \in \mathcal{K}$ hat eine Auflösung $\com{P} \to \com{A} $ + nach links mit $\com{P} \in \mathcal{P}$. + \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} + +\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} + +Im Folgenden nehmen wir an, dass $\mathcal{P}$ die Eigenschaft (1) erfüllt. + +\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. + + 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 + abgeschlossen unter speziellen direkten Colimites ist, folgt mit dem Dual von + \ref{lemma:bounded-compl-in-complete-class}, dass $\com{P} $ K-projektiv ist. + + Erneut nach \ref{kor:k-proj-closed} sind die Komplexe in $\underset{\rightarrow}{\mathcal{P}}$ damit ebenfalls + $K$-projektiv. +\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 + 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 $n \ge 0$. +\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. Dann + setze $\com{P} = \com{P}_{n-1}$, $\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 $f = a_{n-1}f_{n-1}$. + + Da $\com{B} = \tau_{\le n}\com{A} $ nach oben beschränkt ist und $C_f^{i} = P^{i+1} \oplus B^{i}$. + + Mit (1) existiert ein Quasiisomorphismus $g\colon \com{Q} \to \com{C}_f[-1] $ mit $\com{Q}[1] \in \mathcal{P}$. +\end{proof} + \newpage \section{Adjunktion von abgeleitetem Hom und Tensorprodukt}