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zz(y1ZyCdcaM-nEpqw0x7@_Bqpju$~0G=GgGh&Ta;6r+0|ns)`AXF*%&eaebKS+_l@q+*{`SL@x>olCz}-O=j${ErVPghp{Ebt5`~>TTVE>xd*y%L zr^~p*#WBOdVP%C;dllm7f6pon|EpGu5MDk%A~1+>en@F5&k^bhV1-0jSNokO|DfLZ zvurHGL>b*DYK1izrXWY?;CvO;7tmneSD%iSb`2?-fAF)L8x?)N=K=OO(V@ zenV|9599Hy0C@#p@Z^siDWLq~nKbSYo8O%jSrKGi5vW_SoO1t5HMnSt*LuJIvkNHJ zV`ze`wIOW^@pTKZmq6Z@!N@Zx#53@p?P?pwvI4PLkL_K6`%*v(W2C~fiYv2jbs1oA zdKOS(9g(KGJ$+;!wuZL+ z)uN5M?$`}2mWIw4sv%=l>q6uQT~))vRzB9E?& I_{\text{min}}$ ist die Abbildung $M_i \to M_{i-1}$ surjektiv. + \item Für $n > 1$ ist die Abbildung $M_n \to M_{n-1}$ surjektiv. \end{enumerate} + %Weiter sagen wir ein inverses System $(M_i)_{i \in I}$ in $\mathcal{A}b$ genüge der Bedingung + %(R), wenn folgende Eigenschaften erfüllt sind: + %\begin{enumerate}[(i)] + % \item $I$ genügt Bedingung (S). + % \item $M_1 = 0$. + % \item Für $i > I_{\text{min}}$ ist die Abbildung $M_i \to M_{i-1}$ surjektiv. + %\end{enumerate} + \label{def:cond-r} \end{definition} +\begin{bsp} + Spezielle inverse Systeme erfüllen (R). +\end{bsp} + \begin{lemma} - Sei $I$ eine Indexmenge, die Bedingung (S) genügt und seien - $(A_i)_{i \in I}$, $(B_{i})_{i \in I}$, $(C_i)_{i \in I}$ und $(D_i)_{i \in I}$ + Seien + $(A_n)_{n \in \N}$, $(B_{n})_{n \in \N}$, $(C_n)_{n \in \N}$ und $(D_n)_{n \in \N}$ inverse Systeme in $\mathcal{A}b$, die (R) erfüllen und seien \begin{equation} \begin{tikzcd} - (A_i)_{i \in I} \arrow{r}{f_i} & (B_i)_{i \in I} \arrow{r}{g_i} & - (C_i)_{i \in I} \arrow{r}{h_i} & (D_i)_{i \in I} + (A_n)_{n \in \N} \arrow{r}{(f_n)_{n \in \N}} & (B_n)_{n \in \N} \arrow{r}{(g_n)_{n \in \N}} & + (C_n)_{n \in \N} \arrow{r}{(h_n)_{n \in \N}} & (D_n)_{n \in \N} \end{tikzcd} \label{eq:0.11-inv-systems} \end{equation} - Morphismen von inversen Systemen mit $g_i \circ f_i = 0 = h_i \circ g_i$ - für $i \in I$ und sei + Morphismen von inversen Systemen mit $g_n \circ f_n = 0 = h_n \circ g_n$ + für $n \in \N$ und sei \[ \begin{tikzcd} A \arrow{r}{f} & B \arrow{r}{g} & C \arrow{r}{h} & D \end{tikzcd} - \] der Limes von \eqref{eq:0.11-inv-systems}. Für $i \in I$ mit $i > I_{\text{min}}$ - seien $A_i'$, $B_i'$, $C_i'$ und $D_i'$ die jeweiligen Kerne - der Übergangsabbildungen $A_i \to A_{i-1}$, $B_i \to B_{i-1}$, $C_i \to C_{i-1}$ - und $D_i \to D_{i-1}$. + \] der Limes von \eqref{eq:0.11-inv-systems}. Für $n \in \N$ mit $n > 1$ + seien $A_n'$, $B_n'$, $C_n'$ und $D_n'$ die jeweiligen Kerne + 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 $j \in I$, s.d. für alle $i > j$ die Folge + Sei weiter $N \in \N$, s.d. für alle $n > N$ die Folge \[ \begin{tikzcd} - A_i' \arrow{r} & B_i' \arrow{r} & C_i' \arrow{r} & D_i' + A_n' \arrow{r} & B_n' \arrow{r} & C_n' \arrow{r} & D_n' \end{tikzcd} \] exakt ist. Dann ist die natürliche Abbildung \[ - \text{ker } g / \text{im } f \longrightarrow \text{ker } g_j / \text{im } f_j + \text{ker } g / \text{im } f \longrightarrow \text{ker } g_N / \text{im } f_N \] ein Isomorphismus. \label{0.11} \end{lemma} \begin{proof} - Durch Umbenennung können wir ohne Einschränkung annehmen, dass $I = \N$. Sei - $j \in \N$ mit der beschriebenen Eigenschaft. Dann betrachte das kommutative Diagramm: + Sei + $N \in \N$ mit der beschriebenen Eigenschaft. Dann betrachte das folgende kommutative Diagramm und + mache Diagrammjagd. \begin{equation} \begin{tikzcd} - A \arrow{r}{f} \arrow{d} & \text{im } f \arrow{r} - & \text{ker } g \arrow{r} \arrow{d} + A \arrow[twoheadrightarrow]{r}{f} \arrow{d} & \text{im } f \arrow[hookrightarrow]{r} + & \text{ker } g \arrow[hookrightarrow]{r} \arrow{d} & B \arrow{r}{g} \arrow{d} & C \arrow{r}{h} \arrow{d} & D \arrow{d} \\ - A_j \arrow{r}{f_j} & \text{im } f_j \arrow{r} - & \text{ker } g_j \arrow{r} - & B_j \arrow{r}{g_j} - & C_j \arrow{r}{h_j} - & D_j \\ - A_{j+1} \arrow{r}{f_{j+1}} \arrow{u}{p_{A}} & \text{im } f_{j+1} \arrow{r} \arrow{u} - & \text{ker } g_{j+1} \arrow{r} \arrow{u} - & B_{j+1} \arrow{r}{g_{j+1}} \arrow{u}{p_B} - & C_{j+1} \arrow{r}{h_{j+1}} \arrow{u}{p_C} - & D_{j+1} \arrow{u}{p_D} \\ - \text{ker } p_{A_{j+1}} \arrow[from=4-1, to=4-4] \arrow{u} & & - & \text{ker } p_{B_{j+1}} \arrow{r} \arrow{u} - & \text{ker } p_{C_{j+1}} \arrow{r} \arrow{u} - & \text{ker } p_{D_{j+1}} \arrow{u} \\ + A_N \arrow[twoheadrightarrow]{r}{f_N} & \text{im } f_N \arrow[hookrightarrow]{r} + & \text{ker } g_N \arrow[hookrightarrow]{r} + & B_N \arrow{r}{g_N} + & C_N \arrow{r}{h_N} + & D_N \\ + A_{N+1} \arrow[twoheadrightarrow]{r}{f_{N+1}} \arrow[twoheadrightarrow]{u}{p_{A}} + & \text{im } f_{N+1} \arrow[hookrightarrow]{r} \arrow{u} + & \text{ker } g_{N+1} \arrow[hookrightarrow]{r} \arrow{u} + & B_{N+1} \arrow{r}{g_{N+1}} \arrow[twoheadrightarrow]{u}{p_B} + & C_{N+1} \arrow{r}{h_{N+1}} \arrow[twoheadrightarrow]{u}{p_C} + & D_{N+1} \arrow[twoheadrightarrow]{u}{p_D} \\ + \text{ker } p_{A} \arrow[from=4-1, to=4-4] \arrow[hookrightarrow]{u} & & + & \text{ker } p_{B} \arrow{r} \arrow[hookrightarrow]{u} + & \text{ker } p_{C} \arrow{r} \arrow[hookrightarrow]{u} + & \text{ker } p_{D} \arrow[hookrightarrow]{u} \\ \end{tikzcd} \label{eq:0.11-diag} \end{equation} - Injektivität: Sei $(b_i)_{i \in \N} \in \text{ker } g$, sodass $b_j \in \text{im }f_j$. - Dann existiert ein $a_j \in A_j$, sodass $f_j(a_j) = b_j$. Da $p_A$ surjektiv, - existiert ein $x \in A_{j+1}$, sodass $p_A(x) = a_j$. Sei - $y = f_{j+1}(x)$. Weil \eqref{eq:0.11-diag} kommutativ, - ist $p_{B}(y) = b_j$. Da $(b_i)_{i \in \N}$ ein kompatibles - System ist, gilt zudem $p_{B}(b_{j+1}) = b_j$. Also ist - $b_{j+1} - y \in \text{ker } p_{B}$. Weil $y, b_{j+1} \in \text{ker } g_{j+1}$, + Injektivität: Sei $(b_n)_{n \in \N} \in \text{ker } g$, sodass $b_N \in \text{im }f_N$. + Dann existiert ein $a_N \in A_N$, sodass $f_N(a_N) = b_N$. Da $p_A$ surjektiv ist, + existiert ein $x \in A_{N+1}$, sodass $p_A(x) = a_N$. Sei + $y = f_{N+1}(x)$. Weil \eqref{eq:0.11-diag} kommutativ ist, + folgt + \[ + p_B(y) = p_B(f_{N+1}(x)) = f_N(p_A(x)) = f_N(a_N) = b_N + .\] Da $(b_n)_{n \in \N}$ ein kompatibles + System ist, gilt zudem $p_{B}(b_{N+1}) = b_N$. Also ist + $b_{N+1} - y \in \text{ker } p_{B}$. Weil $y, b_{N+1} \in \text{ker } g_{N+1}$, existiert aufgrund der Exaktheit der unteren Zeile ein $\tilde{x} \in \text{ker } p_A$, - sodass $f_{j+1}(\tilde{x}) = b_{j+1} - y$. Nun - setze $a_{j+1} \coloneqq \tilde{x} + x$. Dann ist - $f_{j+1}(a_{j+1}) = b_{j+1}$ und $p_{A}(a_{j+1}) = a_j$, denn - $\tilde{x} \in \text{ker } p_{A}$. Konstruiere so induktiv eine kompatible - Familie $(a_{i})_{i\ge j}$ mit $f(a_i) = (b_i)_{i \ge j}$. Für $i < j$ setze - $a_i \coloneqq p_{A_{i+1}}(a_{i+1})$. Die Kommutativität von \eqref{eq:0.11-diag} - liefert dann ein kompatibles System $(a_i)_{i \in \N}$ mit - $f(a_{i}) = (b_{i})_{i \in \N}$ - - Surjektivität: Sei $b \in \text{ker } g_j$. Weil $p_B$ surjektiv, existiert dann ein - $y \in B_{j+1}$, sodass $p_B(y) = b$. Sei $z = g_{j+1}(y)$. + sodass $f_{N+1}(\tilde{x}) = b_{N+1} - y$. Nun + setze $a_{N+1} \coloneqq \tilde{x} + x$. Dann ist + \[ + f_{N+1}(a_{N+1}) = f_{N+1}(\tilde{x} + x) = b_{N+1} - y + y = b_{N+1} + \] + und + \[ + p_{A}(a_{N+1}) = p_{A}(\tilde{x} + x) = p_{A}(x) = a_N + ,\] denn $\tilde{x} \in \text{ker } p_{A}$. Konstruiere so induktiv eine kompatible + Familie $(a_{n})_{n\ge N}$ mit $f(a_n) = b_n$ für alle $n \ge N$. Für $n < N$ setze + $a_n \coloneqq p_{A_{n+1}}(a_{n+1})$. Die Kommutativität von \eqref{eq:0.11-diag} + liefert dann, dass $(a_n)_{n \in \N}$ ein kompatibles System ist mit $f(a_{n}) = b_n$ für alle $n \in \N$. + + Surjektivität: Sei $b \in \text{ker } g_N$. Weil $p_B$ surjektiv ist, existiert ein + $y \in B_{N+1}$, sodass $p_B(y) = b$. Sei $z = g_{N+1}(y)$. Aufgrund der Kommutativität von \eqref{eq:0.11-diag} ist dann - $p_C(z) = p_C(g_{j+1}(y)) = g_j(p_B(y)) = g_j(b) = 0$, also - folgt $z \in \text{ker } p_C$. Da $h_{j+1} \circ g_{j+1} = 0$ folgt - $h_{j+1}(z) = h_{j+1}(g_{j+1}(z)) = 0$. Da die untere Zeile exakt ist, existiert nun - ein $\tilde{y} \in \text{ker } p_B$, s.d. $g_{j+1}(\tilde{y}) = z$. Also ist - $y - \tilde{y} \in \text{ker } g_{j+1}$ und $p_B(y - \tilde{y}) = p_B(y) = b$. - Setze $b_{j+1} \coloneqq y - \tilde{y}$ und $b_j \coloneqq b$. + \[ + p_C(z) = p_C(g_{N+1}(y)) = g_N(p_B(y)) = g_N(b) = 0 + ,\] + also + folgt $z \in \text{ker } p_C$. Da $h_{N+1} \circ g_{N+1} = 0$ folgt + \[ + h_{N+1}(z) = h_{N+1}(g_{N+1}(z)) = 0 + .\] + 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 + $y - \tilde{y} \in \text{ker } g_{N+1}$ und + \[ + p_B(y - \tilde{y}) = p_B(y) = b + .\] + Setze $b_{N+1} \coloneqq y - \tilde{y}$ und $b_N \coloneqq b$. Dann konstruiere induktiv eine kompatible - Familie $(b_i)_{i \ge j}$ mit $b_i \in \text{ker } g_{i}$. Für $i < j$ setze wie - oben $b_i \coloneqq p_{B_{i+1}}(b_{i+1})$. Erneut liefert die Kommutativität von - \eqref{eq:0.11-diag} ein kompatibles System $(b_i)_{i \in \N} \in \text{ker } g$ mit - $b_j = b$. + Familie $(b_n)_{n \ge N}$ mit $b_n \in \text{ker } g_{n}$ für alle $n \ge N$. Für $n < N$ setze wie + oben $b_n \coloneqq p_{B_{n+1}}(b_{n+1})$. Erneut liefert die Kommutativität von + \eqref{eq:0.11-diag}, dass $(b_n)_{n \in \N} \in \text{ker } g$ ein kompatibles System mit $b_N = b$ ist. \end{proof} +\begin{bem} + Falls in der Situation von \ref{0.11}, $N= 1$ gewählt werden kann, folgt wegen $A_1 = B_1 = C_1 = D_1 = 0$, + dass die Folge + \[ + \begin{tikzcd} + A \arrow{r}{f} & B \arrow{r}{g} & C + \end{tikzcd} + \] exakt ist. +\end{bem} -\begin{lemma} - Die Klasse der exakten Komplexe in $\mathcal{K}(\mathcal{A}b)$ ist abgeschlossen unter speziellen inversen Limites. +\begin{korollar} + Die Klasse $\mathcal{E}$ der exakten Komplexe in $\mathcal{K}(\mathcal{A}b)$ ist abgeschlossen unter speziellen inversen Limites. \label{lemma:exact-comp-complete-inv} -\end{lemma} +\end{korollar} \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 (R) aus \ref{0.11}. Sei also $i \in \Z$ beliebig. Dann erfüllt + Sei $(\com{S}_n)_{n \in \N}$ ein $\mathcal{E}$-spezielles inverses System in $\mathcal{K}(\mathcal{A}b)$. Für $i \in \Z$ + erfüllt $(S_n^{i})_{n \in \N}$ die Bedingung (R) aus \ref{def:cond-r}. 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 + \] die Bedingungen von \ref{0.11} für $N = 1$, + da nach Voraussetzung für alle $n > 1$ der Komplex $\text{ker}(\com{S}_n \to \com{S}_{n-1})$ + exakt ist. Also ist die Folge \[ \begin{tikzcd} \lim S_n^{i-1} \arrow{r} \arrow{d}{=} & \lim S_n^{i} \arrow{d}{=} \arrow{r} & \lim S_n^{i+1} \arrow{d}{=}\\ (\lim S_n)^{i-1} \arrow{r} & (\lim S_n)^{i} \arrow{r} & (\lim S_n)^{i+1} \end{tikzcd} - \] exakt. Da $i \in \Z$ beliebig, folgt $\lim S_n$ exakt. + \] exakt. Da $i \in \Z$ beliebig war, folgt dass $\lim S_n$ exakt ist. \end{proof} \begin{satz} @@ -1126,9 +1203,10 @@ $\mathcal{A}b$. \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 + $(F(\com{S}_n))_{n \in \N}$ ein $\mathcal{J}$-spezielles inverses 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 + \item $F(S_1) = F(0) = 0$, da $F$ mit inversen Limites vertauscht und die Null der (inverse) + Limes des leeren Diagramms ist. \item Für $n > 1$ ist nach Voraussetzung \[ @@ -1144,7 +1222,7 @@ $\mathcal{A}b$. \] 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 $\lim \com{S}_n \in F^{-1}(\mathcal{J})$. + 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}[] @@ -1157,26 +1235,26 @@ $\mathcal{A}b$. \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 + 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 $\com{\text{Hom}}(\com{T}, \com{A})$ exakt ist. Dann - ist $\mathcal{E}_{\com{T}} = \com{\text{Hom}}(\com{T}, -)^{-1}(\mathcal{J})$. $\com{\text{Hom}}(\com{T}, -)$ erfüllt + ist $\mathcal{H}_{\com{T}} = \com{\text{Hom}}(\com{T}, -)^{-1}(\mathcal{J})$. + $\mathcal{E}$ mit $\com{\text{Hom}}(\com{T}, -)$ 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 + $\mathcal{E}$ abgeschlossen unter speziellen inversen Limites. + \item Wegen \ref{lemma:inner-hom-commutes-with-limits} vertauscht $\com{\text{Hom}} (\com{T}, -)$ 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 $\bigcap_{\com{T} \in \mathcal{I}} \mathcal{E}_{\com{T} }$ + Also ist $\mathcal{H}_{\com{T}}$ und damit $\bigcap_{\com{T} \in \mathcal{I}} \mathcal{H}_{\com{T} }$ abgeschlossen unter speziellen inversen Limites. - - Das Insbesondere folgt wenn $\mathcal{I} = \mathcal{J}$ gesetzt wird. + Das Insbesondere folgt wenn $\mathcal{I} = \mathcal{E}$ gesetzt wird. \end{proof} -Wenn wir alle Pfeile umdrehen erhalten wir die folgende Definition und Ergebnisse: +Für die Klasse der K-injektiven Komplexe betrachten wir die duale Version von \ref{def:special-inv-system}: \begin{definition}[Spezielles direktes System] Sei $\mathcal{P} \subset \mathcal{K}$ eine Klasse von Komplexen. @@ -1198,8 +1276,8 @@ Wenn wir alle Pfeile umdrehen erhalten wir die folgende Definition und Ergebniss \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: +Durch Umdrehen aller Pfeile, erhalten wir die duale Version von \ref{lemma:bounded-compl-in-complete-class} +und insbesondere die folgenden Ergebnisse: % brauche ich nicht %\begin{lemma}

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