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zAfcM9PJMR0FMzMU9?frN1unnTccuq!GGD^MKH&acTL+-{_+B?5q&S~zIiej8?-U9W zwxpjHHW7N=3@Mzj6DMyJ3ZwPiUTOIxP(K>gkcwe74IjQDMYzbi{l5kFySv3Vcdu{n U6_uAqP(YXum4QJ{Lmu^i0b 1$ ist die Abbildung $M_n \to M_{n-1}$ surjektiv. +\end{enumerate} + +\begin{lemma} + Sei $I$ eine Indexmenge, die Bedingung (I) genügt und seien + $(A_i)_{i \in I}$, $(B_{i})_{i \in I}$, $(C_i)_{i \in I}$ und $(D_i)_{i \in I}$ + 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} + \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 + \[ + \begin{tikzcd} + A \arrow{r}{f} & B \arrow{r}{g} & C \arrow{r}{h} & D + \end{tikzcd} + \] 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 $D_i \to D_{i-1}$. + + Sei weiter $j \in I$, s.d. für alle $i > j$ die Folge + \[ + \begin{tikzcd} + A_i' \arrow{r} & B_i' \arrow{r} & C_i' \arrow{r} & D_i' + \end{tikzcd} + \] exakt ist. + + Dann ist die natürliche Abbildung + \[ + \text{ker } g / \text{im } f \longrightarrow \text{ker } g_j / \text{im } f_j + \] 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: + \begin{equation} + \begin{tikzcd} + A \arrow{r}{f} \arrow{d} & \text{im } f \arrow{r} + & \text{ker } g \arrow{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} \\ + \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 $y, b_{j+1} \in \text{ker } g_{j+1}$, + 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 $z = g_{j+1}(y)$. + 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 $b_j \coloneqq b$. + 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 + $b_j = b$. +\end{proof} + \section{K-injektive und K-projektive Auflösungen} Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die @@ -469,12 +582,15 @@ Ebenfalls analog gilt: \subsection{Existenz von K-projektiven und K-injektiven Auflösungen} +\subsubsection{Linksauflösungen} + Sei $\mathcal{P}$ eine Klasse von Komplexen in $\mathcal{K}$. Die folgenden Bedingungen an $\mathcal{P}$ sind äquivalent: +% TODO: wirklich notwendig die beschränkung nach oben?? \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 $\com{P} \in \mathcal{P}$. + nach links mit $\com{P} \in \mathcal{P}$ und $\com{P} $ nach oben beschränkt. \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 $i \le n$. @@ -508,6 +624,8 @@ Im Folgenden nehmen wir an, dass $\mathcal{P}$ die Eigenschaft (1) erfüllt. Erneut nach \ref{kor:k-proj-closed} sind die Komplexe in $\underset{\rightarrow}{\mathcal{P}}$ damit ebenfalls $K$-projektiv. + + \label{bsp:bounded-above-projectives} \end{bsp} Unser Ziel ist nun für einen gegebenen Komplex $\com{A}$ eine Auflösung nach Links durch einen Komplex @@ -518,24 +636,373 @@ folgende Lemma führt diese Konstruktion aus. 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 $n \ge 0$. + \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. Dann + 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 $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 $C_f^{i} = P^{i+1} \oplus B^{i}$. + 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 $g''_i\colon Q^{i} \to B^{i-1}$. + + Betrachte für $i \in \Z$ das folgende kommutative Diagramm: + \[ + \begin{tikzcd} + \cdots \arrow{r} & Q^{i} \arrow{r}{d_{Q}} \arrow{d}{(g', g'')} + & Q^{i+1} \arrow{r} \arrow{d}{(g', g'')} & \cdots\\ + \cdots \arrow{r} & P^{i} \oplus B^{i-1} \arrow{r}{d_{C_f[-1]}} & + P^{i+1} \oplus B^{i} \arrow{r} & \cdots + \tag{$*$} \label{eq:1} + \end{tikzcd} + \] In Matrixnotation ist + \begin{align*} + d_{C_f} &= \begin{pmatrix} d_{P}[1] & 0 \\ f[1] & 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} + .\end{align*} + Auswerten 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''} + .\end{align} + Aus \eqref{eq:g'-comp-hom} folgt, dass $g'\colon \com{Q} \to \com{P} $ ein + Komplexhomomorphismus ist. Setze nun + $h\colon \com{C}_{-g'} \to \com{B} $ durch + \[ + h(x,y) = g''[1](x) + f(y) + .\] + Betrachte nun für $i \in \Z$ das folgende Diagramm: + \[ + \begin{tikzcd} + \cdots \arrow{r} & Q^{i+1} \oplus P^{i} \arrow{r}{d_{C_{-g'}}} \arrow{d}{h} & Q^{i+2} \oplus P^{i+1} \arrow{r} \arrow{d}{h} + & \cdots \\ + \cdots \arrow{r} & B^{i} \arrow{r}{d_{B}} & B^{i+1} \arrow{r} & \cdots + \end{tikzcd} + .\] 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} \\ + &= \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{}}{=} + \begin{pmatrix} + d_B g'' & d_B f + \end{pmatrix} \\ + &= d_B h + .\end{salign*} + Also ist $h$ 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$ + \[ + C_h^{i} = C_{-g}^{i+1} \oplus B^{i} = (Q^{i+2} \oplus P^{i+1}) \oplus B^{i} + = Q^{i+2} \oplus (P^{i+1} \oplus B^{i}) + = Q^{i+2} \oplus C_f^i + = C_{-g}^{i}[1] + .\] 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] + = \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 + \end{pmatrix} + = \begin{pmatrix} + -d_Q & 0 & 0 \\ + g' & -d_P & 0 \\ + g'' & f & d_B + \end{pmatrix} + .\end{align*} + Analog folgt + \begin{align*} + d_{C_{-g}[1]} = + \begin{pmatrix} + d_Q[1] & 0 \\ + -g & 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 & 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. + + 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 + $\text{coker } p_{n-1} = \com{Q}[1] \in \mathcal{P}$ und wir haben gradweise + zerfallende exakte Folgen: + \[ + \begin{tikzcd} + 0 \arrow{r} & P^{i} \arrow{r}{p_{n-1}} & Q^{i+1} \oplus P^{i} \arrow{r} + & Q^{i+1} \arrow{r} & 0 + \end{tikzcd} + .\] + Also ist $(\com{P}_n)_{n \ge -1}$ 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 + \[ + \begin{tikzcd} + \com{P}_{n-1} \arrow{r}{p_{n-1}} \arrow{d}{f_{n-1}} + & \com{P}_{n} \arrow{d}{f_n = h} \\ + \tau_{\le n-1}\com{A} \arrow{r}{a_{n-1}} & \tau_{\le n}\com{A} + \end{tikzcd} + \] und $(f_n)_{n \ge -1}$ ist ein direktes System. +\end{proof} + +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. + + \label{satz:existence-left-resolutions} +\end{satz} + +\begin{proof} + Sei $\com{A} \in \mathcal{K}$ und $(\com{P}_n)_{n \ge -1}$, $(f_n)_{n \ge -1}$ 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 + $\com{P} \coloneqq \colim \com{P}_n$ in $\underset{\rightarrow}{\mathcal{P}}$. + + Wir erhalten ebenfalls + \[ + f\coloneqq \colim f_n \colon \com{P} \longrightarrow \colim \tau_{\le n}\com{A} + = \com{A} + .\] Da $\colim$ exakt, folgt für $i \in \Z$: + \[ + H^{i}(f) = H^{i}(\colim f_n) = \colim \underbrace{H^{i}(f_n)}_{\text{Isomorphismus}} + .\] Also ist $f$ ein Quasiisomorphismus. +\end{proof} + +\begin{korollar}[] + Angenommen direkte Colimites existieren in $\mathcal{A}$ und + $\colim$ ist exakt. + + Dann hat jeder Komplex in $\mathcal{K}$ eine K-projektive Linksauflösung. +\end{korollar} + +\begin{proof} + Wähle $\mathcal{P}$ wie in Beispiel \ref{bsp:bounded-above-projectives} und wende + \ref{satz:existence-left-resolutions} an. +\end{proof} + +\subsubsection{Rechtsauflösungen} + +Sei $\mathcal{J}$ eine Klasse von Komplexen in $\mathcal{K}$. Im Folgenden nehmen wir an, +dass $\mathcal{J}$ 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 + $\com{I}$ 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 + der nach unten beschränkten Komplexe mit in $\mathcal{A}$ injektiven Objekten wählen. +\end{bsp} + +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 + 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. + \label{lemma:constr-inv-system} +\end{lemma} + +\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. + + \label{satz:existence-right-resolutions} +\end{satz} - Mit (1) existiert ein Quasiisomorphismus $g\colon \com{Q} \to \com{C}_f[-1] $ mit $\com{Q}[1] \in \mathcal{P}$. +\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. + + 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, + 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. +\end{satz} + +\begin{proof} + Seien $(\com{I}_n)_{n \ge -1}$ und $(f_n)_{n \ge -1}$ 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: + \[ + \begin{tikzcd} + \com{I} \arrow{r} \arrow{d}{f} & \com{I}_n \arrow{r}{p_n} \arrow{d}{f_n} & \com{I_{n-1}} \arrow{d}{f_{n-1}} \\ + \com{A} \arrow{r} & \tau^{\ge -n} \com{A} \arrow{r} & \tau^{\ge -(n-1)} \com{A} + \end{tikzcd} + \] Wende nun $H^{i}(-)$ auf dieses Diagramm an: + \begin{equation} + \begin{tikzcd} + H^{i}(\com{I}) \arrow{r} \arrow{d}{H^{i}(f)} & H^{i}(\com{I}_n) \arrow{r}{H^{i}(p_n)} + \arrow{d}{H^{i}(f_n)}[swap]{\sim} & H^{i}(\com{I_{n-1}}) \arrow{d}{H^{i}(f_{n-1})}[swap]{\sim} \\ + H^{i}(\com{A}) \arrow{r} & H^{i}(\tau^{\ge -n} \com{A}) \arrow{r} & H^{i}(\tau^{\ge -(n-1)} \com{A}) + \end{tikzcd} + \label{eq:diag-hi-in} + .\end{equation} + Die rechten beiden vertikalen Pfeile sind Isomorphismen, da $f_k$ ein + Quasiisomorphismus ist für alle $k \ge -1$. + + 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 + damit ist + $H^{i}(p_n)\colon H^{i}(\com{I}_n) \to H^{i}(\com{I}_{n-1})$ + ein Isomorphismus. + + Betrachte nun die kurze exakte Folge + \[ + \begin{tikzcd} + 0 \arrow{r} & \text{ker } p_n \arrow{r} & \com{I}_n \arrow{r}{p_n} & \com{I}_{n-1} + \arrow{r} & 0 + \end{tikzcd} + .\] Das liefert eine lange exakte Kohomologiefolge: + \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}) + \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 $H^{i-1}(p_n)$. + Aufgrund der Exaktheit von \eqref{eq:long-ex-hi-in} folgt, dann dass + $H^{i}(\text{ker } p_n) = 0$ für alle $n \ge -i+2$. + + Sei nun $m \in \Z$ beliebig. Dann setze $N \coloneqq -m + 1$. Dann ist + für alle $n > N$: + \[ + H^{m}(\text{ker } p_n) = 0 = H^{m+1}(\text{ker } p_n) + .\] + Also ist die Folge + \begin{equation} + \begin{tikzcd} + \text{ker } p_n^{m-1} \arrow{r} & + \text{ker } p_n^{m} \arrow{r} & + \text{ker } p_n^{m+1} \arrow{r} & + \text{ker } p_n^{m+2} + \end{tikzcd} + \end{equation} + 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} + \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 + $n=N$ liefert nun, dass $H^{i}(f)$ ein Isomorphismus ist. \end{proof} \newpage \section{Adjunktion von abgeleitetem Hom und Tensorprodukt} +Sei $A$ ein kommutativer Ring und im Folgenden $\mathcal{A}$ die Kategorie der $A$-Moduln. + +\subsection{K-flache Komplexe} + +Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Komplexen. + +\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. +\end{definition} + +\begin{satz} + Sei $\com{M} \in \mathcal{K}$ mit $M^{i} = 0$ für $i \neq 0$. Dann ist + $\com{M} $ genau dann K-flach, wenn $M^{0}$ flacher $A$-Modul ist. +\end{satz} + +\begin{proof} + Sei $\com{M} \in \mathcal{K}$ wie im Satz. Dann ist für $\com{S} \in \mathcal{K}$ und + $n \in \Z$: + \[ + (\com{M} \otimes_A \com{S})^{n} = \bigoplus_{i+j=n} M^{i} \otimes_A S^{J} + = M^{0} \otimes S^{n} = (M^{0} \otimes \com{S} )^{n} + \] und für $m \in M^{0}$, $s \in S^{n}$: + \[ + d_{\com{M} \otimes_A \com{S} }^{n}(m \otimes s) + = \underbrace{d_{M}^{0}(m)}_{= 0} \otimes_A s + (-1)^{0} m \otimes_A d_{S}(s) + = m \otimes_A d_S(s) + = d_{M^{0} \otimes_A \com{S} } + .\] Also ist $\com{M} \otimes_A \com{S} = M^{0} \otimes_A \com{S}$. Damit folgt + die Behauptung aus den Definitionen. +\end{proof} + +\begin{satz}[] + Sei $\com{M} \in \mathcal{K} $. Dann sind äquivalent: + \begin{enumerate}[(i)] + \item $\com{M} $ ist K-flach. + \item $\com{\text{Hom}}(\com{M}, \com{I})$ ist K-injektiv für jeden + K-injektiven Komplex $\com{I} \in \mathcal{K}$. + \end{enumerate} +\end{satz} + +\begin{proof} + (i)$\implies$(ii): Sei ^ +\end{proof} + \end{document}