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K-injektiv) sind, dann auch der dritte. \end{enumerate} + \label{satz:k-proj-triangulated} \end{satz} \begin{proof} @@ -387,7 +388,6 @@ inverse bzw. direkte Systeme deren Limites die gewünschten Auflösungen liefern $\mathcal{J}$-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 @@ -814,6 +814,7 @@ Dieses Lemma können wir nun anwenden, um das zentrale Ergebnis zu zeigen: $\colim$ ist exakt. Dann hat jeder Komplex in $\mathcal{K}$ eine K-projektive Linksauflösung. + \label{satz:existence-k-proj-resolution} \end{korollar} \begin{proof} @@ -1002,7 +1003,122 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom \end{satz} \begin{proof} - (i)$\implies$(ii): Sei ^ + (i)$\implies$(ii): Sei $\com{I}$ K-injektiv und $\com{S}$ exakt. Dann ist + \[ + \com{\text{Hom}} (\com{S} , \com{\text{Hom}}(\com{M}, \com{I})) + \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} + \com{\text{Hom}}(\underbrace{\com{S} \otimes_A \com{M}}_{\text{exakt}}, \underbrace{\com{I}}_{\text{K-injektiv}} ) + .\] 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 + \[ + \com{\text{Hom}}(\com{S} \otimes_A \com{M}, I) = \com{\text{Hom}}(\com{S} \otimes_A \com{M}, \com{I}) + \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \com{\text{Hom}}(\underbrace{\com{S}}_{\text{exakt}}, \underbrace{\com{\text{Hom}}(\com{M}, \com{I})}_{\text{K-injektiv}}) + \] exakt. +\end{proof} + +\begin{satz}[] + Sei $\com{M} \in \mathcal{K}$ K-projektiv. Dann ist $\com{M} $ K-flach. +\end{satz} + +\begin{proof} + Sei $\com{M} $ K-projektiv und $\com{S} $ exakt. Sei weiter $\com{I} $ K-injektiv. Dann folgt + \[ + \com{\text{Hom}} (\com{M} \otimes_A \com{S}, \com{I}) \stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} + \com{\text{Hom}}(\com{M}, \com{\text{Hom}}(\com{S}, \com{I} ) + .\] 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}. + \label{satz:k-proj-is-k-flat} +\end{proof} + +\begin{satz}[] + Sei $\com{M} \in \mathcal{K}$ K-flach und exakt. Dann ist $\com{M} \otimes_A \com{N} $ exakt + für alle $\com{N} \in \mathcal{K}$. +\end{satz} + +\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 + \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}) + \label{eq:cohom-groups} + .\end{equation} + Wegen \ref{satz:k-proj-is-k-flat} ist $\com{P} $ K-flach, also 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}. +\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} + .\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 + +\begin{satz}[] + Sei $\com{P} \in \mathcal{K}$ K-projektiv und exakt. Dann ist $\com{\text{Hom}}(\com{P} , \com{M} )$ exakt für alle + $\com{M} \in \mathcal{K}$. +\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. +\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} $ + 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}[] + 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. +\end{satz} + \end{document}