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\label{satz:existence-derived-functors} +\end{satz} + \section{Grundlagen} +Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die +Komplexkategorie mit Komplexhomomorphismen bis auf Homotopie als Abbildungen. + +\begin{definition} + Seien $\com{M}, \com{N} \in \mathcal{K}(A\text{-Mod})$. Dann sei + $\com{M} \otimes_A \com{N} \in \mathcal{K}(A\text{-Mod})$ definiert durch + \[ + (\com{M} \otimes_A \com{N})^{n} = \bigoplus_{i \in \Z} M^{i} \otimes_A N^{n-i} + \] mit Differentialen + \[ + d^{n}(m \otimes n) = d_{\com{M} }(m) \otimes n + (-1)^{i} m \otimes d_{\com{N} }(n) + \] für $m \in M^{i}, n \in N^{n-i}$. +\end{definition} + +\begin{definition} + Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe. Dann sei + $\com{\text{Hom}}(\com{A}, \com{B}) \in \mathcal{K}(\mathcal{A}b)$ definiert durch + \[ + \text{Hom}^{n}(\com{A}, \com{B}) = \prod_{i \in \Z} \text{Hom}(A^{i}, B^{i+n}) + \] mit Differentialen + \[ + d^{n}(f) = d_{\com{B} } \circ f - (-1)^{n} f \circ d_{\com{A}} + \] für $f \in \text{Hom}^{n}(\com{A}, \com{B})$. +\end{definition} + +\begin{satz}[Adjunktion der Hom- und Tensorproduktkomplexe] + Seien $\com{M}, \com{N}, \com{P} \in \mathcal{K}(A\text{-Mod})$. Dann existiert + ein natürlicher Isomorphismus, der funktoriell in allen Variablen ist: + \[ + \com{\text{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) + = \com{\text{Hom}}(\com{M},\com{\text{Hom}} (\com{N} , \com{P} )) + .\] + \label{satz:adjunction-hom-tor-comp} +\end{satz} + +\begin{proof} + +\end{proof} + +\begin{lemma}[] + Es gilt + \[ + H^{i}(\com{\text{Hom}}(\com{A}, \com{B}) = \text{Mor}_{\mathcal{K}}(\com{A}, \com{B}[i]) + .\] + \label{hom-compl-cohomgroups} +\end{lemma} + +\begin{proof} + +\end{proof} % TODO: Bedingung (I) an Indexmengen +Folgendes Kriterium für die Exaktheit von Komplexen ist hilfreich: + +\begin{lemma}[] + Sei $\com{A} \in \mathcal{K}$ und $\mathcal{G}$ eine Klasse von Objekten + von $\mathcal{A}$, sodass jedes Objekt von $\mathcal{A}$ eine Einbettung + in ein Objekt aus $\mathcal{G}$ besitzt. Angenommen + $\com{\text{Hom}}(\com{A}, E) \in \mathcal{K}(\mathcal{A}b)$ ist exakt für alle + $E \in \mathcal{K}$. Dann ist $\com{A} $ exakt. + \label{lemma:0.10} +\end{lemma} + +\begin{proof} + Keine Ahnung. + % TODO : einfuegen +\end{proof} + Wir benötigen im Folgenden mehrmals eine bestimmte Bedingung an inverse Systeme in $\mathcal{A}b$. Wir sagen ein inverses System $(M_n)_{n \ge \N}$ genügt Bedingung (R), wenn folgende Eigenschaften erfüllt sind: @@ -131,9 +203,6 @@ $\mathcal{A}b$. Wir sagen ein inverses System $(M_n)_{n \ge \N}$ genügt Bedingu \section{K-injektive und K-projektive Auflösungen} -Sei im Folgenden $\mathcal{A}$ eine feste abelsche Kategorie und $\mathcal{K} = \mathcal{K}(\mathcal{A})$ die -Komplexkategorie mit Komplexhomomorphismen bis auf Homotopie als Abbildungen. - \begin{definition}[K-injektiv bzw. K-projektiv] Ein Komplex $\com{X} \in \K$ heißt K-injektiv (bzw. K-projektiv), wenn für alle $\com{S} \in \K$, der Komplex $\com{\mathrm{Hom}}(\com{S}, \com{X})$ (bzw. $\com{\mathrm{Hom}}(\com{X}, \com{S})$) exakt ist. @@ -183,9 +252,15 @@ $\text{Hom}(X^{0}, N) \to \text{Hom}(X^{0}, P)$ surjektiv und damit $X^{0}$ proj ,,$\impliedby$'': Sei nun $X^{0}$ projektiv, $\com{S} \in \mathcal{K}$ exakt und $f\colon \com{X} \to \com{S}$ Komplexhomomorphismus. Dann betrachte \[ -diag +\begin{tikzcd} + 0 \arrow[from=1-1,to=1-3] \arrow{d} & & X^{0} \arrow{r} + \arrow[dashed, from=1-3,to=2-1]{}{k^{0}} + \arrow[dashed]{dl} \arrow{d}{f^{0}} & 0 \arrow{d} \\ + S^{-1} \arrow{r}{d^{-1}} & \text{im }d^{-1} \arrow{r} & S^{0} \arrow{r}{d^{0}} & S^{1} +\end{tikzcd} .\] -Da $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} k^{0}$. +Da $d^{0}f^{0} = 0$ faktorisiert $f^{0}$ über $\text{ker } d^{0} = \text{im }d^{-1}$. Weil +$X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} k^{0}$. \end{proof} \begin{satz}[] @@ -363,6 +438,7 @@ Durch Umdrehen aller Pfeile erhalten wir analog: \item Für jeden Quasiisomorphismus $u\colon \com{I} \to \com{S} $ in $\mathcal{K}$ existiert ein $v\colon \com{S} \to \com{I} $, s.d. $vu = \text{id}_{\com{I} }$ in $\mathcal{K}$. \end{enumerate} + \label{satz:mork=mord-for-k-inj} \end{satz} \subsection{Spezielle inverse und direkte Systeme} @@ -874,6 +950,7 @@ Durch Umdrehen der Pfeile erhalten wir die duale Aussagen von \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. + \label{satz:existence-k-inj-resolution} \end{satz} \begin{proof} @@ -1000,6 +1077,7 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom \item $\com{\text{Hom}}(\com{M}, \com{I})$ ist K-injektiv für jeden K-injektiven Komplex $\com{I} \in \mathcal{K}$. \end{enumerate} + \label{satz:homs-of-k-flat-are-k-inj} \end{satz} \begin{proof} @@ -1021,8 +1099,60 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom \] exakt. \end{proof} +\begin{satz}[] + \begin{enumerate}[(a)] + \item Falls $\com{M}, \com{N} \in \mathcal{K}$ K-flach sind, dann ist + auch $\com{A} \otimes_A \com{B} $ K-flach. + \item $\com{M} \in \mathcal{K}$ ist K-flach genau dann wenn $\com{M}[1]$ + K-flach ist. + \item Falls zwei Punkte eines ausgezeichneten Dreiecks in $\mathcal{K}$ K-flach + sind, + dann auch der dritte. + \end{enumerate} + Insbesondere ist die volle Unterkategorie der K-flachen Komplexe in $\mathcal{K}$ + eine triangulierte Unterkategorie. + \label{satz:k-flat-triangulated} +\end{satz} + +\begin{proof} + \begin{enumerate}[(a)] + \item Seien $\com{M}, \com{N} \in \mathcal{K}$ K-flach und $\com{S} $ exakt. Dann + ist + \[ + (\com{M} \otimes_A \com{N}) \otimes \com{S} = + \com{M} \otimes_A (\com{N} \otimes_A \com{S}) + \] und die rechte Seite ist exakt. + \item Seien $\com{S}, \com{M} \in \mathcal{K}$. + Dann sind $- \otimes_A \com{S} $ und $\com{M} \otimes_A -$ nach + \ref{satz:tor-is-triangulated} ein triangulierter Funktor, also folgt + \[ + \com{M}[1] \otimes_A \com{S} = + (\com{M} \otimes_A \com{S})[1] + = \com{M} \otimes_A \com{S}[1] + .\] Da Verschieben Exaktheit erhält folgt daraus die Äquivalenz. + \item Sei $(\com{M}, \com{N}, \com{P}, u, v, w)$ ein ausgezeichnetes Dreieck + in $\mathcal{K}$ mit $\com{M}$ und $\com{N} $ K-flach. Sei weiter + $\com{S} $ exakt. Da $- \otimes_A \com{S} $ nach \ref{satz:tor-is-triangulated} + ein triangulierter Funktor ist, erhalten wir das ausgezeichnete Dreieck + $(\com{M} \otimes_A \com{S}, \com{N} \otimes_A \com{S}, \com{P} \otimes_A \com{S}, u \otimes \text{id}_{\com{S} }, v \otimes \text{id}_{\com{S} }, w \otimes \text{id}_{\com{S} })$ + und damit für $i \in \Z$ die exakte Folge + \[ + \begin{tikzcd} + H^{i}(\com{N} \otimes_A \com{S}) \arrow{r} & + H^{i}(\com{P} \otimes_A \com{S}) \arrow{r} & + H^{i+1}(\com{M} \otimes_A \com{S}) + \end{tikzcd} + .\] Da die äußeren Terme nach Voraussetzung $0$ sind, folgt die + Exaktheit von $\com{P} \otimes_A \com{S} $ und damit, dass $\com{P} $ + K-flach ist. + Der allgemeine Fall folgt nun mit \ref{TR2}. + \end{enumerate} + +\end{proof} + \begin{satz}[] Sei $\com{M} \in \mathcal{K}$ K-projektiv. Dann ist $\com{M} $ K-flach. + \label{satz:k-proj-is-k-flat} \end{satz} \begin{proof} @@ -1032,12 +1162,12 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom \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 $\com{N} \in \mathcal{K}$. + \label{satz:tor-exact-for-k-flat} \end{satz} \begin{proof} @@ -1088,6 +1218,7 @@ Umdrehen der Pfeile liefert 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. + \label{satz:derived-hom} \end{satz} \begin{proof} @@ -1095,7 +1226,7 @@ Umdrehen der Pfeile liefert 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} $ + \item Für alle $\com{N} \in \mathcal{K}$ existiert ein Quasiisomorphismus $\com{N} \to \com{I} $ wegen \ref{satz:existence-k-inj-resolution} 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} @@ -1119,6 +1250,92 @@ Umdrehen der Pfeile liefert \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. + \label{satz:derived-tor} +\end{satz} + +\begin{proof} + Erneut in der Notation von \ref{satz:existence-derived-functors} wähle $\mathcal{L}$ + als die volle Unterkategorie der K-flachen Komplexe von $\mathcal{K}$. Dann ist + für $\com{N}$ beliebig: + \begin{enumerate}[(i)] + \item $\mathcal{L}$ ist trianguliert nach \ref{satz:k-flat-triangulated}. + \item Für alle $\com{M} \in \mathcal{K}$ existiert nach + \ref{satz:existence-k-proj-resolution} und \ref{satz:k-proj-is-k-flat} + ein Quasiisomorphismus + $\com{P} \to \com{M}$ mit $\com{P} \in \mathcal{L}$. + \item Nach \ref{satz:tor-exact-for-k-flat} ist + $\com{N} \otimes_A -|_{\mathcal{L}}$ exakt. + \end{enumerate} + Also existiert $\com{N} \otimes_A^{\text{L}} -$ und analog für + $- \otimes_A^{L} \com{N}$. +\end{proof} + +\subsection{Adjunktion} + +Jetzt können wir alles zusammentragen und erhalten: + +\begin{satz} + Seien $\com{M}, \com{N}, \com{P} \in \mathcal{D}$. Dann existiert ein natürlicher + Isomorphismus, der funktoriell in allen Variablen ist: + \[ + \text{R}\com{\text{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) + = \text{R}\com{\text{Hom}}(\com{M}, \text{R}\com{\text{Hom}}(\com{N}, \com{P})) + .\] + \label{satz:adjunction-rhom-rtor} \end{satz} +\begin{proof} + Nach \ref{satz:derived-hom} und \ref{satz:derived-tor} sind alle Terme wohldefiniert + und wir können mit \ref{satz:existence-k-proj-resolution} und + \ref{satz:k-proj-is-k-flat} ohne Einschränkung annehmen, dass $\com{N} $ K-flach ist, + und mit \ref{satz:existence-k-inj-resolution}, dass $\com{P} $ K-injektiv ist. + + Dann folgt + \begin{align*} + \text{R}\com{\text{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) + &= \text{R}\com{\text{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ + &= \com{\text{Hom}}(\com{M} \otimes_A \com{N}, \com{P} ) \\ + &\stackrel{\ref{satz:adjunction-hom-tor}}{=} + \com{\text{Hom}}(\com{M}, \com{\text{Hom}}(\com{N}, \com{P})) \\ + &= \text{R}\com{\text{Hom}}(\com{M}, \com{\text{Hom}}(\com{N}, \com{P})) \\ + &= \text{R}\com{\text{Hom}}(\com{M}, \text{R}\com{\text{Hom}}(\com{N}, \com{P})) + .\end{align*} + Das letzte Gleichheitszeichen gilt, weil nach \ref{satz:homs-of-k-flat-are-k-inj} + $\com{\text{Hom}}(\com{N}, \com{P})$ K-injektiv ist. +\end{proof} + +\begin{korollar}[] + Seien $\com{M}, \com{N}, \com{P} \in \mathcal{D}$. Dann existiert ein natürlicher + Isomorphismus, der funktoriell in allen Variablen ist: + \[ + \text{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) + = \text{Mor}_{\mathcal{D}}(\com{M}, \text{R}\com{\text{Hom}}(\com{N} , \com{P} ) + .\] Insbesondere gilt folgende Funktoradjunktion in $\mathcal{D}$: + \[ + - \otimes_A^{\text{L}} N \dashv \text{R}\com{\text{Hom}}(\com{N}, -) + .\] +\end{korollar} + +\begin{proof} + Wir können wieder annehmen, dass $\com{N}$ K-flach und $\com{P} $ K-injektiv ist. + Dann betrachte: + \begin{salign*} + \text{Mor}_{\mathcal{D}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) + &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} + \text{Mor}_{\mathcal{K}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ + &\stackrel{\ref{hom-compl-cohomgroups}}{=} + H^{0}\com{\text{Hom}}(\com{M} \otimes_A^{\text{L}} \com{N}, \com{P}) \\ + &= H^{0}\com{\text{Hom}}(\com{M} \otimes_A \com{N}, \com{P}) \\ + &\stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} + H^{0}\com{\text{Hom}}(\com{M}, \com{\text{Hom}}(\com{N}, \com{P})) \\ + &\stackrel{\ref{hom-compl-cohomgroups}}{=} + \text{Mor}_{\mathcal{K}}(\com{M}, \com{\text{Hom}}(\com{N}, \com{P})) \\ + &\stackrel{\ref{satz:mork=mord-for-k-inj}}{=} + \text{Mor}_{\mathcal{D}}(\com{M}, \com{\text{Hom}}(\com{N}, \com{P})) \\ + &= \text{Mor}_{\mathcal{D}}(\com{M}, \text{R}\com{\text{Hom}}(\com{N}, \com{P})) + .\end{salign*} + Das letzte Gleichheitszeichen gilt erneut, weil nach \ref{satz:homs-of-k-flat-are-k-inj} + $\com{\text{Hom}}(\com{N}, \com{P})$ K-injektiv ist. +\end{proof} + \end{document}