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Das wird uns erlauben das klassische Adjunktionsresultat \[ - \otimes_A M \dashv \text{Hom}_{A\text{-Mod}}(M, -) -\] für $M$ ein $A$-Modul auf die abgeleiteten Funktoren zu übertragen. +\] für einen $A$-Modul $M$ auf die abgeleiteten Funktoren zu übertragen. \begin{definition} Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe in einer beliebigen abelschen Kategorie $\mathcal{A}$. @@ -538,7 +538,7 @@ wird uns erlauben das klassische Adjunktionsresultat \text{Hom}^{n}(\com{A}, \com{B}) = \prod_{i \in \Z} \text{Hom}(A^{i}, B^{i+n}) \] mit Differentialen \[ - d^{n}(f) = d_{\com{B} } \circ f - (-1)^{n} f \circ d_{\com{A}} + d^{n}(f) = d_{\com{B} } f - (-1)^{n} f d_{\com{A}} \] für $f \in \text{Hom}^{n}(\com{A}, \com{B})$. \end{definition} @@ -553,30 +553,33 @@ wird uns erlauben das klassische Adjunktionsresultat \] für $m \in M^{i}, n \in N^{n-i}$. \end{definition} -\begin{lemma}[] - Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe. Dann gilt für $n \in \Z$: +Die Kohomologiegruppen von $\com{\text{Hom}}(\com{X}, \com{Y})$ für Komplexe +$\com{X}, \com{Y} \in \mathcal{K}$ lassen sich explizit bestimmen: + +\begin{lemma} + Seien $\com{X}, \com{Y} \in \mathcal{K}$ zwei Komplexe. Dann gilt für $n \in \Z$: \[ - H^{n}(\com{\text{Hom}}(\com{A}, \com{B}) = \text{Mor}_{\mathcal{K}}(\com{A}, \com{B}[n]) + H^{n}(\com{\text{Hom}}(\com{X}, \com{Y}) = \text{Mor}_{\mathcal{K}}(\com{X}, \com{Y}[n]) .\]\label{hom-compl-cohomgroups} \end{lemma} \begin{proof} - Sei $(f^i)_{i \in \Z} \in \prod_{i \in \Z} \text{Hom}(A^{i}, B^{i+n})$. Dann ist: + Sei $(f^i)_{i \in \Z} \in \prod_{i \in \Z} \text{Hom}(X^{i}, Y^{i+n})$. Dann ist: \[ - (f^{i})_{i \in \Z} \in \text{ker } d^{n} \iff (-1)^{n} d_{\com{B}}^{i+n} f^i = f^{i+1} d_{\com{A} }^{i} + (f^{i})_{i \in \Z} \in \text{ker } d^{n} \iff (-1)^{n} d_{\com{Y}}^{i+n} f^i = f^{i+1} d_{\com{X} }^{i} \text{ für } i \in \Z - .\] Wegen $d_{\com{B}[n]} = (-1)^{n}d_{\com{B}}$, induziert $(f^i)_{i \in \Z}$ also genau dann - einen Komplexhomomorphismus $\com{A} \to \com{B}[n]$, wenn $(f^{i})_{i \in \Z} \in \text{ker } d^{n}$. + .\] Wegen $d_{\com{Y}[n]} = (-1)^{n}d_{\com{Y}}$, induziert $(f^i)_{i \in \Z}$ also genau dann + einen Komplexhomomorphismus $\com{X} \to \com{Y}[n]$, wenn $(f^{i})_{i \in \Z} \in \text{ker } d^{n}$. - Weiter ist $(f^{i})_{i \in \Z} \in \text{im } d^{n-1}$ genau dann wenn eine Familie - $(k^{i})_{i \in \Z}\in \prod_{i \in \Z}^{} \text{Hom}(A^{i}, B^{i+n-1})$ existiert, sodass + Weiter ist $(f^{i})_{i \in \Z} \in \text{im } d^{n-1}$, genau dann wenn eine Familie + $(k^{i})_{i \in \Z}\in \prod_{i \in \Z}^{} \text{Hom}(X^{i}, Y^{i+n-1})$ existiert, sodass %\[ - % f^{i} = d_{\com{B}}^{i+n-1} k^{i} - (-1)^{n-1} k^{i+1} d_{\com{A} }^{i} - % = d_{\com{B}}^{i+n-1} k^{i} + (-1)^{n} k^{i+1} d_{\com{A} }^{i} + % f^{i} = d_{\com{Y}}^{i+n-1} k^{i} - (-1)^{n-1} k^{i+1} d_{\com{X} }^{i} + % = d_{\com{Y}}^{i+n-1} k^{i} + (-1)^{n} k^{i+1} d_{\com{X} }^{i} %.\] \[ - (-1)^{n}f^{i} = (-1)^{n} d_{\com{B}}^{i+n-1} k^{i} + k^{i+1} d_{\com{A} }^{i} - .\] Erneut wegen $d_{\com{B} [n]} = (-1)^{n} d_{\com{B} }$, ist also für $(f_i)_{i \in \Z} \in \text{ker } d^{n}$ - der induzierte Komplexhomomorphismus $f\colon \com{A} \to \com{B} $ genau dann nullhomotop, + (-1)^{n}f^{i} = (-1)^{n} d_{\com{Y}}^{i+n-1} k^{i} + k^{i+1} d_{\com{X} }^{i} + .\] Erneut wegen $d_{\com{Y} [n]} = (-1)^{n} d_{\com{Y} }$, ist also für $(f_i)_{i \in \Z} \in \text{ker } d^{n}$ + der induzierte Komplexhomomorphismus $f\colon \com{X} \to \com{Y} $ genau dann nullhomotop, wenn $(f^{i})_{i \in \Z} \in \text{im }d^{n-1}$. \end{proof} @@ -598,7 +601,6 @@ wird uns erlauben das klassische Adjunktionsresultat \end{satz} \begin{proof} - \end{proof} % TODO: Bedingung (I) an Indexmengen @@ -617,47 +619,87 @@ wird uns erlauben das klassische Adjunktionsresultat \section{K-injektive und K-projektive Auflösungen} -In diesem Abschnitt möchten wir, in der Notation von \ref{satz:existence-derived-functors}, jeweils eine -Unterkategorie $\mathcal{L}$ finden, die die Bedingungen von \ref{satz:existence-derived-functors} -für die Funktoren $\com{\text{Hom}}(\com{A}, -) $ bzw. $\com{\text{Hom}}(- , \com{A})$ erfüllt. Dazu definieren -wir folgende Klasse von Komplexen: +Sei $\com{Y} \in \mathcal{K}$. +Um die Bedingungen von \ref{satz:existence-derived-functors} für +$\com{\text{Hom}}(\com{Y}, -)$ (bzw. $\com{\text{Hom}}(-, \com{Y})$) +zu erfüllen, benötigen wir +eine Unterkategorie $\mathcal{L}$ von Komplexen in $\mathcal{K}$, sodass + +\begin{enumerate}[(i)] + \item für jeden Komplex $\com{X} \in \mathcal{K}$ ein Quasiisomorphismus + $\com{X} \to \com{I}$ mit $\com{I} \in \mathcal{L}$ + (bzw. $\com{P} \to \com{X} $ mit $\com{P} \in \mathcal{L}$) + existiert und + \item $\com{\text{Hom}}(\com{Y}, -)$ (bzw. $\com{\text{Hom}}(-, \com{Y})$) Exaktheit + von Komplexen aus $\mathcal{L}$ erhält. +\end{enumerate} + +Dazu definieren wir: -\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. +\begin{definition}[K-injektiv] + Ein Komplex $\com{I} \in \mathcal{K}$ heißt K-injektiv, wenn der Funktor + $\com{\text{Hom}}(-, \com{I})$ Exaktheit erhält. Eine K-injektive + Auflösung eines Komplexes $\com{X} \in \mathcal{K}$ ist + ein Quasiisomorphismus $\com{X} \to \com{I} $ mit + $\com{I} \in \mathcal{K}$ K-injektiv. \end{definition} +\begin{definition}[K-projektiv] + Ein Komplex $\com{P} \in \mathcal{K}$ heißt K-projektiv, wenn der Funktor + $\com{\text{Hom}}(\com{P}, -)$ Exaktheit erhält. Eine K-projektive + Auflösung eines Komplexes $\com{X} \in \mathcal{K}$ ist + ein Quasiisomorphismus $\com{P} \to \com{X} $ mit + $\com{P} \in \mathcal{K}$ K-projektiv. +\end{definition} + +Das Ziel dieses Abschnitts ist das folgende Resultat: + +\begin{satz} + Sei $R$ ein Ring und $\mathcal{A}$ die Kategorie der $R$ (links-)Moduln. Dann + hat jeder Komplex in $\mathcal{A}$ eine K-injektive und eine K-projektive + Auflösung. +\end{satz} + +Die Vorgehensweise orientiert sich dabei an \cite{spaltenstein}. + +\subsection{Elementare Eigenschaften von K-injektiven und K-projektiven Komplexen} + +Zunächst werden einige grundlegenden Eigenschaften von K-injektiven und K-projektiven +Komplexen entwickelt. + \begin{bem} - Mit \ref{hom-compl-cohomgroups} ist $\com{X} $ genau dann K-injektiv (bzw. K-projektiv), wenn - $\forall \com{S} \in K$ exakt: $\text{Mor}_{\mathcal{K}}(\com{S}[i] , \com{X}) = 0$ - (bzw. $\text{Mor}_{\mathcal{K}}(\com{X}, \com{S}[i]) = 0)$ $\forall i \in \Z$. Da Verschieben Exaktheit erhält - folgt also + Sei $\com{X} \in \mathcal{K}$. Mit \ref{hom-compl-cohomgroups} ist $\com{X}$ + genau dann K-injektiv, wenn für jeden exakten Komplex + $\com{S} \in \mathcal{K}$ gilt, dass + $\text{Mor}_{\mathcal{K}}(\com{S}[i] , \com{X}) = 0$ $\forall i \in \Z$. + Da Verschieben Exaktheit erhält folgt also \[ - \com{X} \text{ K-projektiv} \iff \text{Mor}_{\mathcal{K}}(\com{X} , \com{S} ) = 0 \quad \forall \com{S} \text{ exakt} - \] - \[ - \com{X} \text{ K-injektiv} \iff \text{Mor}_{\mathcal{K}}(\com{S} , \com{X} ) = 0 \quad \forall \com{S} \text{ exakt} + \com{X} \text{ K-injektiv} \iff \text{Mor}_{\mathcal{K}}(\com{S} , \com{X} ) = 0 \quad \forall \com{S} \in \mathcal{K} \text{ exakt} .\] + Durch Umdrehen der Pfeile erhalten wir + \[ + \com{X} \text{ K-projektiv} \iff \text{Mor}_{\mathcal{K}}(\com{X} , \com{S} ) = 0 \quad \forall \com{S} \in \mathcal{K} \text{ exakt} + \] \end{bem} -\subsection{Elementare Eigenschaften von K-injektiven und K-projektiven Komplexen} - \begin{bem} Ein exakter K-projektiver oder K-injektiver Komplex $\com{X}$ ist zusammenziehbar, d.h. ist nullhomotop. \begin{proof} Betrachte $\mathrm{id}_{\com{X}} \in \mathrm{Hom}^{0}(\com{X}, \com{X}) = \mathrm{Mor}_{\K}(\com{X} , \com{X}) = \mathrm{H}^0 \com{\mathrm{Hom}}(\com{X}, \com{X}) = 0$. Also ist $\mathrm{id}_{\com{X} } = 0$ und damit - $\com{X} = 0$ in $\K$. + $\com{X} \stackrel{\sim }{=} 0$ in $\K$. \end{proof} \end{bem} \begin{satz} - Sei $\com{X} \in \K$ mit $X^{i} = 0$ $\forall i \neq 0$. Dann ist $\com{X} $ K-injektiv (bzw. K-projektiv) genau - dann wenn $A^{0}$ injektiv (bzw. projektiv) in $\mathcal{A}$ ist. + Sei $\com{X} \in \K$ mit $X^{i} = 0$ $\forall i \neq 0$. Dann ist $\com{X} $ K-projektiv (bzw. K-injektiv) genau + dann wenn $X^{0}$ projektiv (bzw. injektiv) in $\mathcal{A}$ ist. \label{satz:single-degree-compl-k-proj} \end{satz} \begin{proof} + Wir zeigen nur den K-projektiven Fall. Der K-injektive folgt dann durch Umdrehen + aller Pfeile. ,,$\implies$'': Sei $\com{X} $ K-projektiv und $\com{S} = 0 \to M \to N \to P \to 0$ kurze exakte Folge in $\mathcal{A}$. Sei $f\colon X^{0} \to P$. Das induziert einen Komplexhomomorphismus $\com{X} \to \com{S}$ \[\begin{tikzcd} @@ -1908,9 +1950,12 @@ Jetzt können wir alles zusammentragen und erhalten: $\com{\text{Hom}}(\com{N}, \com{P})$ K-injektiv ist. \end{proof} +% TODO: zitate richtig machen \begin{thebibliography}{9} \bibitem{hartshorne} Robin Hartshorne. Residues and duality. \emph{Lecture Notes in Math.}. 20, Springer-Verlag (1966) +\bibitem{spaltenstein} +N. Spaltenstein. Resolutions of unbounded complexes. \emph{Composito Mathematica 65}. (1988) \end{thebibliography} \end{document}