From 54a1df7cc89bde8351e14cc2f5683980941fa598 Mon Sep 17 00:00:00 2001 From: flavis Date: Thu, 31 Mar 2022 16:29:15 +0200 Subject: [PATCH] first version semicomplete --- sose2022/bachelorarbeit/arbeit.cls | 209 ++++++++ sose2022/bachelorarbeit/arbeit.pdf | Bin 380828 -> 352999 bytes sose2022/bachelorarbeit/arbeit.tex | 762 ++++++++++++++++++++++------- 3 files changed, 805 insertions(+), 166 deletions(-) create mode 100644 sose2022/bachelorarbeit/arbeit.cls diff --git a/sose2022/bachelorarbeit/arbeit.cls b/sose2022/bachelorarbeit/arbeit.cls new file mode 100644 index 0000000..6fa8437 --- /dev/null +++ b/sose2022/bachelorarbeit/arbeit.cls @@ -0,0 +1,209 @@ +\ProvidesClass{arbeit} +\LoadClass[a4paper, titlepage]{article} + +\RequirePackage[utf8]{inputenc} +\RequirePackage[T1]{fontenc} +\RequirePackage{textcomp} +\RequirePackage[german]{babel} +\RequirePackage{amsmath, amssymb, amsthm} +\RequirePackage{mdframed} +\RequirePackage{fancyhdr} +\RequirePackage{geometry} +\RequirePackage{import} +\RequirePackage{pdfpages} +\RequirePackage{transparent} +\RequirePackage{xcolor} +\RequirePackage{array} +\RequirePackage[shortlabels]{enumitem} +\RequirePackage{tikz} +\RequirePackage{pgfplots} +\RequirePackage[nobottomtitles]{titlesec} +\RequirePackage{listings} +\RequirePackage{mathtools} +\RequirePackage{forloop} +\RequirePackage{totcount} +\RequirePackage{calc} +\RequirePackage{wasysym} +\RequirePackage{environ} + +\usetikzlibrary{quotes, angles} +\pgfplotsset{ + compat=1.15, + default 2d plot/.style={% + grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2} + }, +} + +% PAGE GEOMETRY +\geometry{ + bottom=35mm +} + +% PARAGRAPH no indent but skip +\setlength{\parskip}{3mm} +\setlength{\parindent}{0mm} + +\newtheorem{satz}{Satz}[section] +\newtheorem{lemma}[satz]{Lemma} +\newtheorem{korollar}[satz]{Korollar} +\theoremstyle{definition} +\newtheorem{definition}[satz]{Definition} + +%\theoremstyle{definition} +%\newmdtheoremenv{satz}{Satz}[section] +%\newmdtheoremenv{lemma}[satz]{Lemma} +%\newmdtheoremenv{korollar}[satz]{Korollar} +%\newmdtheoremenv{definition}[satz]{Definition} + +\newtheorem{bsp}[satz]{Beispiel} +\newtheorem{bem}[satz]{Bemerkung} +\newtheorem{aufgabe}{Aufgabe} + +% enable aufgaben counting +\regtotcounter{aufgabe} + +% temporary calculation counter +\newcounter{var} + +\newcommand{\N}{\mathbb{N}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} + +% HEADERS + +\pagestyle{fancy} + +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} +\pdfsuppresswarningpagegroup=1 + +% code listings, define style +\lstdefinestyle{mystyle}{ + commentstyle=\color{gray}, + keywordstyle=\color{blue}, + numberstyle=\tiny\color{gray}, + stringstyle=\color{black}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +% activate my colour style +\lstset{style=mystyle} + +% better stackrel +\let\oldstackrel\stackrel +\renewcommand{\stackrel}[2]{% + \oldstackrel{\mathclap{#1}}{#2} +}% + +% integral d sign +\makeatletter \renewcommand\d[2][]{\ensuremath{% + \,\mathrm{d}^{#1}#2\@ifnextchar^{}{\@ifnextchar\d{}{\,}}}} +\makeatother + +% contradiction +\newcommand{\contr}{\text{\Large\lightning}} + +% disjoint unions: provides cupdot and bigcupdot +\makeatletter +\def\moverlay{\mathpalette\mov@rlay} +\def\mov@rlay#1#2{\leavevmode\vtop{% + \baselineskip\z@skip \lineskiplimit-\maxdimen + \ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}} +\newcommand{\charfusion}[3][\mathord]{ + #1{\ifx#1\mathop\vphantom{#2}\fi + \mathpalette\mov@rlay{#2\cr#3} + } + \ifx#1\mathop\expandafter\displaylimits\fi} +\makeatother + +\newcommand{\cupdot}{\charfusion[\mathbin]{\cup}{\cdot}} +\newcommand{\bigcupdot}{\charfusion[\mathop]{\bigcup}{\cdot}} + +\ExplSyntaxOn + +% S-tackrelcompatible ALIGN environment +% some might also call it the S-uper ALIGN environment +% uses regular expressions to calculate the widest stackrel +% to put additional padding on both sides of relation symbols +\NewEnviron{salign} +{ + \begin{align} + \lec_insert_padding:V \BODY + \end{align} +} +% starred version that does no equation numbering +\NewEnviron{salign*} +{ + \begin{align*} + \lec_insert_padding:V \BODY + \end{align*} +} + +% some helper variables +\tl_new:N \l__lec_text_tl +\seq_new:N \l_lec_stackrels_seq +\int_new:N \l_stackrel_count_int +\int_new:N \l_idx_int +\box_new:N \l_tmp_box +\dim_new:N \l_tmp_dim_a +\dim_new:N \l_tmp_dim_b +\dim_new:N \l_tmp_dim_needed + +% function to insert padding according to widest stackrel +\cs_new_protected:Nn \lec_insert_padding:n + { + \tl_set:Nn \l__lec_text_tl { #1 } + % get all stackrels in this align environment + \regex_extract_all:nnN { \c{stackrel}{(.*?)}{(.*?)} } { #1 } \l_lec_stackrels_seq + % get number of stackrels + \int_set:Nn \l_stackrel_count_int { \seq_count:N \l_lec_stackrels_seq } + \int_set:Nn \l_idx_int { 1 } + \dim_set:Nn \l_tmp_dim_needed { 0pt } + % iterate over stackrels + \int_while_do:nn { \l_idx_int <= \l_stackrel_count_int } + { + % calculate width of text + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 1 }$} + \dim_set:Nn \l_tmp_dim_a {\box_wd:N \l_tmp_box} + % calculate width of relation symbol + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 2 }$} + \dim_set:Nn \l_tmp_dim_b {\box_wd:N \l_tmp_box} + % check if 0.5*(a-b) > minimum padding, if yes updated minimum padding + \dim_compare:nNnTF + { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } > { \l_tmp_dim_needed } + { \dim_set:Nn \l_tmp_dim_needed { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } } + { } + \quad + % increment list index by three, as every stackrel produces three list entries + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + } + % replace all relations with align characters (&) and add the needed padding + \regex_replace_all:nnN + { (<&|&<|\c{iff}&|&\c{iff}|\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}{.*?}{.*?}|\c{stackrel}{.*?}{.*?}&|&\c{neq}|\c{neq}&) } + { \c{kern} \u{l_tmp_dim_needed} \1 \c{kern} \u{l_tmp_dim_needed} } + \l__lec_text_tl + \l__lec_text_tl + } +\cs_generate_variant:Nn \lec_insert_padding:n { V } +\ExplSyntaxOff diff --git a/sose2022/bachelorarbeit/arbeit.pdf b/sose2022/bachelorarbeit/arbeit.pdf index 23213a29b122b2155f4eed5c910f99a566869c21..07720783eadee577b75de197cf5ca7b6fea971a3 100644 GIT binary patch delta 293987 zcmV)RK(oJ`pBLxF6%Zv*L`E$!E;R}z(A34)3OA7t6#_IMlkl)8e}#{|Zo@DPgm*s$ 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+\begin{definition}[Derivierte Kategorie] + Wir bezeichnen die triangulierte Lokalisierung $\mathcal{K}_{\mathcal{Q}is}$ + als die derivierte Kategorie $\mathcal{D} = \mathcal{D}(\mathcal{A})$ von $\mathcal{A}$. +\end{definition} + +\begin{bem}[] + Im Folgenden bezeichne $Q = Q_{\mathcal{A}}\colon \mathcal{K}(\mathcal{A}) \to \mathcal{D}(\mathcal{A})$ + den kanonischen Lokalisierungsfunktor. +\end{bem} + +Um zu verstehen, wann zwei Morphismen in $\mathcal{K}$ den selben Morphismus in $\mathcal{D}$ induzieren, hilft +das folgende Lemma: \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 + Seien $\com{A}, \com{B} \in \mathcal{K}$ und $f\colon \com{A} \to \com{B}$. Dann ist + $f = 0$ in $\mathcal{D}$ genau dann wenn ein Quasiisomorphismus $s\colon \com{B} \to \com{C} $ existiert, sodass + $sf = 0$ in $\mathcal{K}$. + + \label{derived-cat-morphism-null} +\end{lemma} + +\begin{proof} + Es ist $f\text{id}^{-1} = 0$ genau dann wenn ein kommutatives Diagram existiert: \[ \begin{tikzcd} - A \arrow{r}{f} & B \arrow{r}{g} & C \arrow{r}{h} & D + & \com{B} \arrow[dashed]{d} & \\ + \com{A} \arrow{ur}{f} \arrow{dr}{0} & \com{C} & \com{B} \arrow[dashed]{l}{s} \arrow{ul}{\text{id}} \arrow{dl}{\text{id}} \\ + & \com{B} \arrow[dashed]{u} & \end{tikzcd} - \] 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 $D_i \to D_{i-1}$. + \] mit $s$ Quasiisomorphismus. +\end{proof} - Sei weiter $j \in I$, s.d. für alle $i > j$ die Folge +Die derivierte Kategorie erlaubt es Ableitungen von Funktoren allgemeiner zu formulieren. Wir führen hier +nur die Situation der Rechtsableitung kovarianter Funktoren aus. +Seien dafür $\mathcal{A}$ und $\mathcal{B}$ abelsche Kategorien und +$F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{A})$ ein triangulierter Funktor. + +Falls $F$ exakt ist werden Quasiisomorphismen auf Quasiisomorphismen geschickt und $F$ induziert daher +einen Funktor von $\mathcal{D}(\mathcal{A})$ nach $\mathcal{D}(\mathcal{B})$. +Im Allgemeinen ist das nicht der Fall. Dennoch möchten wir einen Funktor von $\mathcal{D}(\mathcal{A})$ +nach $\mathcal{D}(\mathcal{B})$ konstruieren, der $F$ so nahe wie möglich kommt. + +\begin{definition}[Abgeleiteter Funktor] + Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien. Der rechts abgeleitete Funktor von $F$ ist + ein triangulierter Funktor \[ - \begin{tikzcd} - A_i' \arrow{r} & B_i' \arrow{r} & C_i' \arrow{r} & D_i' - \end{tikzcd} - \] exakt ist. + \text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{A}) + \] zusammen mit einer natürlichen Transformation + \[ + \xi\colon Q_{\mathcal{B}} \circ F \to \text{R}F \circ Q_{\mathcal{A}} + \] von Funktoren von $\mathcal{K}(\mathcal{A})$ nach $\mathcal{D}(\mathcal{B})$ mit + der folgenden universellen Eigenschaft: Für jeden triangulierten Funktor + \[ + G\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B}) + \] + und jede natürliche Transformation + \[ + \zeta\colon Q_{\mathcal{B}} \circ F \to G \circ Q_{\mathcal{A}} + \] existiert eine eindeutige natürliche Transformation + \[ + \eta\colon \text{R}F \to G + \] von Funktoren von $\mathcal{D}(\mathcal{A})$ nach $\mathcal{D}(\mathcal{B})$, sodass + \[ + \xi = (\eta \circ Q_{\mathcal{A}}) \circ \xi + .\] +\end{definition} - Dann ist die natürliche Abbildung +\begin{bem}[] + \begin{enumerate}[(1)] + \item Falls $\text{R}F$ existiert, ist dieser bis auf eindeutige natürliche Transformationen eindeutig. + \item Wir schreiben $R^{i}F$ für $H^{i}(\text{R}F)$ und wenn $F$ induziert ist von einem + links-exakten Funktor $F\colon \mathcal{A} \to \mathcal{B}$ und $\mathcal{A}$ genügend Injektive hat, sind + das genau die klassischen abgeleiteten Funktoren von $F$. + \end{enumerate} +\end{bem} + +\begin{satz} + Seien $\mathcal{A}, \mathcal{B}$ abelsche Kategorien und + $F\colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{B})$ ein triangulierter Funktor. Angenommen es + existiert eine triangulierte Unterkategorie $\mathcal{L} \subset \mathcal{K}(\mathcal{A})$, s.d. + \begin{enumerate}[(i)] + \item Jeder Komplex in $\mathcal{K}(\mathcal{A})$ besitzt einen Quasiisomorphismus in einen + Komplex aus $\mathcal{L}$. + \item $F|_{\mathcal{L}}$ ist exakt. + \end{enumerate} + Dann existiert der Funktor $\text{R}F\colon \mathcal{D}(\mathcal{A}) \to \mathcal{D}(\mathcal{B})$ und + eine natürliche Transformation $\xi\colon Q_{\mathcal{B}} \circ F \to \text{R}F \circ Q_{\mathcal{A}}$, sodass + für alle $\com{I} \in \mathcal{L}$ die Abbildung \[ - \text{ker } g / \text{im } f \longrightarrow \text{ker } g_j / \text{im } f_j - \] ein Isomorphismus. - \label{0.11} -\end{lemma} + \xi(\com{I})\colon Q_{\mathcal{B}}(F(\com{I}) \to \text{R}F(Q_{\mathcal{A}}(\com{I})) + \] ein Isomorphismus in $\mathcal{D}(A)$ ist. + \label{satz:existence-derived-functors} +\end{satz} + +\begin{bem} + Die Voraussetzungen von \ref{satz:existence-derived-functors} sind im Allgemeinen schwer zu erfüllen. Ziel + dieser Arbeit ist es herauszuarbeiten, dass im Falle von $\mathcal{A} = A\text{-Mod}$ für einen kommutativen + Ring $A$, die Bedingungen für die im Folgenden definierten Funktoren $\com{\text{Hom}}(\com{M}, -)$, + $\com{\text{Hom}}(-, \com{M})$ und $\com{M} \otimes_A -$ erfüllt sind. +\end{bem} + +\begin{definition} + Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe in einer beliebigen abelschen Kategorie $\mathcal{A}$. + 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{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{lemma}[] + Seien $\com{A}, \com{B} \in \mathcal{K}$ zwei Komplexe. Dann gilt für $n \in \Z$: + \[ + H^{n}(\com{\text{Hom}}(\com{A}, \com{B}) = \text{Mor}_{\mathcal{K}}(\com{A}, \com{B}[n]) + .\]\label{hom-compl-cohomgroups} +\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 - $f(a_{i}) = (b_{i})_{i \in \N}$ + Sei $(f^i)_{i \in \Z} \in \prod_{i \in \Z} \text{Hom}(A^{i}, B^{i+n})$. Dann ist: + \[ + (f^{i})_{i \in \Z} \in \text{ker } d^{n} \iff (-1)^{n} d_{\com{B}}^{i+n} f^i = f^{i+1} d_{\com{A} }^{i} + \text{ für } i \in \Z + .\] 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 $(f^{i})_{i \in \Z} \in \text{ker } d^{n}$. + + 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 + %\[ + % f^{i} = d_{\com{B}}^{i+n-1} k^{i} - (-1)^{n-1} k^{i+1} d_{\com{A} }^{i} + % = d_{\com{B}}^{i+n-1} k^{i} + (-1)^{n} k^{i+1} d_{\com{A} }^{i} + %.\] + \[ + (-1)^{n}f^{i} = (-1)^{n} d_{\com{B}}^{i+n-1} k^{i} + k^{i+1} d_{\com{A} }^{i} + .\] 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, + wenn $(f^{i})_{i \in \Z} \in \text{im }d^{n-1}$. +\end{proof} - 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 - $b_j = b$. +\begin{lemma}[Tensorprodukt ist trianguliert] + Sei $\com{M} \in \mathcal{K}(A\text{-Mod})$. + Dann ist $- \otimes_A \com{M}$ (und $\com{M} \otimes_A -$) ein triangulierter Funktor von $\mathcal{K}(A\text{-Mod})$ + nach $\mathcal{K}(A\text{-Mod})$. + \label{satz:tor-is-triangulated} +\end{lemma} + +\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} +% TODO: Bedingung (I) an Indexmengen + +% TODO: verstehe ich nicht aber habe alternative +%\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 +% $E \in \mathcal{G}$. Dann ist $\com{A} $ exakt. +% \label{lemma:0.10} +%\end{lemma} + +\newpage + \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: + \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. @@ -281,7 +560,7 @@ $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} .\] \item Sei $(\com{X}, \com{Y}, \com{Z}, u, v, w)$ ein ausgezeichnetes Dreieck mit $\com{X}, \com{Y} $ K-projektiv und $\com{S} $ exakt. Nach Anwenden von $\text{Mor}_{\mathcal{K}}(-, \com{S})$ - \ref{hom-cohom-func} ist dann + ist dann mit \ref{hom-cohom-func} \[ \underbrace{\text{Mor}_{\mathcal{K}}(\com{X}[1], \com{S})}_{= 0} \to \text{Mor}_{\mathcal{K}}(\com{Z}, \com{S} ) @@ -289,7 +568,7 @@ $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} \underbrace{\text{Mor}_{\mathcal{K}}(\com{Y}, \com{S} )}_{= 0} \] exakt und die äußeren Terme $0$ nach Voraussetzung und (i). Also folgt $\text{Mor}_{\mathcal{K}}(\com{Z}, \com{S}) = 0$, also $\com{Z} $ K-projektiv. Der allgemeine Fall folgt nun - mit \ref{TR2}. + mit \hyperref[TR2]{TR2}. \end{enumerate} \end{proof} @@ -306,7 +585,7 @@ $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} \text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{D}}(\com{P} , \com{S} ) \] ein Isomorphismus. \end{enumerate} - \label{satz:mork=mord-fuer-kproj} + \label{satz:mork=mord-for-kproj} \end{satz} \begin{proof} @@ -380,7 +659,7 @@ $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} \com{P} \arrow[dashed]{ur}{g} \arrow{r}{\text{id}^{-1}f} & \com{Y} \end{tikzcd} .\] Da $s$ Quasiisomorphismus, ist $s$ Isomorphismus in $\mathcal{D}$, also existiert ein - $g\colon \com{P} \to \com{X} $ in $\mathcal{D}$, sodass das Diagramm kommutiert. \ref{satz:mork=mord-fuer-kproj} + $g\colon \com{P} \to \com{X} $ in $\mathcal{D}$, sodass das Diagramm kommutiert. \ref{satz:mork=mord-for-kproj} (iii) liefert das gewünschte Diagramm in $\mathcal{K}$. (ii)$\implies$(iii): Betrachte @@ -391,7 +670,7 @@ $X^0$ projektiv, existiert $k^{0}\colon A^{0} \to S^{-1}$, s.d. $f^{0} = d^{-1} \end{tikzcd} .\] Da $s$ Quasiisomorphismus existiert mit (ii) ein $f\colon \com{P} \to \com{S}$, s.d. $sf = \text{id}_{\com{P} }$. - (iii)$\implies$(ii): Erneut mit \ref{satz:mork=mord-fuer-kprof} genügt es zu zeigen, dass für + (iii)$\implies$(ii): Erneut mit \ref{satz:mork=mord-for-kproj} genügt es zu zeigen, dass für $\com{S} \in \mathcal{K}$ $\text{Mor}_{\mathcal{K}}(\com{P} , \com{S} ) \to \text{Mor}_{\mathcal{D}}(\com{P} , \com{S} ) $ bijektiv ist. Injektivität: Sei $f\colon \com{P} \to \com{S} $ mit $\text{id}^{-1}f = 0$ in $\mathcal{D}$. Dann @@ -443,8 +722,14 @@ Durch Umdrehen aller Pfeile erhalten wir analog: \subsection{Spezielle inverse und direkte Systeme} -Um für einen Komplex $\com{A} $ eine K-injektive bzw. K-projektive Auflösung zu erhalten, konstruieren wir bestimmte -inverse bzw. direkte Systeme deren Limites die gewünschten Auflösungen liefern. Dazu benötigen wir folgenden Begriff: +In der Notation von +\ref{satz:existence-derived-functors} +möchten wir die Klasse der K-injektiven bzw. K-projektiven Komplexe als $\mathcal{L}$ +für die Funktoren +$\com{\text{Hom}}(\com{M}, -)$ bzw. $\com{\text{Hom}}(-, \com{M})$ für $\com{M} \in \mathcal{K}$ verwenden. + +Nun stellt sich die Aufgabe für jeden Komplex $\com{A} \in \mathcal{K}$ eine K-injektive bzw. eine K-projektive +Auflösung zu konstruieren. Dies machen wir mit sogenannten speziellen inversen bzw. direkten Systemen. \begin{definition}[Spezielles inverses System] Sei $\mathcal{J} \subset \mathcal{K}$ eine Klasse von Komplexen. @@ -507,6 +792,124 @@ abgeschlossen unter speziellen inversen Limites ist. Dazu benötigen wir die fol $\com{A} = \lim \com{S}_n \in \mathcal{J}$. \end{proof} +Unser Ziel ist nun zu zeigen, dass die Klasse der exakten Komplexe abgeschlossen ist bezüglich spezieller +inverser Limites. Da der inverse Limes im Allgemeinen nicht exakt ist, benötigen wir ein technisches Hilfswerkzeug. + +Wir benötigen im Folgenden mehrmals eine bestimmte Bedingung an inverse Systeme in +$\mathcal{A}b$. + +\begin{definition} + Wir sagen eine totalgeordnete Indexmenge $(I, \le)$ genüge der Bedingung (S), wenn eine ordnungserhaltende + Bijektion $\iota\colon I \to \N$ existiert. Für $i \in I$ bezeichne, falls dieser existiert, mit $i+1$ bzw. $i-1$, den + von $\iota$ induzierten Vorgänger bzw. Nachfolger von $i$. + + 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} +\end{definition} + +\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 $(D_i)_{i \in I}$ + 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} + + \begin{lemma} Die Klasse der exakten Komplexe in $\mathcal{K}(\mathcal{A}b)$ ist abgeschlossen unter speziellen inversen Limites. @@ -515,7 +918,7 @@ abgeschlossen unter speziellen inversen Limites ist. Dazu benötigen wir die fol \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 P aus \ref{0.11}. Sei also $i \in \Z$ beliebig. Dann erfüllt + erfüllt $(S_n^{i})_{n \in \N}$ die Bedingung (R) aus \ref{0.11}. 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 $n > 1$ $\text{ker}(\com{S}_n \to \com{S}_{n-1})$ @@ -629,13 +1032,13 @@ Ebenfalls analog gilt: \begin{satz} Sei $\mathcal{B}$ eine weitere abelsche Kategorie und $\mathcal{J} \subset \mathcal{K}(B)$ abgeschlossen - unter speziellen inversen Colimites. Angenommen direkte Colimites existieren in $\mathcal{A}$ und sei + unter speziellen inversen Limites. Angenommen direkte Colimites existieren in $\mathcal{A}$ und sei $F\colon \mathcal{K}(A) \to \mathcal{K}(B)$ ein kontravarianter Funktor, der direkte Colimites in inverse Limites überführt und gradweise zerfallende kurze exakte Folgen erhält. Dann ist $F^{-1}(\mathcal{J}) \subset \mathcal{K}(\mathcal{A})$ abgeschlossen unter speziellen direkten Colimites. - \label{satz:complete-inv-system-functor} + \label{satz:complete-dir-system-functor} \end{satz} \begin{korollar}[] @@ -722,8 +1125,11 @@ folgende Lemma führt diese Konstruktion aus. 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 $f = a_{n-1}f_{n-1}$. - + und $f = a_{n-1}f_{n-1}\colon \com{P} \to \com{B}$. Es gilt dann + \begin{equation} + f d_P = d_B f + \label{eq:f-comp-hom} + \end{equation} 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 @@ -732,15 +1138,16 @@ folgende Lemma führt diese Konstruktion aus. 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: - \[ + \begin{equation} \begin{tikzcd} \cdots \arrow{r} & Q^{i} \arrow{r}{d_{Q}} \arrow{d}{(g', g'')} & Q^{i+1} \arrow{r} \arrow{d}{(g', g'')} & \cdots\\ \cdots \arrow{r} & P^{i} \oplus B^{i-1} \arrow{r}{d_{C_f[-1]}} & P^{i+1} \oplus B^{i} \arrow{r} & \cdots - \tag{$*$} \label{eq:1} + \label{eq:1} \end{tikzcd} - \] In Matrixnotation ist + \end{equation} + In Matrixnotation ist \begin{align*} d_{C_f} &= \begin{pmatrix} d_{P}[1] & 0 \\ f[1] & d_{B} \end{pmatrix} \intertext{Also folgt} @@ -775,7 +1182,7 @@ folgende Lemma führt diese Konstruktion aus. \begin{pmatrix} d_B g'' & f d_P \end{pmatrix} \\ - &\stackrel{\eqref{}}{=} + &\stackrel{\eqref{eq:f-comp-hom}}{=} \begin{pmatrix} d_B g'' & d_B f \end{pmatrix} \\ @@ -1070,6 +1477,28 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom die Behauptung aus den Definitionen. \end{proof} +Im Folgenden benötigen wir folgendes Kritierium für die Exaktheit von Komplexen: + +\begin{lemma} + Sei $\com{A} \in \mathcal{K}$. Angenommen für jeden K-injektiven Komplex $\com{I} \in \mathcal{K}$ ist + $\com{\text{Hom}}(\com{A}, \com{I})$ exakt. Dann ist $\com{A}$ exakt. + \label{lemma:0.10} +\end{lemma} + +\begin{proof} + Sei $\com{B} \in \mathcal{K}$ beliebig. Dann existiert nach \ref{satz:existence-k-inj-resolution} ein + K-injektiver Komplex $\com{I} \in \mathcal{K}$ und ein Quasiisomorphismus $\com{B} \to \com{I}$. Dann + gilt $\com{B} \stackrel{\sim }{=} \com{I} $ in $\mathcal{D}$ und wir erhalten + \[ + \text{Mor}_{\mathcal{D}}(\com{A}, \com{B}) \stackrel{\sim }{=} + \text{Mor}_{\mathcal{D}}(\com{A} , \com{I} ) \stackrel{\ref{satz:mork=mord-for-k-inj}}{=} + \text{Mor}_{\mathcal{K}}(\com{A} , \com{I} ) = 0 + .\] + Mit Yoneda folgt nun, dass $\com{A} \stackrel{\sim }{=} 0$ in $\mathcal{D}$. Da per Definition + $H^{i}(-)\colon \mathcal{K} \to \mathcal{A}b$ über den kanonischen Funktor $Q\colon \mathcal{K} \to \mathcal{D}$ + faktorisiert, folgt $H^{i}(\com{A}) = 0$ für $i \in \Z$, also $\com{A}$ exakt. +\end{proof} + \begin{satz}[] Sei $\com{M} \in \mathcal{K} $. Dann sind äquivalent: \begin{enumerate}[(i)] @@ -1178,10 +1607,10 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom 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} + \label{eq:cohom-groups-1} .\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}. + $\com{M} \otimes_A \com{P} $ exakt ist. Damit folgt die Behauptung aus \eqref{eq:cohom-groups-1}. \end{proof} \begin{satz}[] @@ -1200,7 +1629,7 @@ Um das Tensorprodukt abzuleiten, benötigen wir noch eine weitere Klasse von Kom 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} + \label{eq:cohom-groups-2} .\end{equation} Da $\com{P} $ K-projektiv und $\com{I} $ exakt, folgt die Exaktheit der rechten Seite und damit die Behauptung. \end{proof} @@ -1210,6 +1639,7 @@ 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 $\com{M} \in \mathcal{K}$. + \label{satz:hom-exact-for-k-proj} \end{satz} \subsection{Abgeleitete $\com{\text{Hom}}$ Funktoren} @@ -1295,7 +1725,7 @@ Jetzt können wir alles zusammentragen und erhalten: \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}}{=} + &\stackrel{\ref{satz:adjunction-hom-tor-comp}}{=} \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}))