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+\usepackage{amsmath, amssymb, amsthm} +\usepackage{tikz-cd} + +\newcommand{\spec}{\operatorname{Spec }} + +\begin{document} + +\section{Projektive Moduln und Algebren} + +\begin{satz}[Projektiv ist lokal frei] + Sei $A$ ein Ring und $M$ ein $A$-Modul. Die folgenden Eigenschaften sind äquivalent: + \begin{enumerate}[(i)] + \item $M$ ist endlich erzeugter projektiver $A$-Modul. + \item Es existieren Elemente $\{f_i\}_{i \in I}$ von $A$ mit $\sum_{i \in I} (f_i) = A$, sodass + $M_{f_i}$ freier, endlich erzeugter $A_{f_i}$ Modul ist. + \end{enumerate} + \label{satz:projectiveislocallyfree} +\end{satz} + +\begin{proof} + Siehe Theorem 4.6 in \cite{lenstra}. +\end{proof} + +\begin{satz} + Sei $B$ endliche, projektive $A$-Algebra. Dann gilt + \begin{enumerate}[(a)] + \item $A \to B$ ist genau dann injektiv, wenn $[B : A] \ge 1$. + \item $A \to B$ ist genau dann surjektiv, wenn $[B : A ] \le 1$. + \item $A \to B$ ist genau dann ein Isomorphismus, wenn $[ B : A ] = 1$. + \end{enumerate} + \label{satz:rings-degree} +\end{satz} + +\begin{satz}[] + Sei $B$ eine $A$-Algebra und $C$ eine treuflache $A$-Algebra, sodass + $B \otimes_A C$ projektive, separable $C$-Algebra ist. Dann ist $B$ projektive, separable $A$-Algebra. + \label{satz:4.14} +\end{satz} + +\begin{proof} + Vortrag 8. Theorem 4.14 in \cite{lenstra}. +\end{proof} + +\section{Endlich étale Morphismen} + +\begin{definition}[Affiner Morphismus] + Sei $f\colon Y \to X$ ein Morphismus von Schemata. $f$ ist \emph{affin}, wenn + eine offene affine Überdeckung $\{U_i\}_{i \in I}$ von $X$ existiert, sodass + $f^{-1}(U_i)$ affin ist für alle $i \in I$ +\end{definition} + +\begin{bem} + $f\colon Y \to X$ ist genau dann affin, wenn für jede offene affine Menge $U \subseteq X$, + $f^{-1}(U)$ affin ist. +\end{bem} + +\begin{definition} + Sei $f\colon Y \to X$ ein affiner Morphismus von Schemata. $f$ ist \emph{endlich und lokal frei}, wenn + eine offene affine Überdeckung $\{U_i\}_{i \in I}$ existiert mit $U_i = \text{Spec }A_i$, sodass + $f^{-1}(U_i) = \text{Spec }B_i$, wobei $B_i$ eine endliche und freie $A_i$-Algebra ist. +\end{definition} + +\begin{lemma}[] + Sei $f\colon A \to B$ ein Ringhomomorphismus und $S \subseteq A$ ein multiplikatives System. Dann + ist $f(S)$ ein multiplikatives System von $B$ und + \[ + S^{-1}B \simeq f(S)^{-1}B + \] als $S^{-1}A$-Algebren. + \label{lemma:localisation} +\end{lemma} + +\begin{proof} + Die Formel $\frac{b}{s} \mapsto \frac{b}{f(s)}$ induziert den Isomorphismus. +\end{proof} + +\begin{lemma} + Sei $f\colon A \to B$ ein Ringhomomorphismus und $\varphi\colon \text{Spec }B \to \text{Spec }A$ der induzierte + Morphismus affiner Schemata. Sei $g \in A$. Dann ist + \[ + \varphi^{-1}(D(g)) = D(f(g)) + .\] Insbesondere gilt + \[ + \varphi^{-1}(\text{Spec }A_g) = \text{Spec }B_g + .\] + \label{lemma:d(f)} +\end{lemma} + +\begin{proof} + Die erste Gleichung ist aus Algebra 2 bekannt und gilt allgemeiner für Morphismen lokal geringter Räume. Die + zweite Gleichung folgt aus der Ersten, wenn der Isomorphismus $D(g) = \text{Spec }A_g$ eingesetzt wird, unter + Verwendung des Ringisomorphismus + \[ + B_g = B \otimes_A A_g \simeq B_{f(g)} + .\] +\end{proof} + +\begin{satz} + Sei $f\colon Y \to X$ ein Morphismus von Schemata. Dann ist + $f$ genau dann endlich und lokal frei, wenn für jede offene affine Menge $U = \text{Spec }A$ von $X$, + $f^{-1}(U)$ affin mit + $f^{-1}(U) = \text{Spec }B$ und $B$ eine endliche, projektive $A$-Algebra ist. + \label{satz:morph-local-free-char} +\end{satz} + +\begin{proof} + ($\Rightarrow$) + %Sei $D$ endliche freie $C$-Algebra. Dann ist $D_g$ endliche freie $C_g$-Algebra für $g \in C$. Außerdem ist + %$\text{Spec }C_g = D(g) \subseteq \text{Spec }C$ eine Basis von $\text{Spec }C$. + Sei $X = \bigcup_{i \in I} U_i$ mit $U_i = \text{Spec }A_i$ offen, $f^{-1}(U_i) = \text{Spec }B_i$ + und $B_i$ endliche freie $A_i$-Algebra. Nun sei $g \in A_i$ beliebig. + Mit \ref{lemma:d(f)} ist $f^{-1}((A_i)_g) = \text{Spec }(B_i)_g$ und + da Lokalisieren Freiheit und Rang von Moduln erhält, ist $(B_i)_{g}$ endliche freie $(A_i)_g$-Algebra. +Da die Mengen der Form $D(g) = \text{Spec }(A_i)_g$ mit +$g \in A_i$ eine Basis von $U_i = \text{Spec }A_i$ bilden, können wir + ohne Einschränkung annehmen, dass $\{U_i\}_{i \in I}$ bereits eine Basis von $X$ ist, insbesondere + existiert ein $J \subseteq I$, sodass $U = \bigcup_{j \in J} U_j$. + + Ebenfalls + ist $\{D(g)\}_{g \in A}$ eine Basis von $U$, also existiert für alle $j \in J$ eine Familie + $\{g_{jk}\}_{k \in K_j}$, sodass $U_j = \bigcup_{k \in K_j} D(g_{jk})$. Außerdem + ist $D(g_{jk}) \subseteq U_j$ also + \[ + \text{Spec }A_{g_{jk}} = D(g_{jk}) = D(g_{jk}) \cap U_j = \text{Spec }(A_{j})_{g_{jk}} + .\] Damit folgt $f^{-1}(\text{Spec }A_{g_{jk}}) = \text{Spec }B_{g_{jk}}$ mit $B_{g_{jk}}$ + endliche freie $A_{g_{jk}}$-Algebra. Außerdem ist $U = \bigcup D(g_{jk})$, also + $\sum_{} (g_{jk}) = A$. Also folgt mit \ref{satz:projectiveislocallyfree}, + dass $B$ eine endliche, projektive $A$-Algebra ist. + + ($\Leftarrow$) Sei $X = \bigcup_{i \in I} U_i$ mit $U_i = \text{Spec }A_i$ offen. Dann + ist $f^{-1}(U_i) = \text{Spec }B_i$ mit einer endlichen, projektiven $A_i$-Algebra $B_i$. Nach + \ref{satz:projectiveislocallyfree} + existieren $\{g_{ij}\}_{j \in J_i} \subseteq A_i$, sodass $\sum_{j \in J_i} (g_{ij}) = A_i$ und + $(B_i)_{g_{ij}}$ endliche, freie $(A_{i})_{g_{ij}}$-Algebra. Es folgt + \[ + U_i = \bigcup_{j \in J_i} D(f_{ij}) = \bigcup_{j \in J_i} \text{Spec } (A_i)_{g_{ij}} + .\] Da die $\{U_i\}_{i \in I}$ eine Überdeckung von $X$ sind, folgt die Behauptung. +\end{proof} + +\begin{bem}[Grad] + Sei $f\colon Y \to X$ ein endlicher, lokal freier Morphismus von Schemata und $U = \text{Spec }A$ offen in $X$ + mit $f^{-1}(U) = \text{Spec }B$. Nach 4.9 existiert eine stetige Funktion + \[ + [B : A] \colon U = \text{Spec }A \to \Z, \mathfrak{p} \mapsto \text{rang}_{A_{\mathfrak{p}}} B_{\mathfrak{p}} + .\] + Sei $U' = \text{Spec }A'$ offen in $X$ mit $f^{-1}(U') = \text{Spec }B'$. Dann stimmen + $[B' : A']$ und $[B : A]$ auf $U \cap U'$ überein, d.h. wir erhalten eine Funktion + \[ + \text{deg}(f) = [Y : X] \colon \operatorname{sp}(X) \to \Z + ,\] wobei $\operatorname{sp}(X)$ den unterliegenden topologischen Raum von $X$ bezeichne. +\end{bem} + +\begin{proof} + Sei zunächst $U \subseteq U'$ und $x \in U$. Da $\{D(g)\}_{g \in A'}$ eine Basis von $U'$ ist, existiert ein + $g \in A'$, sodass $x \in D(g) \subseteq U$. Dann ist + \[ + \text{Spec }A'_{g} = D(g) = D(g) \cap U = \text{Spec }A_{g_{|U}} + .\] Also folgt + \[ + [B' : A'](x) = [B'_g : A'_g](x) = [B_{g_{|U}} : A_{g_{|U}} ](x) = [ B : A ](x) + .\] Im Allgemeinen sei $x \in U \cap U'$. Dann existiert eine offene affine Menge $x \in V \subseteq U \cap U'$ + und wir können zweimal den Spezialfall für $V \subseteq U$ und $V \subseteq U'$ anwenden. +\end{proof} + +\begin{lemma} + Sei $f\colon Y \to X$ endlich und lokal frei. Dann ist + $[Y : X]$ lokalkonstant, das heißt eine stetige Abbildung $\operatorname{sp}(X) \to \Z$, wobei $\Z$ die diskrete + Topologie trägt. + Insbesondere ist die Menge + \[ + \{x \in \operatorname{sp}(X) \mid [Y : X](x) = n\} + \] offen und abgeschlossen in $X$ und $[Y : X]$ ist konstant, falls $X$ zusammenhängend ist. +\end{lemma} + +\begin{proof} + Lokalisieren ist exakt, erhält also den Rang von freien Moduln und $X$ ist lokal frei. +\end{proof} + +\begin{definition}[Surjektive Morphismen] + Ein Morphismus $Y \to X$ von Schemata heißt \emph{surjektiv}, falls die zugrundeliegende Abbildung + zwischen den topologischen Räumen surjektiv ist. +\end{definition} + +\begin{satz} + Sei $f\colon Y \to X$ endlich und lokal frei. Dann gilt + \begin{enumerate}[(a)] + \item $Y = \emptyset \iff [Y : X] = 0$. + \item $Y \to X$ Isomorphismus $\iff [Y : X] = 1$. + \item $Y \to X$ surjektiv $\iff [Y : X] \ge 1 \iff$ für alle offenen affinen Teilmengen $U = \operatorname{Spec }A$ von + $X$ ist $f^{-1}(U) = \spec B$, wobei $B$ eine treuprojektive $A$-Algebra ist. + \end{enumerate} + \label{satz:degree} +\end{satz} + +\begin{proof} + Alle Eigenschaften sind lokal auf $X$, das heißt oE sei $X = \spec A$ affin. Da $f$ affin ist, folgt + $Y = \spec B$, wobei $B$ endliche und projektive $A$-Algebra ist. + \begin{enumerate}[(a)] + \item $Y = \emptyset \iff B = 0 \iff B_{\mathfrak{p}} = 0$ + $\forall \mathfrak{p} \in \spec A \iff \text{rang}_{A_{\mathfrak{p}}} B_{\mathfrak{p}} = 0$ + $\forall \mathfrak{p} \in \spec A$ + $\iff [B : A] = 0$. + \item Das ist \ref{satz:rings-degree}(c). + \item Die zweite Äquivalenz gilt nach Definition von treuprojektiv. Für die erste: + Sei $\spec B \to \spec A$ surjektiv und sei $\mathfrak{p} \in \spec A$ mit Urbild + $\mathfrak{q} \in \spec{B}$. Also ist $B \neq 0$ und damit $B_{\mathfrak{q}} \neq 0$. + Sei $\varphi\colon A \to B$ der induzierte Ringhomomorphismus, + $S = \varphi(A \setminus \mathfrak{p})$ und $T = B \setminus \mathfrak{q}$. Wegen + $\mathfrak{p} = f(\mathfrak{q}) = \varphi^{-1}(\mathfrak{q})$, folgt + $S \subseteq T$. Und damit + \[ + B_{\mathfrak{q}} = T^{-1}B \simeq (S^{-1}T)^{-1}(S^{-1}B) = (S^{-1}T)^{-1} B_{\mathfrak{p}}, + \] also ist $B_{\mathfrak{q}}$ eine Lokalisierung von $B_{\mathfrak{p}}$. Also folgt + auch $B_{\mathfrak{p}} \neq 0$, also + $[ B : A ](\mathfrak{p}) > 0$. Rückrichtung: Mit \ref{satz:rings-degree}(a) ist $A \to B$ injektiv + und weil $B$ endliche $A$-Algebra, ist $B$ ganze Ringerweiterung von $A$, also folgt + die Aussage aus \cite{macdonald} Theorem 5.10. + \end{enumerate} +\end{proof} + +\begin{definition}[Endlich étaler Morphismus] + Sei $f\colon Y \to X$ ein affiner Morphismus von Schemata. $f$ ist \emph{endlich étale}, falls + eine offene affine Überdeckung $\{U_i\}_{i \in I}$ existiert mit + $U_i = \spec A_i$, sodass $f^{-1}(U_i) = \spec B_i$, wobei $B_i$ eine freie, separable $A_i$-Algebra ist. +\end{definition} + +\begin{bem} + Jeder endlich étale Morphismus ist endlich und lokal frei, denn separable Algebren sind per Definition endlich. +\end{bem} + +%\begin{lemma} +% Seien $M, N$ $A$-Moduln und $M$ endlich präsentiert, d.h. es existiert eine exakte Folge +% \[ +% A^{m} \to A^{n} \to M \to 0 +% .\] Sei weiter $S \subset A$ ein multiplikatives System. Dann ist der natürliche $A$-Modul Homomorphismus +% \[ +% S^{-1}\operatorname{Hom}_A(M, N) \to \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N) +% \] ein Isomorphismus. +% \label{lemma:localisation-finitely-pres} +%\end{lemma} +% +%\begin{proof} +% $S^{-1}A$ ist ein flacher $A$-Modul und $\operatorname{Hom}_A(-, N)$ bzw. $\operatorname{Hom}_{S^{-1}A}(-, S^{-1}N)$ +% sind linksexakt. So erhalten wir exakte Folgen +% \[ +% 0 \to \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N) +% \to (S^{-1}N)^{n} \to (S^{-1}N)^{m} +% \] und +% \[ +% 0 \to S^{-1}\operatorname{Hom}_{A}(M, N) \to (S^{-1}N)^{n} \to (S^{-1}N)^{m} +% .\] Das 5-er Lemma liefert das Ergebnis. +%\end{proof} + +\begin{lemma}[] + Sei $M$ endlich präsentierter $A$-Modul, das heißt es existiere eine exakte Folge + \[ + A^{m} \to A^{n} \to M \to 0 + .\] Sei weiter $C$ eine $A$-Algebra. Dann ist der natürliche $A$-Modulhomomorphismus + \[ + \operatorname{Hom}_A(M, A) \otimes_A C \to \operatorname{Hom}_{C}(M \otimes_A C, C) + \] ein Isomorphismus. + \label{lemma:tensor-and-hom} +\end{lemma} + +\begin{proof} + Da Tensorieren rechtsexakt ist und $\operatorname{Hom}_{C}(-, C)$ linksexakt, erhalten wir durch anwenden + in dieser Reihenfolge auf die exakte Folge $A^{m} \to A^{n} \to M \to 0$, die exakte Folge + \[ + 0 \to \operatorname{Hom}_{C}(M \otimes_A C, C) \to C^{n} \to C^{m} + .\] Andererseits liefert zunächst anwenden von $\operatorname{Hom}_{A}(-, A)$ die exakte Folge + \[ + 0 \to \operatorname{Hom}_{A}(M, A) \to A^{n} \to A^{m} + .\] Tensorieren mit $C$ liefert die exakte Folge + \[ + \underbrace{\operatorname{Tor}_{1}^{A}(A^{m}, C)}_{= 0} \to \operatorname{Hom}_{A}(M, A) \otimes_A C \to C^{n} \to C^{m} + .\] Der linke Term verschwindet, weil $A^{m}$ flach ist. Untereinanderschreiben der beiden Folgen mit + den natürlichen Homomorphismen und Auffüllen mit $0$ nach links liefert ein kommutatives Diagramm und mit dem 5-er Lemma + die Behauptung. +\end{proof} + +\begin{bem} + \ref{lemma:tensor-and-hom} wendet sich insbesondere dann an, wenn $M$ endlich erzeugter projektiver Modul ist. +\end{bem} + +\begin{korollar} + Sei $B$ endliche, projektive $A$-Algebra und $S \subset A$ ein multiplikatives System. Dann + ist der natürliche $A$-Modulhomomorphismus + \[ + \operatorname{Hom}_{A}(B, A) \to \operatorname{Hom}_{S^{-1}A}(S^{-1}B, S^{-1}A) + \] ein Isomorphismus. + \label{lemma:localisation-finitely-pres} +\end{korollar} + +\begin{korollar} + Ein Morphismus von Schemata $f\colon Y \to X$ ist genau dann endlich étale, wenn + eine Basis von offenen affinen Mengen $\{U_i\}_{i \in I}$ von $X$ existiert, sodass + $U_i = \spec A_i$ und $f^{-1}(U_i) = \spec B_i$, wobei $B_i$ freie, separable $A_i$-Algebra ist. + + \label{bem:finite-etale-basis} +\end{korollar} + +\begin{proof} + Die Rückrichtung ist klar. Für die Hinrichtung beachte, dass eine endliche, projektive $A$-Algebra $B$ genau dann + separabel ist, wenn der von der Spur induzierte $A$-Modulhomomorphismus $B \to \operatorname{Hom}_A(B, A)$ + ein Isomorphismus ist. Diese Eigenschaft bleibt nach \ref{lemma:localisation-finitely-pres} + durch Lokalisieren erhalten. +\end{proof} + +\begin{lemma} + Sei $B$ eine endliche, projektive $A$-Algebra und + $\phi\colon B \to \operatorname{Hom}_A(B, A)$ die von der Spur induzierte Abbildung. Dann ist + $B$ genau dann separabel über $A$, wenn die folgenden äquivalenten Bedingungen erfüllt sind: + \begin{enumerate}[(i)] + \item $\phi$ ist ein Isomorphismus. + \item Die induzierte Abbildung + $B_{\mathfrak{p}} \to \operatorname{Hom}_{A_{\mathfrak{p}}}(B_{\mathfrak{p}}, A_{\mathfrak{p}})$ + ist ein Isomorphismus für alle $\mathfrak{p} \in \spec A$. + \end{enumerate} + \label{lemma:separable-is-local} +\end{lemma} + +\begin{proof} + Isomorphismus zu sein ist eine lokale Eigenschaft und $B$ ist endlich erzeugter, projektiver $A$-Modul, das + heißt insbesondere endlich präsentiert, also folgt die Behauptung mit \ref{lemma:localisation-finitely-pres}. +\end{proof} + +\begin{satz} + Sei $f\colon Y \to X$ ein Morphismus von Schemata. Dann ist $f$ genau dann endlich étale, wenn + für jede offene affine Teilmenge $U = \spec A$ von $X$, $f^{-1}(U)$ affin mit $f^{-1}(U) = \spec B$ und + $B$ eine projektive, separable $A$-Algebra ist. + \label{satz:equiv-finite-etale} +\end{satz} + +\begin{proof} + ($\Rightarrow$) Sei $U = \spec A$ offen in $X$ und sei $f^{-1}(U) = \spec B$. + Dann ist $B$ nach \ref{satz:morph-local-free-char} endliche, projektive $A$-Algebra. + Sei $\mathfrak{p} \in \spec A$ und $x \in U$ der zugehörige Punkt. Dann + ist $A_{\mathfrak{p}} = \mathcal{O}_{U,x} = \mathcal{O}_{V,x}$ für jede offene Menge $x \in V \subseteq U$. + Nach \ref{bem:finite-etale-basis} existiert + $x \in V \subseteq U$ offen affin mit $V = \text{Spec }A'$ und $f^{-1}(V) = \text{Spec }B'$, wobei + $B'$ freie, separable $A'$-Algebra. Dann ist nach \ref{lemma:localisation-finitely-pres} + $B'_{\mathfrak{p}} \to \operatorname{Hom}_{A'_{\mathfrak{p}}}(B'_{\mathfrak{p}}, A'_{\mathfrak{p}})$ ein + Isomorphismus, also auch + $B_{\mathfrak{p}} \to \operatorname{Hom}_{A_{\mathfrak{p}}}(B_{\mathfrak{p}}, A_{\mathfrak{p}})$. Also + nach \ref{lemma:separable-is-local} ist $B$ separable $A$-Algebra. + + ($\Leftarrow$) Da endlich étale eine auf $X$ lokale Eigenschaft ist, sei oE $X = \spec A$. Dann ist + $Y = f^{-1}(X) = \spec B$ mit $B$ projektive, separable $A$-Algebra. Nach \ref{satz:projectiveislocallyfree} + existieren $\{f_i\}_{i \in I}$, sodass $\sum_{i \in I} (f_i) = A$ und + $B_{f_i}$ endliche, freie $A_{f_i}$-Algebra. Da Lokalisieren Separabilität erhält, ist $B_{f_i}$ auch separabel + über $A_{f_i}$ und es folgt die Behauptung wegen $X = \bigcup_{i \in I} D(f_i)$. +\end{proof} + +\begin{satz}[Basiswechsel] + Sei $f\colon Y \to X$ endlich étale und $g\colon W \to X$ ein Morphismus von Schemata. Dann ist + $Y \times_{X} W \to W$ endlich étale. + \label{satz:basischange} +\end{satz} + +\begin{proof} + Endlich étale ist eine lokale Eigenschaft auf $W$ und damit insbesondere auf $X$. Genauer: + Sei $w \in W$ beliebig. Dann existiert eine offene affine Menge $g(w) \in U \subseteq X$ und + offene affine Mengen $U_i \subseteq W$, sodass $g^{-1}(U) = \bigcup_{i \in I} U_i$. Da $w \in g^{-1}(U)$ existiert + ein $i_0 \in I$, sodass $w \in U_{i_0}$. Durch Ersetzen von $W$ durch $U_{i_0}$ und $X$ durch $U$ können + wir also oE annehmen, dass $W = \spec C$ und $X = \spec A$. Da $f$ affin, ist damit + auch $Y = f^{-1}(X) = \spec B$ affin. Dann ist $Y \times_{X} W = \spec B \otimes_A C$. + + Es bleibt also folgende Aussage zu zeigen: Sei $B$ projektive, separable $A$-Algebra und $C$ eine $A$-Algebra. Dann + ist $B \otimes_A C$ projektive, separable $C$-Algebra. + + Zunächst ist $B \otimes_A C$ projektiv, denn da $B$ endlich erzeugter, projektiver $A$-Modul ist, + existiert ein $A$-Modul $Q$, sodass + $A^{n} \simeq B \oplus Q$ als $A$-Moduln für ein $n \ge 0$. Da der natürliche Isomorphismus + $A^{n} \otimes_A C \to C^{n}$ auch $C$-linear ist, folgt durch + Tensorieren mit $C$ + \[ + C^{n} \simeq A^{n} \otimes_A C \simeq (B \oplus Q) \otimes_A C = (B \otimes_A C) \oplus (Q \otimes_A C) + .\] Also ist $B \otimes_A C$ projektiver $C$-Modul. + + Für die Separabilität ist zu zeigen, dass der von der Spurabbildung induzierte Homomorphismus + $B \otimes_A C \to \operatorname{Hom}_{C}(B \otimes_A C, C)$ ein Isomorphismus ist. Das folgt aus + \ref{lemma:tensor-and-hom} und dem kommutativen Diagramm: + \[ + \begin{tikzcd} + B \otimes_A C \arrow{r} & \operatorname{Hom}_C(B\otimes_A C, C) \\ + B \otimes_A C \arrow{u}{\operatorname{id}} \arrow[swap]{r}{\sim} + & \operatorname{Hom}_A(B, A) \otimes_A C \arrow[swap]{u}{\sim} + \end{tikzcd} + .\] +\end{proof} + +\begin{satz} + Sei $f\colon Y \to X$ affiner Morphismus von Schemata und $g\colon W \to X$ ein + surjektiver, endlicher und lokal freier Morphismus. Dann ist $Y \to X$ genau dann endlich étale, wenn + $Y \times_X W \to W$ endlich étale ist. +\end{satz} + +\begin{proof} + Die Hinrichtung gilt für beliebige Basiswechsel nach \ref{satz:basischange}. Zur Rückrichtung: Sei + $U \subseteq X$ affin und $U = \spec A$. + Also ist $f^{-1}(U) = \spec B$, da $f$ affin. Es genügt nun zu zeigen, dass $B$ projektive, separable $A$-Algebra ist. + + Nach \ref{satz:degree} ist $g^{-1}(U) = \spec C$, wobei + $C$ eine endliche, treuprojektive $A$-Algebra ist. Sei $\mathfrak{m}$ ein Maximalideal von $A$. Angenommen + $C = \mathfrak{m}C$. Dann folgt insbesondere $C_{\mathfrak{m}} = \mathfrak{m}C_{\mathfrak{m}}$, also + da $C$ endlich erzeugter $A_{\mathfrak{m}}$-Modul folgt mit Nakayama $C_{\mathfrak{m}} = 0$, also + $[ C : A ](\mathfrak{m}) = 0 < 1$. Widerspruch. Da $C$ insbesondere flach, folgt mit Algebra 2, dass + $C$ treuflache $A$-Algebra ist. + + Nun sei $p\colon Y \times_X W \to W$. Dann ist $p^{-1}(\spec C) = \spec B \otimes_A C$. Da + $p$ endlich étale, folgt mit \ref{satz:equiv-finite-etale}, dass $B \otimes_A C$ eine projektive separable + $C$-Algebra ist. Mit der Treuflachheit von $C$ und \ref{satz:4.14} folgt nun die Behauptung. +\end{proof} + +\end{document} diff --git a/sose2022/galois/vortrag_affin.pdf b/sose2022/galois/vortrag_affin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d3e92738618ec918fe7319c35a8f0377681da140 GIT binary patch literal 225566 zcma&tLz5hqzWQp zw2XAD0MdU;BWnOQ0tNy*LrVY;4}f0A#MaE&oPe2?gPq|241iwD!rIxyk$_&z+Q8XF z#Kg$X*aX1G2XJzBG%>INxNj`$Xgh7Pq4~bnDdeT*Mj+_G*=9^!aYQUiZWLQwy^u}> zAuYDDb+#vN-Mp;c0t+SpK_@1b7`x(UOE7_FWA{3R;Oq7Jjft^eB6D?O&dc%LQ$=R}^7~Sr zg##nzw|_pke_l8I15>e#m=9w#irrd%aIV&Psr$M#{Wz`A*NknZ%;_1J1D~nAF@yOX z_&%8IAu`HAD*s>o|TUPH85PuB6p$ zubU{njPYaC_T?Sfl{Njjex}MhaGMMvX2JjmDb>ngku)t(iujk(P5)paY1W)g7M*d; zY<3*FPekEHIPwzVePL{75z1E3cz!V2=agtoU#q}YN%Smh=|CVQ66F?O{wX06_0RjW zA7hy&W06VFqewwdCg&$+rZ%noWJFch(VlO_&8Q8>a)n~B1h)Vjk-9|B;``Xuv4;{b zjZ9k}#%e$vAG{y`b&;RE$-BV=X)~V?%L0p;Bz8*cJ%tTabwq}05EdIDML&h1=a_)8 zqX_VxBipx?npgcZduzwWi8q<@dt`e|()g5`R;R=SAx04d3C(%nHw{a!auy5~YT}PU 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b/sose2022/galois/vortrag_affin.tex @@ -0,0 +1,834 @@ +\documentclass[a4paper]{../bachelorarbeit/arbeit} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[german]{babel} +\usepackage{amsmath, amssymb, amsthm} +\usepackage{tikz-cd} + +\makeatletter +\newcommand{\colim@}[2]{% + \vtop{\m@th\ialign{##\cr + \hfil$#1\operator@font colim$\hfil\cr + \noalign{\nointerlineskip\kern1.5\ex@}#2\cr + \noalign{\nointerlineskip\kern-\ex@}\cr}}% +} +\newcommand{\colim}{% + \mathop{\mathpalette\colim@{\rightarrowfill@\textstyle}}\nmlimits@ +} +\makeatother + +\newcommand{\spec}{\operatorname{Spec }} + +\begin{document} + +\section{Projektive Moduln und Algebren} + +\begin{satz}[Projektiv ist lokal frei] + Sei $A$ ein Ring und $M$ ein $A$-Modul. Die folgenden Eigenschaften sind äquivalent: + \begin{enumerate}[(i)] + \item $M$ ist endlich erzeugter projektiver $A$-Modul. + \item $M$ ist endlich präsentiert und $M_{\mathfrak{p}}$ ist freier + $M_{\mathfrak{p}}$-Modul für alle $\mathfrak{p} \in \spec A$. + \item Es existieren Elemente $\{f_i\}_{i \in I}$ von $A$ mit $\sum_{i \in I} (f_i) = A$, sodass + $M_{f_i}$ freier, endlich erzeugter $A_{f_i}$ Modul ist. + \end{enumerate} + \label{satz:projectiveislocallyfree} +\end{satz} + +\begin{proof} + Siehe Theorem 4.6 in \cite{lenstra}. +\end{proof} + +\begin{satz} + Sei $B$ eine $A$-Algebra und $C$ eine treuflache $A$-Algebra, sodass + $B \otimes_A C$ projektive, separable $C$-Algebra ist. Dann ist $B$ projektive, separable $A$-Algebra. + \label{satz:4.14} +\end{satz} + +\begin{proof} + Vortrag 8. Theorem 4.14 in \cite{lenstra}. +\end{proof} + +\begin{satz} + Sei $A$ ein Ring und $B$ projektive separable $A$-Algebra. Dann + existiert eine $B$-Algebra $C$ und ein $B$-Algebraisomorphismus + $B \otimes_A B \to B \times C$. + \label{satz:4.16} +\end{satz} + +\begin{proof} + Vortrag 8. Proposition 4.16 in \cite{lenstra}. +\end{proof} + +\section{Endlich étale Morphismen} + +\begin{definition} + Sei $f\colon A \to B$ ein Ringhomomorphismus. + $f$ ist \emph{endlich und lokal frei}, wenn eine Familie $(f_i)_{i \in I}$ existiert, sodass + $A = \sum_{i \in I} (f_i)$ und $B_{f_i}$ endliche, freie $A_{f_{i}}$ Algebra ist für alle $i \in I$. + + %Sei $f\colon Y \to X$ ein affiner Morphismus von Schemata. $f$ ist \emph{endlich und lokal frei}, wenn + %eine offene affine Überdeckung $\{U_i\}_{i \in I}$ existiert mit $U_i = \text{Spec }A_i$, sodass + %$f^{-1}(U_i) = \text{Spec }B_i$, wobei $B_i$ eine endliche und freie $A_i$-Algebra ist. +\end{definition} + +\begin{lemma}[] + Sei $f\colon A \to B$ ein Ringhomomorphismus und $S \subseteq A$ ein multiplikatives System. Dann + ist $f(S)$ ein multiplikatives System von $B$ und + \[ + S^{-1}B \simeq f(S)^{-1}B + \] als $S^{-1}A$-Algebren. + \label{lemma:localisation} +\end{lemma} + +\begin{proof} + Die Formel $\frac{b}{s} \mapsto \frac{b}{f(s)}$ induziert den Isomorphismus. +\end{proof} + +%\begin{lemma} +% Sei $f\colon A \to B$ ein Ringhomomorphismus und $\varphi\colon \text{Spec }B \to \text{Spec }A$ der induzierte +% Morphismus affiner Schemata. Sei $g \in A$. Dann ist +% \[ +% \varphi^{-1}(D(g)) = D(f(g)) +% .\] Insbesondere gilt +% \[ +% \varphi^{-1}(\text{Spec }A_g) = \text{Spec }B_g +% .\] +% \label{lemma:d(f)} +%\end{lemma} +% +%\begin{proof} +% Die erste Gleichung ist aus Algebra 2 bekannt und gilt allgemeiner für Morphismen lokal geringter Räume. Die +% zweite Gleichung folgt aus der Ersten, wenn der Isomorphismus $D(g) = \text{Spec }A_g$ eingesetzt wird, unter +% Verwendung des Ringisomorphismus +% \[ +% B_g = B \otimes_A A_g \simeq B_{f(g)} +% .\] +%\end{proof} + +\begin{bem} + Sei $f\colon A \to B$ ein Ringhomomorphismus. Dann ist $f$ genau dann endlich und lokal frei, + wenn $B$ endliche, projektive $A$-Algebra ist. + \label{satz:morph-local-free-char} +\end{bem} + +\begin{proof} + \ref{satz:projectiveislocallyfree} +\end{proof} + +\begin{satz}[Komposition] + Sei $A$ ein Ring, $B$ endliche, projektive $A$-Algebra und $C$ endliche, projektive $B$-Algebra. Dann + ist $C$ endliche, projektive $A$-Algebra. + \label{satz:composition-projective} +\end{satz} + +\begin{proof} + Sei $A^{n} = B \oplus Q$ und $B^{m} = C \oplus P$. Dann haben wir Isomorphismen von $A$-Moduln + \[ + A^{mn} = (A^{n})^{m} = (B \oplus Q)^{m} = B^{m} \oplus Q^{m} = C \oplus P \oplus Q^{m} + .\] +\end{proof} + +\begin{bem} + Sei $B$ endliche projektive $A$-Algebra und $\mathfrak{p} \in \spec A$. Nach \ref{satz:projectiveislocallyfree} + existiert eine Familie $\{f_i\}_{i \in I}$, sodass $A = \sum_{i \in I} (f_i)$ und + $B_{f_i}$ endliche, freie $A_{f_i}$-Algebra. Da $\mathfrak{p} \subsetneq A$ existiert ein $i \in I$, sodass + $f_i \not\in \mathfrak{p}$. Also ist $A_{\mathfrak{p}} = (A_{f_i})_{\mathfrak{p}}$ und + $B_{\mathfrak{p}} = (B_{f_i})_{\mathfrak{p}}$, also insbesondere $B_{\mathfrak{p}}$ endliche freie + $A_{\mathfrak{p}}$-Algebra. +\end{bem} + +\begin{definition}[Grad] + Sei $B$ endliche projektive $A$-Algebra. Dann ist + \[ + [ B : A ] \colon \spec A \to \Z, \mathfrak{p} \mapsto \text{rank}_{A_{\mathfrak{p}}}B_{\mathfrak{p}} + \] die \emph{Gradabbildung}. +\end{definition} + +\begin{satz} + Sei $B$ endliche, projektive $A$-Algebra. Dann gilt + \begin{enumerate}[(a)] + \item $A \to B$ ist genau dann injektiv, wenn $[B : A] \ge 1$. + \item $A \to B$ ist genau dann surjektiv, wenn $[B : A ] \le 1$. + \item $A \to B$ ist genau dann ein Isomorphismus, wenn $[ B : A ] = 1$. + \end{enumerate} + \label{satz:rings-degree} +\end{satz} + +\begin{proof} + Vortrag 8. +\end{proof} + +\begin{lemma} + Sei $B$ endliche, projektive $A$-Algebra. Dann ist + $[B : A]$ lokalkonstant, das heißt eine stetige Abbildung $\spec A \to \Z$, wobei $\Z$ die diskrete + Topologie trägt. Insbesondere ist die Menge + \[ + \{\mathfrak{p} \in \spec A \mid [B : A](\mathfrak{p}) = n\} + \] offen und abgeschlossen in $\spec A$ und $[B : A]$ ist konstant, falls $\spec A$ zusammenhängend ist. +\end{lemma} + +\begin{proof} + Erneut nach \ref{satz:projectiveislocallyfree} seien $\{f_i\}_{i \in I}$ in $A$, sodass + $A = \sum_{i \in I} (f_i)$ und $B_{f_i}$ endliche, freie $A_{f_i}$-Algebra für alle $i \in I$. Dann + ist $\spec A = \bigcup_{i \in I} D(f_i)$, wobei $D(f_i) = \{ \mathfrak{p} \in \spec A \mid f_i \not\in p\}$. + Per Definition der Zariskitopologie auf $\spec A$ sind die Mengen $D(f_i)$ offen und + $[B : A]|_{D(f_i)}$ ist konstant. +\end{proof} + +\begin{bem}[Offene und abgeschlossene Mengen in $\spec A$] + Sei $A$ ein Ring. Dann sind die offenen und abgeschlossenen Mengen in $\spec A$ + von der Form $D(e)$, wobei $e$ idempotent und eindeutig bestimmt ist. Insbesondere + existiert für alle $n \ge 0$ genau ein Idempotent $e$, sodass + $\{\mathfrak{p} \in \spec A \mid [ B : A](\mathfrak{p}) = n\} = D(e)$. + \label{bem:clopen-sets} +\end{bem} + +\begin{definition}[Treuprojektive Algebren] + Sei $B$ endliche projektive $A$-Algebra. Dann ist $B$ \emph{treuprojektiv}, wenn $[B : A] \ge 1$. +\end{definition} + +%\begin{definition}[Surjektive Algebren] +% Eine $A$-Algebra $B$ heißt $\emph{surjektiv}$, falls die induzierte Abbildung +% $\spec B \to \spec A$ surjektiv ist. +%\end{definition} + +\begin{satz} + Sei $B$ endliche, projektive $A$-Algebra. Dann ist + \begin{enumerate}[(a)] + \item $B = 0 \iff [B : A] = 0$. + \item $A \to B$ Isomorphismus $\iff [B : A] = 1$. + \item $\spec B \to \spec A$ surjektiv $\iff$ $B$ treuprojektive $A$-Algebra. + \end{enumerate} + \label{satz:degree} +\end{satz} + +\begin{proof} + \begin{enumerate}[(a)] + \item $B = 0 \iff B_{\mathfrak{p}} = 0$ + $\forall \mathfrak{p} \in \spec A \iff \text{rang}_{A_{\mathfrak{p}}} B_{\mathfrak{p}} = 0$ + $\forall \mathfrak{p} \in \spec A$ + $\iff [B : A] = 0$. + \item Das ist \ref{satz:rings-degree}(c). + \item + Sei $\spec B \to \spec A$ surjektiv und sei $\mathfrak{p} \in \spec A$ mit Urbild + $\mathfrak{q} \in \spec{B}$. Also ist $B \neq 0$ und damit $B_{\mathfrak{q}} \neq 0$. + Sei $\varphi\colon A \to B$ der induzierte Ringhomomorphismus, + $S = \varphi(A \setminus \mathfrak{p})$ und $T = B \setminus \mathfrak{q}$. Wegen + $\mathfrak{p} = f(\mathfrak{q}) = \varphi^{-1}(\mathfrak{q})$, folgt + $S \subseteq T$. Und damit + \[ + B_{\mathfrak{q}} = T^{-1}B \simeq (S^{-1}T)^{-1}(S^{-1}B) = (S^{-1}T)^{-1} B_{\mathfrak{p}}, + \] also ist $B_{\mathfrak{q}}$ eine Lokalisierung von $B_{\mathfrak{p}}$. Also folgt + auch $B_{\mathfrak{p}} \neq 0$, also + $[ B : A ](\mathfrak{p}) > 0$. Rückrichtung: Mit \ref{satz:rings-degree}(a) ist $A \to B$ injektiv + und weil $B$ endliche $A$-Algebra, ist $B$ ganze Ringerweiterung von $A$, also folgt + die Aussage aus \cite{macdonald} Theorem 5.10. + \end{enumerate} +\end{proof} + +\begin{definition}[Endlich étale Algebren] + Eine $A$-Algebra $B$ ist \emph{endlich étale}, wenn Elemente $\{f_i\}_{i \in I}$ existieren, sodass + $A = \sum_{i \in I} (f_i)$ und $B_{f_i}$ freie, separable $A_{f_i}$-Algebra ist für alle $i \in I$. +\end{definition} + +\begin{bem} + Jede endlich étale Algebra ist auch endlich und lokal frei, denn separable Algebren sind per Definition endlich. + \label{bem:finite-etale-is-locally-free} +\end{bem} + +%\begin{lemma} +% Seien $M, N$ $A$-Moduln und $M$ endlich präsentiert, d.h. es existiert eine exakte Folge +% \[ +% A^{m} \to A^{n} \to M \to 0 +% .\] Sei weiter $S \subset A$ ein multiplikatives System. Dann ist der natürliche $A$-Modul Homomorphismus +% \[ +% S^{-1}\operatorname{Hom}_A(M, N) \to \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N) +% \] ein Isomorphismus. +% \label{lemma:localisation-finitely-pres} +%\end{lemma} +% +%\begin{proof} +% $S^{-1}A$ ist ein flacher $A$-Modul und $\operatorname{Hom}_A(-, N)$ bzw. $\operatorname{Hom}_{S^{-1}A}(-, S^{-1}N)$ +% sind linksexakt. So erhalten wir exakte Folgen +% \[ +% 0 \to \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N) +% \to (S^{-1}N)^{n} \to (S^{-1}N)^{m} +% \] und +% \[ +% 0 \to S^{-1}\operatorname{Hom}_{A}(M, N) \to (S^{-1}N)^{n} \to (S^{-1}N)^{m} +% .\] Das 5-er Lemma liefert das Ergebnis. +%\end{proof} + +\begin{lemma}[] + Sei $M$ endlich präsentierter $A$-Modul, das heißt es existiere eine exakte Folge + \[ + A^{m} \to A^{n} \to M \to 0 + .\] Sei weiter $C$ eine $A$-Algebra und $N$ ein $A$-Modul. + Falls $N$ oder $C$ flach sind, ist der natürliche $C$-Modulhomomorphismus + \[ + \operatorname{Hom}_A(M, N) \otimes_A C \to \operatorname{Hom}_{C}(M \otimes_A C, N \otimes_A C) + \] ein Isomorphismus. + \label{lemma:tensor-and-hom} +\end{lemma} + +\begin{proof} + Da Tensorieren rechtsexakt ist und $\operatorname{Hom}_{C}(-, N \otimes_A C)$ linksexakt, erhalten wir durch anwenden + in dieser Reihenfolge auf die exakte Folge $A^{m} \to A^{n} \to M \to 0$, die exakte Folge + \[ + 0 \to \operatorname{Hom}_{C}(M \otimes_A C, N \otimes_A C) \to (N \otimes_A C)^{n} \to (N \otimes_A C)^{m} + .\] Andererseits liefert zunächst anwenden von $\operatorname{Hom}_{A}(-, N)$ die exakte Folge + \[ + 0 \to \operatorname{Hom}_{A}(M, N) \to N^{n} \to N^{m} + .\] Tensorieren mit $C$ liefert die exakte Folge + \[ + \underbrace{\operatorname{Tor}_{1}^{A}(N^{m}, C)}_{= 0} \to \operatorname{Hom}_{A}(M, N) \otimes_A C + \to (N \otimes_A C)^{n} \to (N \otimes_A C)^{m} + .\] Der linke Term verschwindet, weil $N^{m}$ oder $C$ flach ist. Untereinanderschreiben der beiden Folgen mit + den natürlichen Homomorphismen und Auffüllen mit $0$ nach links liefert ein kommutatives Diagramm und mit dem 5-er Lemma + die Behauptung. +\end{proof} + +\begin{bem} + \ref{lemma:tensor-and-hom} wendet sich insbesondere dann an, wenn $M$ endlich erzeugter projektiver Modul ist. +\end{bem} + +\begin{korollar} + Seien $M$, $N$ $A$-Moduln und $M$ endlich präsentiert. Sei weiter $S \subset A$ ein multiplikatives System. Dann + ist der natürliche $S^{-1}A$-Modulhomomorphismus + \[ + S^{-1}\operatorname{Hom}_{A}(M, N) \to \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N) + \] ein Isomorphismus. + \label{lemma:localisation-finitely-pres} +\end{korollar} + +%\begin{korollar} +% Ein Morphismus von Schemata $f\colon Y \to X$ ist genau dann endlich étale, wenn +% eine Basis von offenen affinen Mengen $\{U_i\}_{i \in I}$ von $X$ existiert, sodass +% $U_i = \spec A_i$ und $f^{-1}(U_i) = \spec B_i$, wobei $B_i$ freie, separable $A_i$-Algebra ist. +% +% \label{bem:finite-etale-basis} +%\end{korollar} +% +%\begin{proof} +% Die Rückrichtung ist klar. Für die Hinrichtung beachte, dass eine endliche, projektive $A$-Algebra $B$ genau dann +% separabel ist, wenn der von der Spur induzierte $A$-Modulhomomorphismus $B \to \operatorname{Hom}_A(B, A)$ +% ein Isomorphismus ist. Diese Eigenschaft bleibt nach \ref{lemma:localisation-finitely-pres} +% durch Lokalisieren erhalten. +%\end{proof} + +\begin{lemma} + Sei $B$ eine endliche, projektive $A$-Algebra. Dann ist $A \to B$ genau dann separabel, wenn + $A_{\mathfrak{p}} \to B_{\mathfrak{p}}$ separabel ist für alle $\mathfrak{p} \in \spec A$. + \label{lemma:separable-is-local} +\end{lemma} + +\begin{proof} + $A \to B$ ist genau dann separabel, wenn die von der Spur induzierte Abbildung + $B \to \operatorname{Hom}_{A}(B, A)$ ein Isomorphismus ist. Das + ist eine lokale Eigenschaft. Da außerdem $B$ endlich erzeugter, projektiver $A$-Modul ist, das + heißt insbesondere endlich präsentiert ist, folgt mit \ref{lemma:localisation-finitely-pres} + \[ + \operatorname{Hom}_{A}(B, A)_{\mathfrak{p}} = + \operatorname{Hom}_{A_{\mathfrak{p}}}(B_{\mathfrak{p}}, A_{\mathfrak{p}}) + \] und damit die Behauptung. +\end{proof} + +\begin{satz} + Sei $B$ eine $A$-Algebra. Dann ist $B$ genau dann endlich étale über $A$, wenn + $B$ projektive, separable $A$-Algebra ist. + \label{satz:equiv-finite-etale} +\end{satz} + +\begin{proof} + ($\Rightarrow$) + Nach \ref{bem:finite-etale-is-locally-free} ist $B$ endliche projektive $A$-Algebra. Weiter sei + $\{f_i\}_{i \in I}$ in $A$, sodass $A = \sum_{i \in I} (f_i)$ und + $B_{f_i}$ endliche separable $A_{f_i}$ Algebra für alle $i \in I$. Nun sei $\mathfrak{p} \in \spec A$. Dann + existiert ein $i \in I$, sodass $f_i \not\in \mathfrak{p}$. Also + folgt $B_{\mathfrak{p}} = (B_{f_i})_{\mathfrak{p}}$ und $A_{\mathfrak{p}} = (A_{f_i})_{\mathfrak{p}}$. + Nach \ref{lemma:separable-is-local} ist also + $A_{\mathfrak{p}} \to B_{\mathfrak{p}}$ separabel. + Da $\mathfrak{p}$ beliebig war, folgt erneut mit \ref{lemma:separable-is-local} die Behauptung. + + ($\Leftarrow$) Sei $B$ projektive, separable $A$-Algebra. Nach \ref{satz:projectiveislocallyfree} + existieren $\{f_i\}_{i \in I}$, sodass $\sum_{i \in I} (f_i) = A$ und + $B_{f_i}$ endliche, freie $A_{f_i}$-Algebra. Da Lokalisieren Separabilität erhält, ist $B_{f_i}$ auch separabel + über $A_{f_i}$ und es folgt die Behauptung. +\end{proof} + +\begin{satz}[Basiswechsel endlich projektive] + Sei $B$ endlich projektive $A$-Algebra und $C$ eine weitere $A$-Algebra. Dann + ist $B \otimes_A C$ endlich projektive $C$-Algebra. + Insbesondere kommutiert das folgende Diagramm + \[ + \begin{tikzcd} + \spec C \arrow[swap]{dr}{[B \otimes_A C : C]} \arrow[from=1-1,to=1-3] & & \spec A \arrow{dl}{[B : A]} \\ + & \Z &. + \end{tikzcd} + \] + \label{satz:basischange-projective} +\end{satz} + +\begin{proof} + Da $B$ endlich erzeugter, projektiver $A$-Modul ist, + existiert ein $A$-Modul $Q$, sodass + $A^{n} \simeq B \oplus Q$ als $A$-Moduln für ein $n \ge 0$. Da der natürliche Isomorphismus + $A^{n} \otimes_A C \to C^{n}$ auch $C$-linear ist, folgt durch + Tensorieren mit $C$ + \[ + C^{n} \simeq A^{n} \otimes_A C \simeq (B \oplus Q) \otimes_A C = (B \otimes_A C) \oplus (Q \otimes_A C) + .\] Also ist $B \otimes_A C$ (endlich erzeugter) projektiver $C$-Modul. + + Für die Grade: Sei $\mathfrak{p} \in \spec C$ und $\mathfrak{q} \coloneqq \mathfrak{p}^{c}$. Sei + weiter $\text{rank}_{A_{\mathfrak{q}}} B_{\mathfrak{q}} = n$. Also + folgt + \[ + (B \otimes_A C)_{\mathfrak{q}} = B \otimes_A C \otimes_A A_{\mathfrak{q}} = B_{\mathfrak{q}} + \otimes_A C = A_{\mathfrak{q}}^{n} \otimes_A C = (A^{n} \otimes_A C)_{\mathfrak{q}} = (C^{n})_{\mathfrak{q}} + .\] Die natürlichen Isomorphismen sind auch $C_{\mathfrak{q}}$ linear, also folgt + $\text{rank}_{C_{\mathfrak{q}}} (B \otimes_A C)_{\mathfrak{q}} = n$. Da + $(B \otimes_A C)_{\mathfrak{p}}$ bzw. $C_{\mathfrak{p}}$ eine Lokalisierung von + $(B \otimes_A C)_{\mathfrak{q}}$ bzw. $C_{\mathfrak{q}}$ ist und Lokalisieren den Grad erhält, folgt die + Behauptung. +\end{proof} + +\begin{satz}[Basiswechsel endlich étale] + Sei $B$ endlich étale $A$-Algebra und $C$ eine weitere $A$-Algebra. Dann ist + $B \otimes_{A} C$ endlich étale $C$-Algebra. + \label{satz:basischange} +\end{satz} + +\begin{proof} + Mit \ref{satz:equiv-finite-etale} genügt es zu zeigen: + Sei $B$ projektive, separable $A$-Algebra und $C$ eine $A$-Algebra. Dann + ist $B \otimes_A C$ projektive, separable $C$-Algebra. Mit + \ref{satz:basischange-projective} genügt es die Separabilität zu zeigen. + + Es ist also zu zeigen, dass der von der Spurabbildung induzierte Homomorphismus + $B \otimes_A C \to \operatorname{Hom}_{C}(B \otimes_A C, C)$ ein Isomorphismus ist. Das folgt aus + \ref{lemma:tensor-and-hom} und dem kommutativen Diagramm: + \[ + \begin{tikzcd} + B \otimes_A C \arrow{r} & \operatorname{Hom}_C(B\otimes_A C, C) \\ + B \otimes_A C \arrow{u}{\operatorname{id}} \arrow[swap]{r}{\sim} + & \operatorname{Hom}_A(B, A) \otimes_A C \arrow[swap]{u}{\sim} + \end{tikzcd} + .\] +\end{proof} + +\begin{satz}[Treuprojektiv ist treuflach] + Sei $B$ treuprojektive $A$-Algebra. Dann ist $B$ treuflach. + \label{satz:faithfully-projective-faithfully-flat} +\end{satz} + +\begin{proof} + Wir zeigen die Kontraposition: Sei $B$ nicht treuflach, dann existiert + nach Algebra 2 ein Maximalideal $\mathfrak{m}$ von $A$, sodass $B = \mathfrak{m}B$. + Dann folgt insbesondere $B_{\mathfrak{m}} = \mathfrak{m}B_{\mathfrak{m}}$. + Da $B_{\mathfrak{m}}$ endlich erzeugter $A_{\mathfrak{m}}$-Modul folgt mit Nakayama $B_{\mathfrak{m}} = 0$, also + $[ B : A ](\mathfrak{m}) = 0 < 1$. +\end{proof} + +\begin{korollar} + Sei $B$ eine $A$-Algebra und $C$ treuprojektive $A$-Algebra. + Dann ist $B$ genau dann endlich étale $A$-Algebra, wenn + $B \otimes_A C$ endlich étale $C$-Algebra ist. + \label{kor:finite-etale} +\end{korollar} + +\begin{proof} + Die Hinrichtung gilt für beliebige Basiswechsel nach \ref{satz:basischange}. + Zur Rückrichtung: Nach \ref{satz:faithfully-projective-faithfully-flat} ist $C$ treuflach. Damit folgt + die Behauptung aus \ref{satz:4.14}. +\end{proof} + +\begin{definition}[Total zerlegbare Algebren] + Eine $A$-Algebra $B$ ist \emph{total zerlegbar}, wenn + $A$ ein Produkt von Ringen $A_n$ ist mit $n \ge 0$ und + $B$ isomorph ist zu $\prod_{n \ge 0} A_n^{n}$, sodass + \[ + \begin{tikzcd} + B \arrow{r}{\sim} & \prod_{n \ge 0}^{} A_n^{n} \\ + A \arrow{u} \arrow{r}{\sim} & \prod_{n \ge 0} A_n \arrow{u} + \end{tikzcd} + \] kommutiert. +\end{definition} + +\begin{satz}[Aufgabe 5.3] + Sei $A$ ein Ring und $B_i$ $A$-Algebren für $1 \le i \le n$. Sei weiter + $B = \prod_{i=1}^{n} B_i$. Dann ist $B$ genau dann endlich étale, wenn + jedes $B_i$ endlich étale ist. In diesem Fall gilt + $[B : A] = \sum_{i=1}^{n} [B_i : A]$. + \label{ex:5.3} +\end{satz} + +\begin{proof} + Als $A$-Modul ist $B \simeq \bigoplus_{i = 1}^{n} B_i$. Also ist $B$ genau dann endlich erzeugt bzw. projektiv, wenn + $B_i$ endlich erzeugt bzw. projektiv ist für $1 \le i \le n$. Ebenso + existiert ein natürlicher $A$-Modulisomorphismus + \[ + \text{Hom}_A(B, A) = \text{Hom}_A\left( \bigoplus_{i=1}^{n} B_i, A \right) + = \prod_{i=1}^{n} \text{Hom}_A(B_i, A) + .\] Das heißt + $B \to \text{Hom}_A(B, A)$ ist genau dann ein Isomorphismus, wenn $B_i \to \text{Hom}_A(B_i, A)$ ein Isomorphismus + ist für $1 \le i \le n$. + + Gradformel: Sei $\mathfrak{p} \in \spec A$. Dann ist + \[ + [B : A](\mathfrak{p})% = \text{rank}_{A_{\mathfrak{p}}} B_{\mathfrak{p}} + = \text{rank}_{A_{\mathfrak{p}}} \left(\bigoplus_{i = 1}^{n} B_i\right)_{\mathfrak{p}} + = \text{rank}_{A_{\mathfrak{p}}} \left( \bigoplus_{i = 1}^{n} (B_{i})_{\mathfrak{p}} \right) + = \sum_{i=1}^{n} \text{rank}_{A_{\mathfrak{p}}} (B_{i})_{\mathfrak{p}} + = \sum_{i=1}^{n} [B_i : A ](\mathfrak{p}) + .\] +\end{proof} + +\begin{lemma} + Seien $A_1, \ldots, A_n$ Ringe und $f \in \prod_{i=1}^{n} A_i$. Dann ist der natürliche Homomorphismus + \[ + \left(\prod_{i=1}^{n} A_i\right)_f \longrightarrow \prod_{i=1}^{n} (A_i)_{f_i} + \] ein Isomorphismus. + \label{lemma:localised-product-ring} +\end{lemma} + +\begin{proof} + Man bediene sich der Endlichkeit. +\end{proof} + +\begin{lemma} + Sei $A$ ein Ring und $\{f_i\}_{i \in I}$ Elemente von $A$, sodass + $\spec A = \coprod_{i \in I} D(f_i)$. Dann ist der natürliche Ringhomomorphismus + \[ + A \longrightarrow \prod_{i \in I} A_{f_i} + \] ein Isomorphismus. + \label{lemma:disjoint-union-of-spec} +\end{lemma} + +\begin{proof} + Sei $\mathfrak{p} \in \spec A$ und $i \in I$. Falls $f_i \not\in \mathfrak{p}$ ist + $(A_{f_i})_{\mathfrak{p}} = A_{\mathfrak{p}}$. Sei nun $f_i \in \mathfrak{p}$. Beh.: $(A_{f_i})_{\mathfrak{p}} = 0$. + + Wir zeigen $f_i$ nilpotent in $A_{\mathfrak{p}}$. + Sei $\mathfrak{q} \in \spec A$ mit $f_i \not\in \mathfrak{q}$. Dann ist + $\{\mathfrak{q}\} \subseteq D(f_i)$ und $D(f_i)$ ist abgeschlossen. Also folgt + $V(\mathfrak{q}) = \overline{\{\mathfrak{q}\}} \subseteq D(f_i)$. Da $\mathfrak{p} \not\in D(f_i)$, folgt + a fortiori $\mathfrak{p} \not\in V(\mathfrak{q})$. M.a.W.: $\mathfrak{q} \not\in \spec A_{\mathfrak{p}}$. + Kontraposition: Für $\mathfrak{q} \in \spec A_{\mathfrak{p}}$ folgt bereits $f_i \in \mathfrak{q}$. + Also ist $f_i$ nilpotent in $A_{\mathfrak{p}}$ und damit + $(A_{f_i})_{\mathfrak{p}} = (A_{\mathfrak{p}})_{f_i} = 0$. + + Es existiert nun genau ein $i_{0} \in I$, sodass $\mathfrak{p} \in D(f_{i_0})$. Lokalisieren + von $A \to \prod_{i \in I} A_i$ ergibt also + \[ + A_{\mathfrak{p}} \to \left(\prod_{i \in I} A_{f_i}\right)_{\mathfrak{p}} + = \prod_{i \in I} (A_{f_i})_{\mathfrak{p}} = A_{\mathfrak{p}} + .\] Das erste Gleichheitszeichen gilt, da Spektren von Ringen quasikompakt sind, das heißt $I$ endlich ist. + Weil Isomorphismus eine lokale Eigenschaft ist, folgt die Behauptung. +\end{proof} + +\begin{lemma} + Sei $f\colon A \to B$ Ringhomomorphismus und $\varphi \colon \spec B \to \spec A$ die induzierte + Abbildung. Dann ist für $a \in A$: + \[ + \varphi^{-1}(D(a)) = D(f(a)) + .\] + \label{lemma:preimage-of-d} +\end{lemma} + +\begin{proof} + $\mathfrak{p} \in \varphi^{-1}(D(a)) \iff \varphi(\mathfrak{p})\in D(a) \iff a \not\in \varphi(\mathfrak{p}) + = f^{-1}(\mathfrak{p}) \iff \mathfrak{p} \not\in D(f(a))$. +\end{proof} + +\begin{satz} + Seien $(A_i)_{i \in I}, (B_i)_{i \in I}$ Ringe mit $I$ endlich und sei $B_i$ endlich étale $A_i$-Algebra + für alle $i \in I$. Dann ist $\prod_{i \in I} B_i$ endlich étale $\prod_{i \in I} A_i$-Algebra. Außerdem + ist jede endlich étale $\prod_{i \in I} A_i$ Algebra von dieser Form. + Weiter ist + \[ + \left[ \prod_{i \in I} B_i : \prod_{i \in I} A_i \right]\Big|_{\spec A_j} = [ B_j : A_j] + .\] + \label{satz:projective-prod} +\end{satz} + +\begin{proof} + Die Folge abelscher Gruppen + \[ + \begin{tikzcd} + \left( \prod_{i \in I} A_i \right)^{n} \arrow{r}\arrow{d}{\simeq} + & \left( \prod_{i \in I} A_i \right)^{m} \arrow{r}\arrow{d}{\simeq} + & \prod_{i \in I} B_i \arrow{r} \arrow{d}{=} & 0 \\ + \prod_{i \in I} A_i^{n} \arrow{r} & \prod_{i \in I} A_i^{m} \arrow{r} & \prod_{i \in I} B_i \arrow{r} & 0 + \end{tikzcd} + \] ist genau dann exakt, wenn + \[ + \begin{tikzcd} + A_i^{n} \arrow{r} & A_i^{m} \arrow{r} & B_i \arrow{r} & 0 + \end{tikzcd} + \] exakt ist für alle $i \in I$. Also ist $\prod_{i \in I} B_i$ genau dann endlich präsentierter + $\prod_{i \in I} A_i$-Modul, wenn $B_i$ endlich präsentierter $A_i$-Modul ist für alle $i \in I$. + + Außerdem ist $\spec A = \coprod_{i \in I} \spec A_i$. Also für $\mathfrak{p} \in \spec A$ + existiert genau ein $i \in I$, sodass $\mathfrak{p} \in \spec A_i$ und + $A_{\mathfrak{p}} = (A_{i})_{\mathfrak{p}}$ und $B_{\mathfrak{p}} = (B_{i})_{\mathfrak{p}}$. + Also ist $A_{\mathfrak{p}} \to B_{\mathfrak{p}}$ genau dann frei separabel für + alle $\mathfrak{p} \in \spec A$, wenn + $(A_i)_{\mathfrak{p}} \to (B_{i})_{\mathfrak{p}}$ frei separabel ist für alle $\mathfrak{p} \in \spec A_i$ und + alle $i \in I$. + + Insgesamt ist $A \to B$ genau dann endlich étale, wenn $A_i \to B_i$ endlich étale ist für alle $i \in I$. + + %Endlich étale ist eine lokale Eigenschaft auf $A$, genauer: Für alle $i \in I$ existieren + %$\{f_{ij}\}_{j \in J_i}$, sodass $A_i = \sum_{j \in J_i} (f_{ij})$ und + %$(B_i)_{f_{ij}}$ separable, freie $(A_{i})_{f_{ij}}$-Algebra für alle $j \in J_i$. Dann + %setze $\tilde{f}_{ij} = (0, \ldots, 0, f_{ij}, 0, \ldots, 0)$. Dann ist mit \ref{lemma:localised-product-ring} + %$B_{\tilde{f}_{ij}} = (B_i)_{f_{ij}}$ separable, freie $A_{\tilde{f}_{ij}} = (A_i)_{f_{ij}}$-Algebra + %und $A = \sum_{i \in I} \sum_{j \in J_i} (\tilde{f}_{ij})$. Also ist $B$ endlich separable $A$-Algebra. + + Sei nun $f\colon \prod_{i \in I} A_i \to B$ endlich étale. Es ist + $\spec A = \coprod_{i \in I} \spec A_i$. Nach \ref{bem:clopen-sets} existieren + also idempotente Elemente $e_i \in A$, sodass $\spec A = \coprod_{i \in I} D(e_i)$. Nach + \ref{lemma:preimage-of-d} ist $\spec B = \coprod_{i \in I} D(f(e_i))$. Wir haben das folgende kommutative + Diagramm + \[ + \begin{tikzcd} + B \arrow{r} & \prod_{i \in I} B_{e_i} \\ + A \arrow{u} \arrow{r} & \prod_{i \in I} A_{e_i} \arrow{u} + \end{tikzcd} + .\] Nach \ref{lemma:disjoint-union-of-spec} sind die horizontalen Pfeile Isomorphismen + und wir sind in der obigen Situation. + + Die Gradformel folgt aus $\spec A = \coprod_{i \in I} \spec A_i$ und durch Berechnung in der Überdeckung aus dem + ersten Absatz. +\end{proof} + +\begin{satz} + Sei $B$ eine $A$-Algebra. Dann ist $B$ genau dann endlich étale, + wenn $B \otimes_A C$ total zerlegbare $C$-Algebra ist für eine treuprojektive $A$-Algebra $C$. + \label{th:5.10} +\end{satz} + +\begin{proof} + Die Rückrichtung ist klar nach \ref{kor:finite-etale}. Hinrichtung: Sei $B$ eine endlich + étale $A$-Algebra und sei zunächst $[ B : A ] = n$ konstant. Dann zeigen wir die Behauptung per Induktion + nach $n$. + + Falls $n = 0$: Dann ist $B = 0$ und wir können $C = A$ setzen. Sei nun $n > 0$. + Nach \ref{satz:4.16} existiert eine $B$-Algebra $B'$ und ein + Isomorphismus von $B$-Algebren $B \otimes_A B \to B \times B'$. + Nach \ref{satz:basischange} ist $B \otimes_A B$ endlich étale $B$-Algebra und + $[ B \otimes_A B : B] = n$. Wenn $B$ natürlicherweise als $B$-Algebra aufgefasst wird, ist $[ B : B ] = 1$, + also nach \ref{ex:5.3} $[ B' : B ] = [ B \times B' : B ] - [ B : B ] = n-1$. Also wendet + sich die Induktionsvoraussetzung an und es gibt eine treuprojektive $B$-Algebra $C$, sodass + $B' \otimes_B C$ total zerlegbare $C$-Algebra ist. + + Dann ist $B \otimes_A C = B \otimes_A B \otimes_B C = (B \times B') \otimes_B C = + (B \otimes_B C) \times (B' \otimes_B C) = C \times (B' \otimes_B C)$. Da $C$ + und $B' \otimes_B C$ als $C$-Algebren total zerlegbar sind, ist auch $B \otimes_A C$ total zerlegbar als + $C$-Algebra. + + Nach \ref{satz:composition-projective} ist $C$ endliche, projektive $A$-Algebra. Da $[ B : A ] \ge 1$ + und $[ C : B ] \ge 1$ ist $\spec C \to \spec B \to \spec A$ surjektiv, also + $C$ treuprojektive $A$-Algebra nach \ref{satz:degree}. + + Im Allgemeinen Fall sei $\spec A = \coprod_{n \ge 0} [ B : A ]^{-1}(\{n\})$. Dann existieren + idempotente Elemente $(e_n)_{n \ge 0}$, sodass $D(e_n) = [B : A]^{-1}(\{n\})$, also + $\spec A = \coprod_{n \ge 0} D(e_n)$. Mit \ref{lemma:disjoint-union-of-spec} + ist also $A = \prod_{n \ge 0}^{} A_{e_n}$, wobei wegen der Quasikompaktheit von $\spec A$ fast alle $e_n = 0$ sind. + Mit \ref{satz:projective-prod} ist also + $B = \prod_{n \ge 0} B_{e_n}$ mit $A_{e_n} \to B_{e_n}$ endlich étale. + + Nun ist $[ B_{e_n} : A_{e_n} ] = n$ und + mit dem ersten Teil existiert eine treuprojektive $A_{e_n}$-Algebra $C_{n}$, sodass + $B_{e_n} \otimes_A C_n$ total zerlegbare $C_n$-Algebra ist. Setze + nun $C = \prod_{n \ge 0} C_{n}$. Nach \ref{satz:projective-prod} ist + $C$ endliche, projektive $A$-Algebra und $[C : A]|_{D(e_n)} = [ C_{e_n} : A_{e_n} ] \ge 1$, also + $[ C : A] \ge 1$ und damit $C$ treuprojektiv. + + Weiter ist + \[ + B \otimes_A C = \left(\prod_{n \ge 0} B_{e_n} \right) \otimes_{\prod_{n \ge 0} A_{e_n} } + \left(\prod_{n \ge 0} C_n\right) + = \prod_{n \ge 0}^{} B_{e_n} \otimes_{A_{e_n}} C_n + \simeq \prod_{n \ge 0} C_n^{n} + \] wobei der letzte Isomorphismus aus der totalen Zerlegbarkeit von $B_{e_n} \otimes_{A_{e_n}} C_n$ folgt. +\end{proof} + +\begin{satz}[] + Sei $B$ endlich étale $A$-Algebra und $C$ endlich étale $B$-Algebra. Dann ist + $C$ endlich étale $A$-Algebra. +\end{satz} + +\begin{proof} + Sei zunächst $B$ total zerlegbar über $A$ von konstantem Grad. Dann + ist $B = A^{n}$ und + nach \ref{satz:projective-prod} $C = \prod_{i=1}^{n} C_i$, wobei $C_i$ endlich étale $A$-Algebra. Dann + ist $C$ endlich étale nach \ref{ex:5.3}. + + Falls $B$ total zerlegbar über $A$ von nicht notwendig konstantem Grad, ist $A = \prod_{n \ge 0} A_n$ + und $B = \prod_{n \ge 0} A_n^{n}$. Dann ist $C = \prod_{n \ge 0} C_n$ mit $C_n$ endlich étale + $A_n^{n}$-Algebra nach \ref{satz:projective-prod}. Wende + nun den ersten Fall auf $A_n \to A_n^{n} \to C_n$ an. Mit \ref{satz:projective-prod} folgt dann die Behauptung. + + Im Allgemeinen: Mit \ref{th:5.10} wähle $D$ treuprojektive $A$-Algebra, sodass + $B \otimes_A D$ total zerlegbare $D$-Algebra. Nach \ref{satz:basischange} ist dann + $C \otimes_A D = C \otimes_B (B \otimes_A D)$ endlich étale $B \otimes_A D$-Algebra. Mit dem obigen Speziallfall + ist also $C \otimes_A D$ endlich étale $D$-Algebra. Da $A \to D$ treuprojektiv war, folgt + mit \ref{kor:finite-etale} die Behauptung. +\end{proof} + +Sei $A$ ein Ring und $E$ eine endliche Menge. Dann schreiben wir $A^{E}$ für $\prod_{e \in E} A$. Sei +$\phi\colon D \to E$ eine Abbildung zwischen endlichen Mengen. Für $d \in D$ sei +$\psi_d \colon A^{E} \to A$ gegeben durch $(a_e)_{e \in E} \mapsto a_{\phi(d)}$. Das induziert +einen Ringhomomorphismus $A^{E} \to A^{D}$. + +\begin{lemma} + Der induzierte Ringhomomorphismus $A^{E} \to A^{D}$ ist endlich étale. + \label{lemma:induced-finite-etale} +\end{lemma} + +\begin{proof} + $A \to A$ und $A \to 0$ sind endlich étale, also mit \ref{satz:projective-prod} auch + $\psi_d \colon A^{E} \to A = A \times 0 \times \ldots \times 0$ + und damit $A^{E} \to A^{D}$ nach \ref{ex:5.3}. +\end{proof} + +\begin{lemma} + Sei $(A, \mathfrak{m})$ ein lokaler Ring. Dann hat $A$ keine nicht-trivialen idempotenten Elemente. + \label{lemma:local-idempotents} +\end{lemma} + +\begin{proof} + Sei $e \in A$ idempotent. Dann ist $e(1-e) = e^2 - e = e - e = 0$. Falls $e \in A^{\times }$, folgt + $1-e = 0$ also $e = 1$. Falls $e \not\in A^{\times}$: Dann ist $e \in \mathfrak{m}$. Angenommen + $1-e \in \mathfrak{m}$, dann ist auch $1 = 1-e +e \in \mathfrak{m}$. Widerspruch. Also ist + $1-e \not\in \mathfrak{m}$. Da $A$ lokal, ist also $1-e \in A^{\times}$ und damit $e = 0$. +\end{proof} + +\begin{lemma} + Sei $A$ ein Ring ohne nicht-triviale Idempotente und seien $E,D$ endliche Mengen. Dann + ist jeder $A$-Algebra Homomorphismus $A^{E} \to A^{D}$ induziert von einer Abbildung $D \to E$. + \label{lemma:no-idempotents} +\end{lemma} + +\begin{proof} + Sei $\psi\colon A^{E} \to A^{D}$ ein $A$-Algebra Homomorphismus. + Setze $f_e \coloneqq (\delta_{\tilde{e}e})_{\tilde{e} \in E} \in A^{E}$. Dann ist $f_e^2 = 1$ und + $f_ef_{\tilde{e}} = 0$ falls $e \neq \tilde{e}$. + + Sei nun $d \in D$ beliebig. Dann gilt + $\psi(f_e)^2 = \psi(f_e^2) = \psi(f_e)$, also $\psi(f_e)_d$ idempotent, also $\psi(f_e)_d \in \{0, 1\}$. + Weiter ist $1 = \psi(1)_d = \psi(\sum_{e \in E} f_e)_d$ und für $e \neq \tilde{e} \in E$ ist + $0 = \psi(f_ef_{\tilde{e}})_d = \psi(f_e)_d \psi(f_{\tilde{e}})_d$. Es existiert also + genau ein $e(d) \in E$, sodass $\psi(f_e)_d = 1$. + + Setze nun $\phi\colon D \to E$, $d \mapsto e(d)$. Nun sei $f = (a_e)_{e \in E} \in A^{E}$ beliebig. Dann + gilt + $f = \sum_{e \in E} a_e f_e$ und + \[ + \psi(f)_d = \sum_{e \in E} a_e \psi(f_e)_d = a_{\phi(d)} + .\] +\end{proof} + +\begin{lemma} + Sei $A$ ein Ring, $M$ ein $A$-Modul und $\mathfrak{p} \in \spec A$. Dann ist + \[ + M_{\mathfrak{p}} = \colim_{f \in A \setminus \mathfrak{p}} M_f + \] wobei $A \setminus \mathfrak{p}$ durch die Teilbarkeitsrelation halbgeordnet ist. + \label{kor:localisation-is-colim} +\end{lemma} + +\begin{proof} + Für eine multiplikative Menge $S$ vertaucht $- \otimes_A S^{-1}A$ mit Kolimites, da das Tensorprodukt linksadjungiert + ist. Es genügt also den Fall $M = A$ zu zeigen. + + Sei $S = A \setminus \mathfrak{p}$. Dann ist $S$ halbgeordnet und gerichtet. + Für alle $f \in S$ ist $\frac{f}{1} \in A_{\mathfrak{p}}^{\times}$, + also existiert eine natürliche Abbildung $A_f \to A_{\mathfrak{p}}$. Für $f, g \in S$ mit + $f \mid g$ kommutieren diese Abbildungen mit $A_f \to A_g$ und induzieren damit + eine Abbildung $\colim_{f \in S} A_f \to A_{\mathfrak{p}}$. Wir zeigen, dass + diese Abbildung bijektiv ist. + + Surjektiv: Sei $x = \frac{a}{f} \in A_{\mathfrak{p}}$. Also $f \in S$ und das Bild von $\frac{a}{f} \in A_f$ + in $\colim_{f \in S} A_f$ ist ein Urbild von $x$. + Injektiv: Sei $x \in \colim_{f \in S} A_f$ mit Bild $0$ in $A_{\mathfrak{p}}$. Dann + existiert ein $f \in S$ und $a \in A$, sodass $\frac{a}{f^{n}} = 0$ in $A_{\mathfrak{p}}$. Also + existiert ein $g \in S$, sodass $ga = 0$. Insbesondere ist $fga = 0$ also $\frac{a}{f} = 0$ in $A_{fg}$. + Außerdem ist $f \mid fg$ und damit $x = 0$. +\end{proof} + +\begin{lemma} + Seien $A, B, C$ Ringe und $f\colon A \to B$, $g\colon A \to C$ total zerlegbar und $h\colon C \to B$ ein + Ringhomomorphismus mit $f = hg$. Sei weiter $\mathfrak{p} \in \spec A$. Dann + existiert ein $a \in A$, sodass $a \not\in \mathfrak{p}$ und $f, g$ und $h$ trivial über $D(a)$ sind. Das heißt + es existieren endliche Mengen $D$ und $E$ und Isomorphismen $\alpha\colon A_a^{D} \to B_a$ + und $\beta\colon A_a^{E} \to C_a$ und eine Abbildung $\phi\colon D \to E$, sodass das Diagramm + \[ + \begin{tikzcd} + B_a & & & \arrow[from=1-4,to=1-1,swap]{}{h} C_a \\ + & \arrow{ul}{\alpha} A_a^{D} & \arrow{l} A_a^{E} \arrow{ur}{\beta} & \\ + A_a \arrow[from=3-1,to=1-1]{}{f} \arrow{ur} & & & \arrow[from=3-4,to=3-1]{}{\operatorname{id}_{A_a}} \arrow{ul} A_a \arrow[from=3-4,to=1-4]{}{g} + \end{tikzcd} + \] kommutiert. + \label{lemma:locally-trivial} +\end{lemma} + +\begin{proof} + Da die Mengen der Form $D(a)$ eine Basis der Topologie von $\spec A$ bilden, kann $a$ so gewählt werden, dass + $[B : A]$ und $[C : A]$ von konstantem Grad auf $D(a)$ sind mit $\mathfrak{p} \in D(a)$. + Durch ersetzen von $A$, $B$ und $C$ + durch $A_a$, $B_a$ und $C_a$ können wir wegen der totalen Zerlegbarkeit von $f$ und $g$ ohne Einschränkung + annehmen, dass $B \simeq A^{D}$ und $C \simeq A^{E}$. + + Der lokale Ring $A_{\mathfrak{p}}$ hat nach \ref{lemma:local-idempotents} keine nicht trivialen idempotenten Elemente. + Also ist die induzierte Abbildung $h \colon A_{\mathfrak{p}}^{E} \to A_{\mathfrak{p}}^{D}$ nach + \ref{lemma:no-idempotents} induziert von einer Abbildung $\phi\colon D \to E$. $\phi$ induziert + nun eine Abbildung $\psi\colon A^{E} \to A^{D}$ und $h$ und $\psi$ haben selbes Bild + in + \begin{salign*} + \operatorname{Hom}_A(A^{E}, A^{D})_{\mathfrak{p}} &\stackrel{\ref{lemma:localisation-finitely-pres}}{=} \operatorname{Hom}_{A_{\mathfrak{p}}}(A_{\mathfrak{p}}^{E}, + A_{\mathfrak{p}}^{D}) + .\end{salign*} + Nach \ref{kor:localisation-is-colim} existiert nun ein $a \in A \setminus \mathfrak{p}$, sodass + $h$ und $\psi$ die selbe Abbildung $A_a^{E} \to A_a^{D}$ induzieren. +\end{proof} + +\begin{satz} + Seien $f\colon A \to B$ und $g\colon A \to C$ endlich étale Ringhomomorphismen und + $h\colon C \to B$ ein Ringhomomorphismus mit $f = gh$. Dann ist $h$ endlich étale. +\end{satz} + +\begin{proof} + Seien zunächst $f$ und $g$ total zerlegbar. Die Eigenschaft endlich étale ist lokal auf $C$ und damit + insbesondere auf $A$. Es genügt also für jedes $\mathfrak{p} \in \spec A$ ein $a \in A$ zu finden, sodass + $h\colon C_a \to B_a$ endlich étale. + Damit folgt die Aussage aus \ref{lemma:locally-trivial} und \ref{lemma:induced-finite-etale}. + + Im Allgemeinen seien $D_1$ und $D_2$ treuprojektive $A$-Algebren, sodass + $D_1 \to B \otimes_A D_1$ und $D_2 \to C \otimes_A D_2$ total zerlegbar sind. Dann + ist mit \ref{satz:composition-projective} und \ref{satz:basischange-projective} + auch $A \to D_2 \to D_1 \otimes_A D_2$ endlich und projektiv. Da + $[ D_1 : A ] \ge 1$ folgt erneut mit \ref{satz:basischange-projective}, dass + $[ D_1 \otimes_A D_2 : D_2] \ge 1$. Also + sind mit \ref{satz:degree} $\spec D_1 \otimes_A D_2 \to \spec D_2$ und $\spec D_2 \to \spec A$ surjektiv, also + auch $\spec D_1 \otimes_A D_2 \to \spec A$. Damit ist $D = D_1 \otimes_A D_2$ treuprojektive $A$-Algebra. + + Da $D_1 \to B \otimes_A D_1$ total zerlegbar ist und $- \otimes_A D_2$ mit endlichen Produkten kommutiert, ist auch + $D = D_1 \otimes_A D_2 \to B \otimes_A D_1 \otimes_A D_2 = B \otimes_A D$ total zerlegbar. Analog + ist $D \to C \otimes_A D$ total zerlegbar und + \[ + \begin{tikzcd} + C \otimes_A D \arrow[from=1-1,to=1-3] & & B \otimes_A D \\ + & \arrow{ul} D \arrow{ur} & + \end{tikzcd} + \] kommutiert, also wendet sich der obige Speziallfall an und $C \otimes_A D \to B \otimes_A D$ ist endlich étale. + + Da $A \to D$ treuprojektiv, folgt mit \ref{satz:basischange-projective}, dass $C \to C \otimes_A D$ treuprojektiv ist. + Anwenden von \ref{kor:finite-etale} auf $C \otimes_A D \to B \otimes_A D = B \otimes_C (C \otimes_A D)$, liefert + $h\colon C \to B$ endlich étale. +\end{proof} + +\end{document}