From e4d915b718aae2e2945f843419d5204712fcff44 Mon Sep 17 00:00:00 2001 From: Johannes Loher Date: Tue, 8 Aug 2017 20:09:19 +0200 Subject: [PATCH] =?UTF-8?q?Bereachnung=20der=20Randabbildung=20im=20Koszuk?= =?UTF-8?q?omplex=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Ausarbeitung.tex | 1 - chapters/chapter3.tex | 60 ++++++++++++++++++++++++++++++++----------- custom_commands.tex | 3 +++ 3 files changed, 48 insertions(+), 16 deletions(-) diff --git a/Ausarbeitung.tex b/Ausarbeitung.tex index 7f1a2da..2b9877c 100644 --- a/Ausarbeitung.tex +++ b/Ausarbeitung.tex @@ -19,7 +19,6 @@ \input{theorem_environments} \input{custom_commands} - \allowdisplaybreaks{} \begin{document} \include{title} diff --git a/chapters/chapter3.tex b/chapters/chapter3.tex index a78dd32..0ba0df7 100644 --- a/chapters/chapter3.tex +++ b/chapters/chapter3.tex @@ -62,29 +62,59 @@ Im Folgenden sei $A$ ein kommutativer Ring. \begin{equation*} K_p(\bmx) \cong \bigwedge\nolimits^p(A^r). \end{equation*} + Die Randabbildung $d\colon K_p(\bmx) \to K_{p - 1}(\bmx)$ ist durch folgende Formel gegeben: + \begin{equation} + \label{eq:koszul-komplex-randabbildung} + d(e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p)\text{-mal}}) = \sum_{k=1}^p {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}} + \end{equation} + Dabei verwenden wir die Konvention, dass das Symbol unter \enquote{$\widehat{\;}$} weggelassen wird. \begin{proof} - Für $p\in \Z$ sei $I_p^r = \{\bmi \subset \{1,\ldots,r\} \mid \#\bmi = p\}$. Wir haben folgendes zu zeigen: + Für $p\in \Z$ sei $I_p^r = \lbrace\bmi \subset \lbrace1,\ldots,r\rbrace \mid \#\bmi = p\rbrace$. Wir haben folgendes zu zeigen: \begin{equation*} - K_p(\bmx) = \bigoplus_{\bmi \in I_p^r}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right) + K_p(\bmx) = \bigoplus_{\bmi \in I_p^r}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right) \end{equation*} - Dies beweisen wir durch Induktion über $r$. Im Fall $r = 1$ gilt $I_0^1 = \{\emptyset\}$, $I_1^1 = \{\{1\}\}$ und für $p\notin \{0, 1\}$ gilt $I_p^1 = \emptyset$. Damit folgt: + Dies beweisen wir durch Induktion über $r$. Im Fall $r = 1$ gilt $I_0^1 = \lbrace\emptyset\rbrace$, $I_1^1 = \lbrace\lbrace1\rbrace\rbrace$ und für $p \notin \lbrace0, 1\rbrace$ gilt $I_p^1 = \emptyset$. Damit folgt: \begin{align*} - K_0(\bmx) &= K_0(x_1) = \bigoplus_{\bmi \in I_0^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - K_1(\bmx) &= K_1(x_1) = \bigoplus_{\bmi \in I_1^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - K_p(\bmx) &= K_p(x_1) = 0 = \bigoplus_{\bmi \in I_p^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right) \qquad \text{für }p \notin \{0, 1\} + K_0(\bmx) &= K_0(x_1) = \bigoplus_{\bmi \in I_0^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + K_1(\bmx) &= K_1(x_1) = \bigoplus_{\bmi \in I_1^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + K_p(\bmx) &= K_p(x_1) = 0 = \bigoplus_{\bmi \in I_p^1}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right) && \text{für }p \notin \lbrace0, 1\rbrace \end{align*} + Für die Randabbildung $d\colon K_1(x_1) \to K_0(x_1)$ im Grad $1$ gilt + \begin{equation*} + d(e_{x_{i_1}}) = x_1 \cdot 1 = {(-1)}^{1 + 1} x_1 \widehat{e_{x_{i_1}}}\otimes 1 = \sum_{k=1}^1 {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(1 - 1 + 1)\text{-mal}}. + \end{equation*} + In den anderen Graden ist die Randabbildung nach \cref{defn:koszul-komplex-einfach} immer $0$ und auch die Formel aus \cref{eq:koszul-komplex-randabbildung} ergibt $0$. + Sei also nun $r > 1$ und die Behauptung für alle $s\in \N$ mit $0 \le s < r$ bereits bewiesen. Sei außerdem $\bmx' = (x_1,\ldots,x_{r-1})$. Dann gilt: \begin{align*} K_p(\bmx) &= \bigoplus_{i + j = p} K_i(\bmx') \otimes_A K_j(x_r)\\ - &= \bigoplus_{j \in \{0,1\}} K_{p - j}(\bmx') \otimes_A K_j(x_r)\\ - &= \quad\phantom{\oplus}\bigoplus_{\bmi \in I_p^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r - 1\} \setminus \bmi}K_0(x_i)\right)\right) \otimes_A K_0(x_r)\\ - &\phantom{=}\quad \oplus \bigoplus_{\bmi \in I_{p - 1}^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r - 1\} \setminus \bmi}K_0(x_i)\right)\right) \otimes_A K_1(x_r)\\ - &= \quad\phantom{\oplus}\bigoplus_{\bmi \in I_p^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - &\phantom{=}\quad \oplus \bigoplus_{\bmi \in I_{p - 1}^{r - 1}}\left( \left(\bigotimes_{i \in \bmi \cup \{r\}}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus (\bmi \cup \{r\})}K_0(x_i)\right)\right)\\ - &= \quad \phantom{\oplus} \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \notin \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - &\phantom{=}\quad \oplus \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \in \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - &= \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \in \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right)\\ - &= \bigoplus_{\bmi \in I_p}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \{1, \ldots, r\} \setminus \bmi}K_0(x_i)\right)\right) + &= \bigoplus_{j \in \lbrace0,1\rbrace} K_{p - j}(\bmx') \otimes_A K_j(x_r)\\ + &= \poplus\bigoplus_{\bmi \in I_p^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r - 1\rbrace \setminus \bmi}K_0(x_i)\right)\right) \otimes_A K_0(x_r)\\ + &\peq \oplus \bigoplus_{\bmi \in I_{p - 1}^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r - 1\rbrace \setminus \bmi}K_0(x_i)\right)\right) \otimes_A K_1(x_r)\\ + &= \poplus\bigoplus_{\bmi \in I_p^{r - 1}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + &\peq \oplus \bigoplus_{\bmi \in I_{p - 1}^{r - 1}}\left( \left(\bigotimes_{i \in \bmi \cup \lbrace r \rbrace}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus (\bmi \cup \lbrace r \rbrace)}K_0(x_i)\right)\right)\\ + &= \poplus \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \notin \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + &\peq \oplus \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \in \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + &= \bigoplus_{\substack{\bmi \in I_{p}^{r}\\r \in \bmi}}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right)\\ + &= \bigoplus_{\bmi \in I_p}\left( \left(\bigotimes_{i \in \bmi}K_1(x_i)\right) \otimes_A \left(\bigotimes_{i \in \lbrace1, \ldots, r\rbrace \setminus \bmi}K_0(x_i)\right)\right) + \end{align*} + Nun betrachten wir die Randabbildung $d\colon K_p(\bmx) \to K_{p - 1}(\bmx)$. Dabei unterscheiden wir die zwei Fälle $r \in \lbrace i_1, \ldots, i_p\rbrace$ und $r \notin \lbrace i_1, \ldots, i_p\rbrace$. Im ersten Fall gilt: + \begin{align*} + & d(e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - 1 - p)\text{-mal}} \otimes \underbrace{1}_{\in K_0(x_r)}) \\ + =& d(e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - 1 - p)\text{-mal}}) \otimes 1 + {(-1)}^p e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - 1 - p)\text{-mal}} \otimes \underbrace{d(1)}_{= 0} \\ + =& \left(\sum_{k=1}^p {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p)\text{-mal}}\right) \otimes 1 \\ + =& \sum_{k=1}^p {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}} + \end{align*} + Im zweiten Fall können wir ohne Einschränkung $r = i_p$ annehmen. Dann gilt: + \begin{align*} + & d(e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p)\text{-mal}}) \\ + =&\pplus d(e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{((r - 1) - (p - 1)\text{-mal}}) \otimes e_{x_{i_p}} \\ + &+ {(-1)}^{p + 1} e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p)\text{-mal}} \otimes d(e_{x_{i_p}}) \\ + =& \pplus \left( \sum_{k=1}^{p - 1} {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}}\right) \otimes e_{x_{i_p}} \\ + & + {(-1)}^{p+1} e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p)\text{-mal}} \otimes (x_{i_p} \cdot 1) \\ + =& \pplus \sum_{\substack{k=1\\k\ne p}}^p {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}} \\ + & + {(-1)}^{p + 1} x_{i_p} e_{x_{i_1}}\otimes \cdots \otimes e_{x_{i_{p - 1}}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}} \\ + =& \sum_{k=1}^p {(-1)}^{k+1} x_{i_k}e_{x_{i_1}}\otimes \cdots \otimes \widehat{e_{x_{i_k}}} \otimes \cdots \otimes e_{x_{i_p}} \otimes \underbrace{1 \otimes \cdots \otimes 1}_{(r - p + 1)\text{-mal}} \end{align*} \end{proof} \end{lem} diff --git a/custom_commands.tex b/custom_commands.tex index 3978ccf..d8b2dc5 100644 --- a/custom_commands.tex +++ b/custom_commands.tex @@ -18,3 +18,6 @@ \DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\map}{map} \DeclareMathOperator{\Supp}{Supp} +\DeclareMathOperator{\pplus}{\phantom{+}} +\DeclareMathOperator{\poplus}{\phantom{\oplus}} +\DeclareMathOperator{\peq}{\phantom{=}}