From c125dff175c93fb933c272b10322c4537ce3c3e0 Mon Sep 17 00:00:00 2001 From: Johannes Loher Date: Wed, 2 Aug 2017 18:24:13 +0200 Subject: [PATCH] initial commit --- .gitignore | 23 +++++ Ausarbeitung.tex | 217 +++++++++++++++++++++++++++++++++++++++ LOGO_UR.eps | Bin 0 -> 444466 bytes bibliography.bib | 10 ++ custom_commands.tex | 18 ++++ theorem_environments.tex | 18 ++++ 6 files changed, 286 insertions(+) create mode 100644 .gitignore create mode 100644 Ausarbeitung.tex create mode 100644 LOGO_UR.eps create mode 100644 bibliography.bib create mode 100644 custom_commands.tex create mode 100644 theorem_environments.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..45b9962 --- /dev/null +++ b/.gitignore @@ -0,0 +1,23 @@ +# Compiled tex files +*.log +*.aux +*.out +*.toc +*.pdf.mk +*.pdf_FAILED +*.synctex.gz +*.dvi +*.fdb_latexmk +*.ps +*.bbl +*.blg +*.bcf +*.run.xml + +# Mac OS X files +.DS_Store +._* + +# misc +Makefile +.*.swp diff --git a/Ausarbeitung.tex b/Ausarbeitung.tex new file mode 100644 index 0000000..d76e5d1 --- /dev/null +++ b/Ausarbeitung.tex @@ -0,0 +1,217 @@ +\documentclass[paper=a4,parskip=half,11pt,ngerman,oneside,bibliography=totoc]{scrreprt} + \usepackage[utf8]{inputenc} + \usepackage[T1]{fontenc} + \usepackage{hyperref} + \hypersetup{pdfinfo={Title={Serres Tor-Formel und Reduktion auf die Diagonale}, Author={Johannes Loher}}} + \usepackage{amsthm} + \usepackage{csquotes} + \usepackage{babel} + \usepackage{amsmath} + \usepackage{amssymb} + \usepackage{lmodern} + \usepackage{mathtools} + \usepackage{enumerate} + \usepackage[backend=biber,style=alphabetic]{biblatex} + \bibliography{bibliography} + \usepackage{cleveref} + + \input{theorem_environments} + \input{custom_commands} + +\begin{document} + +\begin{titlepage} + \begin{center} + \includegraphics[width=0.5\textwidth]{./LOGO_UR}\\[1cm] + \textsc{\LARGE Universität Regensburg}\\[0.5cm] + \textsc{\Large Fakultät für Mathematik}\\[1.5cm] + { \huge \bfseries Serres Tor-Formel und Reduktion auf die Diagonale}\\[1.5cm] + {\LARGE Masterarbeit}\\[1cm] + {\Large von}\\[0.2cm] + {\LARGE \bfseries Johannes Loher}\\[0.2cm] + {\large (Matrikelnummer 1 576 123)}\\[1.5cm] + \begin{tabular}{lr} + \large Betreuer: & \large Prof.~Dr.~Moritz Kerz\\ + \large Gutachter: & \large Prof.~Dr.~Moritz Kerz + \end{tabular} + \vfill + {\large \today} + \end{center} +\end{titlepage} + +\tableofcontents + +\chapter{Einleitung} +\label{cha:einleitung} + +\chapter{Hilbert-Samuel-Polynome} +\label{cha:hilbert-samuel-polynome} + +\section{Polynomartige Funktionen} +\label{sec:polynomartige-funktionen} + +Im Folgenden sei $A = \Z$ oder $A = \{z \in \Z \mid z \ge n_0\}$ für ein gewisses $n_0 \in \Z$. + +\begin{defn} + \label{defn:polynomartige-funktionen} + Eine Funktion $f\colon A \to \Z$ heißt \textbf{polynomartig}, wenn es ein Polynom $P_f \in \Q[X]$ gibt, das folgende Eigenschaft erfüllt: Es gibt ein $m \in \Z$ mit der Eigenschaft $f(n) = P_f(n)$ für alle $n \ge m$. Gibt es solch ein Polynom, so ist es offensichtlich eindeutig durch $f$ bestimmt. Wenn $k$ der Grad von $P_f$ ist, so sagen wir auch, dass $f$ Grad $k$ hat und mit $e_k(f)$ bezeichnen wir dann den höchsten Koeffizienten von $P_f$. +\end{defn} + +\begin{defn} + \label{defn:differenzenoperator} + Wir definieren den \textbf{Differenzenoperator} $\Delta$ durch: + \begin{align*} + \Delta\colon \map(A,\Z) &\to \map(A,\Z)\\ + f &\mapsto (\Delta f \colon A \to \Z,\, n \mapsto f(n+1) - f(n)) + \end{align*} +\end{defn} + +\begin{lem} + \label{lem:kritreium-fuer-polynomartig} + Sei $f\colon A \to \Z$ eine Abbildung, dann sind folgende Aussagen äquivalent: + \begin{enumerate}[(i)] + \item $f$ ist polynomartig vom Grad $r$ + \item $\Delta f$ ist polynomartig vom Grad $r - 1$ + \item Für alle $s \ge r + 1$ gibt es ein $m \in \Z$, sodass für alle $n \ge m$ gilt: $\Delta^s f(n) = 0$. + \end{enumerate} + \begin{proof} + \cite[Chapter II. B 2. Lemma 2.]{chin2012local} %TODO: Vielleicht Beweis ausführen + \end{proof} +\end{lem} + +\section{Das Hilbert-Polynom} +\label{sec:das-hilbert-polynom} + +Im Folgenden sei nun $H=\bigoplus_{n \in \N} H_n$ ein graduierter Ring mit folgenden Eigenschaften: +\begin{enumerate}[a)] + \item $H_0$ ist artinsch. + \item $H$ wird von $H_0$ und einer endlichen Anzahl an Elementen $x_1,\ldots,x_r \in H_1$ erzeugt. +\end{enumerate} +Es gilt dann $H \cong H_0[X_1,\ldots,X_r]/I$, wobei $I \subset H_0[X_1,\ldots,X_r]$ ein homogenes Ideal ist. Da $H_0[X_1,\ldots,X_r]$ nach dem Hilbertschen Basissatz noethersch ist, ist es also auch $H$. + +Sei $M = \bigoplus_{n \in \N} M_n$ ein endlich erzeugter graduierter $H$-Modul. $M$ ist noethersch, also ist für alle $n \in N$ der $H$-Untermodul $M_{\ge n} \coloneqq \bigoplus_{i \ge n} M_n$ endlich erzeugt. Es gilt $M_n \cong M_{\ge n}/M_{\ge n+1}$, also ist auch $M_n$ ein endlich erzeugter $H$-Modul und damit auch ein endlich erzeugter $H/\Ann(M_n)$- Modul. Es gilt +\begin{align*} + H/\Ann(M_n) = H/\Ann(M_{\ge n}/M_{\ge n+1}) = H/\left(\bigoplus_{n\ge 1}H_n\right) \cong H_0, +\end{align*} +also ist $M_n$ ein endlich erzeugter $H_0$-Modul und somit artinsch und noethersch. Folglich ist er von endlicher Länge und wir können folgende Abbildung definieren: +\begin{align*} + \chi(M,\bullet) \colon \N &\to \N\\ + n &\mapsto \chi(M,n) \coloneqq \length_{H_0}(M_n) +\end{align*} +Dabei ist $\length_{H_0}(M_n)$ die Länge von $M_n$ als $H_0$-Modul. + +\begin{thm}[Hilbert] + \label{thm:hilbert-polynomial} + Die Abbildung $\chi(M,\bullet)$ ist polynomartig vom Grad $\le r-1$. + \begin{proof} + Da jeder endlich erzeugte graduierte $H$-Modul auch ein endlich erzeugter graduierter $H_0[X_1,\ldots,X_r]$-Modul ist, dürfen wir $H = H_0[X_1,\ldots,X_r]$ annehmen. + + Wir beweisen die Behauptung nun durch Induktion über $r$. + + Im Fall $r = 0$ ist $M$ ein endlich erzeugter $H_0$-Modul und damit von endlicher Länge. Es gilt $M_n = 0$ für große $n$ und damit ist $\chi(M,\bullet)$ polynomartig vom Grad $-\infty$. + + Wir nehmen nun $r > 0$ an und, dass die Behauptung für $r - 1$ bereits gezeigt wurde. Seien $N$ und $R$ der Kern und der Kokern des Endomorphismus $\Phi\colon M \to M,\, m \mapsto X_r \cdot m$. Das sind graduierte Moduln und wir erhalten die exakten Sequenzen: + \begin{equation*} + 0 \to N_n \to M_n \overset{\Phi}{\to}M_{n+1} \to R_{n+1} \to 0 + \end{equation*} + Es folgt + \begin{equation*} + \Delta\chi(M,n) = \chi(M,n+1) - \chi(M,n) = \chi(R,n+1) -\chi(N,n). + \end{equation*} + Wegen $X_rR = X_rN = 0$ können wir $R$ und $N$ als graduierte $H_0[X_1,\ldots,X_{r-1}]$-Moduln betrachten. Nach der Induktionsvoraussetzung sind dann $\chi(R,\bullet)$ und $\chi(N,\bullet)$ polynomartige Funktionen vom Grad $\le r-2$, also auch $\Delta\chi(M,\bullet)$. Nach \cref{lem:kritreium-fuer-polynomartig} ist dann $\chi(M,\bullet)$ polynomartig vom Grad $\le r - 1$. + \end{proof} +\end{thm} + +\begin{nota} + \label{nota:hilbert-polynomial} + Das Polynom $P_{\chi(M,\bullet)}$ heißt \textbf{Hilbert-Polynom} und wird auch mit $Q(M)$ bezeichnet. Seinen Wert bei einer ganzen Zahl $n$ schreiben wir auch als $Q(M,n)$. Offensichtlich gilt $Q(M) = 0$ genau dann, wenn $\length_{H_0}(M) < \infty$. +\end{nota} + +\section{Das Samuel-Polynom} +\label{sec:das-samuel-polynom} +Sei $A$ ein noetherscher kommutativer Ring und sei $M$ ein endlich erzeugter $A$-Modul und $\mfq$ ein Ideal von $A$. Wir nehmen zusätlich Folgendes an: +\begin{equation} + \label{eq:laenge-kleiner-unendlich} + \length_A(M/\mfq M) < \infty. +\end{equation} +Dies ist äquivalent zu: +\begin{equation} + \label{eq:elemente-in-supp} + \text{Alle Elemente in }\Supp(M) \cap V(\mfq)\text{ sind maximale Ideale.} +\end{equation} +Der Fall, der uns im Folgenden am meisten interessiert, ist, dass $A$ lokal mit maximalem Ideal $\mfm$ ist und $\mfq$ ein Ideal von $A$ mit $\mfm \supset \mfq \supset \mfm^s$ für ein $s > 0$. + +Sei $(M_i)$ eine $\mfq$-gute Filtrierung von $M$, das heißt +\begin{align*} + &M = M_0 \supset M_1 \supset \ldots,\\ + &M_i \supset \mfq M_{i-1},\\ + &M_i = \mfq M_{i-1} \text{ für große i.} +\end{align*} +Wegen $V(\mfq) = V(\mfq^n)$ für alle $n > 0$ sind die $A$-Moduln $M/\mfq^nM$ von endlicher Länge und wegen $M_n \supset \mfq^nM$ auch die $A$-Moduln $M/M_n$. Demnach ist die Abbildung +\begin{align*} + f_M \colon \N &\to \N\\ + n &\mapsto \length_{A}(M/M_n) +\end{align*} +wohldefiniert. + +\begin{thm}[Samuel] + \label{thm:samuel-polynomial} + Die Abbildung $f_M$ ist polynomartig. + \begin{proof} + Wir können $\Ann(M) = 0$ annehmen: Sei $A' = A/\Ann(M)$ und $\mfq'$ das Bild von $q$ in $A'$. Dann ist $(M_i)$ auch eine $\mfq'$-gute Filtrierung von $M$ als $A'$-Modul und es gilt offensichtlich $\length_A(M/M_n) = \length_{A'}(M/M_n)$ und $\Ann_{A'}(M) = 0$. + + Nach \cref{eq:elemente-in-supp} sind dann alle Elemente in $V(\mfq)$ maximale Ideale, also ist $A/\mfq$ artinsch. Sei \begin{equation*} + H = \gr_{\mfq}(A) = \bigoplus_{n\in \N} \mfq^n/\mfq^{n+1} + \end{equation*} + der zu $\mfq$ assozierte graduierte Ring. Dann ist + \begin{equation*} + \gr(M) = \bigoplus_{n\in \N} M_n/M_{n+1} + \end{equation*} + ein graduierte $H$-Modul. Nach Voraussetzung gibt es ein $n_0\in \N$ mit $M_{n+1} = \mfq M_n$ für alle $n \ge n_0$, also wird $\gr(M)$ von + \begin{equation*} + M_0/M_1 \oplus \cdots \oplus M_{n_0}/M_{n_0+1} + \end{equation*} + erzeugt und ist demnach endlich erzeugt. %TODO: Wieso? + Nach \cref{thm:hilbert-polynomial} ist dann die Abbildung $\chi(\gr(M),\bullet)$ polynomartig. Es gilt + \begin{equation*} + \Delta f_M(n) = \length_A(M/M_{n+1}) - \length_A(M/M_n) = \length_A(M_n/M_{n+1}) = \chi(\gr(M), n), + \end{equation*} + also ist $\Delta f_M$ polynomartig und nach \cref{lem:kritreium-fuer-polynomartig} auch $f_M$. + \end{proof} +\end{thm} + +\begin{bem} + \label{bem:samuel-polynom} + Das Polynom $P_{f_M}$ heißt \textbf{Samuel-Polynom} und wird auch mit $P((M_i))$ bezeichnet. Seinen Wert bei einer ganzen Zahl $n$ schreiben wir auch als $P((M_i),n)$. Der Beweis von \cref{thm:samuel-polynomial} zeigt + \begin{equation} + \Delta P((M_i)) = Q(\gr(M)), + \end{equation} + wobei $\gr(M)$ der zur Filtrierung $(M_i)$ assozierte graduierte Modul ist. + + Ist $(M_i)$ die $\mfq$-adische Filtrierung auf $M$, so schreiben wir auch $P_\mfq(M)$ anstatt von $P((\mfq^iM))$. Das nächste Lemma liefert einen Zusammenhang zwischen dem allgemeinen und dem $\mfq$-adischen Fall: +\end{bem} + +\begin{lem} + \label{lem:samuel-polynom-allgemein-und-q-adisch} + Es gilt \begin{equation*} + P_\mfq(M) = P((M_i)) + R, + \end{equation*} + wobei $R \in \Q[X]$ ein Polynom vom Grad $\le \deg(P_\mfq(M)) - 1$, dessen führender Koeffizient~$\ge 0$ ist. + \begin{proof} + + \end{proof} +\end{lem} +\printbibliography + +\chapter*{Eigenständigkeitserklärung} +\addcontentsline{toc}{chapter}{Eigenständigkeitserklärung} +Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst und +keine anderen Hilfsmittel als die angegebenen verwendet habe. + +\vspace{1.5cm} +\begin{tabular}{lp{2em}l} + \hspace{6cm} & & \hspace{6cm} \\\cline{1-1}\cline{3-3} + Ort, Datum & & Unterschrift +\end{tabular} + +\end{document} \ No newline at end of file diff --git a/LOGO_UR.eps b/LOGO_UR.eps new file mode 100644 index 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zL$5#&K{*)Q7>Dc+-gFTnOcaN`GAxZki0U+4!~vpP`S|l=YGe3FU>x57=PL{p8tZHd zf{k&C|95^1gJVn>ha9{DOQR5?It9_4-mC8$_uN-w_j>^REO-T63C2`V))kZRgD7|* z{7x9&xDftn7{2mC7?rvDLikD;e%*yIx*7bBQ#jXNan+bU8f1U@Lim<%0icICuxjvr W7!FS1J8rrHoP%S|?N7l1@cRFSW@NGe literal 0 HcmV?d00001 diff --git a/bibliography.bib b/bibliography.bib new file mode 100644 index 0000000..ea7967d --- /dev/null +++ b/bibliography.bib @@ -0,0 +1,10 @@ +@book{chin2012local, + title={Local Algebra}, + author={Chin, C.W. and Serre, J.P.}, + isbn={9783662042038}, + lccn={00032970}, + series={Springer Monographs in Mathematics}, + url={https://books.google.de/books?id=Ne3tCAAAQBAJ}, + year={2012}, + publisher={Springer Berlin Heidelberg} +} diff --git a/custom_commands.tex b/custom_commands.tex new file mode 100644 index 0000000..b57dc50 --- /dev/null +++ b/custom_commands.tex @@ -0,0 +1,18 @@ +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} + +\newcommand{\length}{\ell} + +\newcommand{\mfa}{\mathfrak{a}} +\newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfm}{\mathfrak{m}} +\newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} + +\DeclareMathOperator{\Ann}{Ann} +%\DeclareMathOperator{\deg}{deg} +\DeclareMathOperator{\gr}{gr} +\DeclareMathOperator{\map}{map} +\DeclareMathOperator{\Supp}{Supp} diff --git a/theorem_environments.tex b/theorem_environments.tex new file mode 100644 index 0000000..ab5e520 --- /dev/null +++ b/theorem_environments.tex @@ -0,0 +1,18 @@ +\newtheorem{thm}{Theorem}[chapter] +\newtheorem{prop}[thm]{Proposition} +\newtheorem{lem}[thm]{Lemma} +\newtheorem{kor}[thm]{Korrolar} +\newtheorem{propdef}[thm]{Proposition/Definition} +\theoremstyle{definition} +\newtheorem{defn}[thm]{Definition} +\newtheorem{nota}[thm]{Notation} +\newtheorem{bem}[thm]{Bemerkung} + +\crefname{thm}{Theorem}{Theoreme} +\crefname{prop}{Proposition}{Propositionen} +\crefname{kor}{Korrolar}{Korrolare} +\crefname{propdef}{Proposition/Definition}{Propositionen/Definitionen} +\crefname{lem}{Lemma}{Lemmata} +\crefname{defn}{Definition}{Definitionen} +\crefname{nota}{Notation}{Notationen} +\crefname{bem}{Bemerkung}{Bemerkungen}