diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..35d55d8 --- /dev/null +++ b/main.tex @@ -0,0 +1,322 @@ +\documentclass[a4paper]{amsart} +%\documentclass[a4paper,twoside]{article} + +\usepackage{color} +%\usepackage{authblk} +%\usepackage{makeidx} % allows index generatio +\makeindex +\usepackage{graphicx} +\usepackage[all]{xy} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{hhline} +\usepackage{tikz-cd} +\usepackage{longtable} +\usepackage{ulem} +%\usepackage{amsfont} +\usepackage[a4paper, left=2cm, right=2cm, top=2cm,bottom=2cm]{geometry} +\usepackage{hyperref} +\usepackage[draft=true]{minted} +\usepackage{xspace} +\usepackage[backend=bibtex, bibencoding=utf8, doi=false, url=false, isbn=false]{biblatex} +\addbibresource{./sources.bib} +%\usepackage{draftwatermark} +%\SetWatermarkText{Early draft -- Do not circulate!} +%\SetWatermarkScale{1.2} +%\usepackage{cleveref} + +\theoremstyle{definition} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{result}[theorem]{Result} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{example}[theorem]{Example}% +\newtheorem{conjecture}[theorem]{Conjecture}% +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{construction}[theorem]{Construction}% +\newtheorem{observation}[theorem]{Observation} + +%\newenvironment{proof}{\noindent + %{\bf Proof:}}{ + %\begin{flushright}$\Box$\end{flushright}} + +\usepackage[all]{xy} +\usepackage{pdflscape} +%\usepackage{nccmath} + +%\newenvironment{mpmatrix}{\begin{smallsize}\begin{pmatrix}}{\end{\pmatrix}\end{smallsize}}% +\setcounter{MaxMatrixCols}{66} + +\newcommand{\Span}{\mathrm{span}\,} +\newcommand{\Tor}{\mathrm{Tor}\,} +\newcommand{\Der}{\mathrm{Der}} +\newcommand{\KK}{\mathbb{K}} +\newcommand{\fsl}{\mathfrak{sl}} +\newcommand{\Eu}{\mathrm{Eu}} +\newcommand{\fgl}{\mathfrak{gl}} +\newcommand{\sq}{\mathrm{sq}} +\newcommand{\Hom}{\mathrm{Hom}} +\newcommand{\Pf}{\mathrm{Pf}\,} +\newcommand{\Jac}{\mathrm{Jac}} +\newcommand{\Grass}{\mathrm{Grass}} +\newcommand{\im}{\mathrm{im}\,} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\Id}{\mathrm{Id}} +\newcommand{\PP}{\mathbb{P}} +\newcommand{\IA}{\mathbb{A}} +\newcommand{\FF}{\mathbb{F}} +\newcommand{\Inf}{\mathrm{Inf}} +\newcommand{\NN}{\mathbb{N}} +\newcommand{\Spec}{\mathrm{Spec}\,} +\newcommand{\CC}{\mathbb C} +\newcommand{\sk}{\mathrm{sk}} +\newcommand{\sym}{\mathrm{sym}} +\newcommand{\QQ}{\mathbb Q} +\newcommand{\bbk}{\mathbf k} +\newcommand{\ZZ}{\mathbb Z} +\renewcommand{\PP}{\mathbb P} +\newcommand{\GL}{\mathrm{GL}} +\newcommand{\SL}{\mathrm{SL}} +\newcommand{\OO}{\mathcal{O}} +\newcommand{\m}{\mathfrak{m}} +\newcommand{\height}{\mathrm{height}\,} +\newcommand{\coker}{\mathrm{coker}\,} +\newcommand{\grade}{\mathrm{grade}\,} +\newcommand{\depth}{\mathrm{depth}\,} +\newcommand{\projdim}{\mathrm{projdim}\,} +\newcommand{\Ext}{\mathrm{Ext}} +\newcommand{\Aut}{\mathrm{Aut}} +\newcommand{\Triv}{\mathrm{Triv}} +\newcommand{\nef}{\mathrm{nef}} +\newcommand{\aut}{\mathrm{aut}} +\newcommand{\rk}{\mathrm{rank}} +\newcommand{\Div}{\mathrm{Div}} +\newcommand{\Kosz}{\mathrm{Kosz}} +\newcommand{\rank}{\mathrm{rank}\,} +\newcommand{\tr}{\mathrm{tr}\,} +\newcommand{\NS}{\mathrm{NS}} +\newcommand{\Num}{\mathrm{Num}} +\newcommand{\Pic}{\mathrm{Pic}} +\newcommand{\Oscar}{OSCAR\xspace} +%\newcommand{\D}{d} +\newcommand{\D}{\operatorname{d}\!} + +%CHANGE FOR WORKING PHASE -- DELETE SUBSEQUENT TWO LINES LATER +\newcommand{\red}[1]{{\textcolor{red}{#1}}} +\newcommand{\blue}[1]{{\textcolor{blue}{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\title{Some thoughts on finite extensions of rings} + +\author{Others} +\address{ + Other authors\dots +} +%\email{brandhorst@math.uni-sb.de} +% +\author{Matthias Zach} +\address{ + Matthias Zach: + RPTU Kaiserslautern-Landau, + 67663 Kaiserslautern, + Germany +} + +%\email{zach@mathematik.uni-kl.de} +\begin{abstract} + Dah. +\end{abstract} +\maketitle + +We are considering complex analytic maps $f \colon (\CC^n, 0) \to (\CC^p, 0)$ which +are \textit{finite}. In particular, we may assume $n \geq p$. By definition, $f$ is +finite if and only if its pullback $\varphi \colon \OO_p \to \OO_n$ is a finite +extension of rings, i.e. $\OO_n = \CC\{x_1,\dots,x_n\}$ is a finite +$\OO_p = \CC\{y_1,\dots,y_p\}$-module via $\varphi$. + +\begin{definition} + Let $\varphi \colon R \to S$ be a morphism of commutative rings with unit. An element + $x \in S$ is called \textit{integral} over $R$ if there exists a \textit{monic} + polynomial + \[ + P = X^r + c_{r-1} \cdot X^{r-1} + \dots + c_1 \cdot X + c_0 \in R[X] + \] + such that $P(x) = 0$ in $S$. +\end{definition} + +\begin{proposition} + \label{prp:Finiteness} + Let $\varphi \colon R \to S$ be a morphism of commutative, Noetherian rings with unit. + The set of elements in $S$ which are integral over $R$ is a subalgebra of $S$. + Moreover, $S$ is finite over $R$ if and only if $S$ is generated + as an $R$-algebra by finitely many integral elements. +\end{proposition} +\noindent +For a proof see \cite[Theorem 4.2 and Corollary 4.5]{Eisenbud95}. + +\medskip + +For complex analytic mappings $f$ as above, we are working with rings of +convergent power series in several variables. In practice, however, we will +mostly encounter maps which admit a \textit{polynomial representative} +\[ + f \colon \CC^n \to \CC^p, \quad (x_1,\dots,x_n) \mapsto (y_1,\dots, y_p) = + (f_1(x), \dots, f_p(x)), \quad f_j(x) \in \CC[x_1,\dots,x_n]. +\] +It seems to be no significant restriction for practical applications to +assume that also this polynomial representative is actually finite. +For finite extensions of polynomial rings we do have algorithmic methods +to compute with the structure of $S$ as an $R$-module. + +\begin{proposition} + Let $\varphi \colon \CC[y] \to \CC[x]$ be a mapping of polynomial + rings given by $y_j \mapsto f_j(x_1,\dots,x_n)$. Let + \[ + J = \langle y_j - f_j(x) : j = 1,\dots, p\rangle \subset \CC[y, x] + \] + and $LJ$ the \textit{leading ideal} of $J$ with respect to a monomial + ordering which is an elimination ordering on the $x$-variables. + Then $\varphi$ is finite if and only if $\langle x_1,\dots, x_n\rangle^N \subset LJ$ + for some $N \in \NN$. In this case, the set of monomials + $\{ x^\alpha : \alpha \in \NN^n, \, x^\alpha \notin LI\}$ + are a generating set for $\CC[x]$ as a $\CC[y]$-module. +\end{proposition} +\noindent +TODO: Find a reference for the proof. +\medskip + +Based on this result we can implement seamless transitions for elements $h \in S$ +between their representation as polynomials $h \in \CC[x]$ +and module elements $v \in R \cdot e_1 + \dots + R \cdot e_m$ with respect to a +chosen system of generators over $R$. + +\medskip +Working with power series in computer algebra is an almost impossible undertaking +beyond simple arithmetic. Recall that in general it must be deemed impossible to even decide +equality of two given power series. Thus, the naive hope at this point would be +that \textit{localizing} at the origins of the domain and codomain of $f$, +respectively, would bring us sufficiently close to working with power +series by restricting ourselves to those which are in fact \textit{rational functions}. + +\begin{proposition} + \label{prp:NonFiniteness} + Let $\varphi \colon R \to S$ be a finite extension of commutative Noetherian rings with + unit. Let $P \subset R$ be a prime ideal and $Q \subset S$ prime with + $\varphi^{-1}(Q) = P$ (i.e. $Q$ is a \textit{prime over} $P$). + Then the naturally induced morphism on the localizations + \[ + \phi \colon R_P \to S_Q + \] + is finite if and only if $Q$ is the only prime in $S$ with that property. +\end{proposition} + +\begin{proof} + According to Proposition + \ref{prp:Finiteness}, we have finiteness of $\phi$ if and only if an arbitrary element + $\frac{a}{b} \in S_Q$ is integral over $R_P$. Now by assumption both $a$ and $b$ are + integral over $R$ and hence over $R_P$ and since $\frac{a}{b} = a \cdot \frac{1}{b}$ + it suffices to consider elements of the form $\frac{1}{b}$ for $b \notin Q$. + + Suppose we found a monic polynomial + \[ + F = X^r + c_{r-1} \cdot X^{r-1} + \dots + c_1 \cdot X^1 + c_0 \in R_P[X] + \] + with $F\left(\frac{1}{b}\right) = 0$ in $S_Q$. Then we can clear denominators to + obtain + \[ + F' = c'_r \cdot X^r + c'_{r-1} \cdot X^{r-1} + \dots + c'_1 \cdot X^1 + c'_0 \in R[X] + \] + with $F'\left(\frac{1}{b}\right) = 0$ and $c'_r \notin P$. + But then, by definition of localization at $Q$, + \[ + u \cdot \left( c'_r + c'_{r-1} \cdot b + \dots + c'_1 \cdot b^{r-1} + c'_0 \cdot b^r + \right) = 0 + \] + in $S$ for some $u \notin Q$. This expression can be split as + \begin{equation} + \label{eqn:Splitting} + -u \cdot c'_r = u \cdot b \cdot \left(c'_{r-1} + c'_{r-2} \cdot b + \dots + c'_1 \cdot b^{r-2} + c'_0 \cdot b^{r-1} \right). + \end{equation} + + Now suppose $Q'\neq Q$ was another prime with $\varphi^{-1}(Q') = P = \varphi^{-1}(Q)$. + Choosing $b \in Q' \setminus Q$ leads to a contradiction in (\ref{eqn:Splitting}): + Clearly, the right hand side, containing $b$ as a factor, is an element in $Q'$. + Comparing with the left hand side, we find that + $c'_r \notin P = \varphi^{-1}(Q) = \varphi^{-1}(Q')$ and, hence, $\varphi(c'_r) \in S$ is + neither an element of $Q$, nor of $Q'$. This leaves us with $u \in Q'$. + + To arrive at a contradiction, let + \[ + \mathrm{ord}_{Q'} \colon S \to \NN_0, \quad a \mapsto \mathrm{max} + \left\{k \in \NN_0 : a \in Q'^{(k)}\right\} + \] + where $Q'^{(k)}$ denotes the $k$-th \textit{symbolic power}, i.e. + the preimage of $\left(Q'_{Q'}\right)^k$ in $S$ under the localization + map $S \to S_{Q'}$. The map $\mathrm{ord}_{Q'}$ is a \textit{valuation} + and it is now easy to see that the valuation of the left hand side of + (\ref{eqn:Splitting}) is strictly less than the valuation of the + right hand side. + + TODO: Find an argument to prove the converse. +% On the contrary, if $Q$ was the only prime over $P$, we can argue as follows. +% Let $b \in S \setminus Q$ be arbitrary and +% \[ +% G = X^r + a_{r-1} \cdot X^{r-1} + a_{r-2} \cdot X^{r-2} + \dots + a_1 \cdot X + a_0 \in +% R[X] +% \] +% a monic polynomial with $G(b) = 0$ in $S$. We may choose $G$ to be of minimal +% degree with this property. Then +% \[ +% G^\vee = 1 + a_{r-1} \cdot X + a_{r-2} \cdot X^2 + \dots + a_1 \cdot X^{r-1} + a_0 \cdot X^r +% \] +% satisfies $G^\vee\left(\frac{1}{b}\right) = 0$ in $S_Q$. Provided that we are allowed +% to divide by $a_0$ in $R_P$, we would thus get a monic polynomial +% in $R_P[X]$ annihilating $\frac{1}{b}$. +% +% Now suppose $a_0 \in P$ so that $\varphi(a_0) \in Q$. Expanding $G(b) = 0$ +% then yields +% is a unit in $ +% Now suppose $\varphi(a_0) \in Q$, so +% $a_0 \in P = \varphi^{1}(Q)$. + +\end{proof} + +\begin{example} + Consider the map + \[ + f \colon \CC \to \CC, \quad x \mapsto y = x - x^3. + \] + Clearly, $\varphi \colon \CC[y] \to \CC[x]$ is finite and, in fact, + $\CC[x]$ is a free $\CC[y]$-module with generating set $1, x, x^2$. + + For the analytification of this map $f \colon (\CC, 0) \to (\CC, 0)$ + the implicit function theorem yields that there exists a local inverse + \[ + y = x - x^3, \quad x = y + \sum_{k = 2}^\infty c_k \cdot y^k \in \CC\{y\} + \] + as a convergent power series in $y$. Therefore $\varphi^h \colon \CC\{y\} \to \CC\{x\}$ + is in fact an \textit{isomorphism} and $\CC\{x\}$ is generated as a $\CC\{y\}$-module + by the unit element $1 \in \CC\{x\}$. + + Thus we find a discrepancy of $3 - 1 = 2$ in the ranks of the two modules over the + respective rings. One could hope that the localization + \[ + \phi \colon \CC[y]_{\langle y\rangle} \to \CC[x]_{\langle x \rangle} + \] + provides us with a reasonable approximation to $\varphi^h$, but + Proposition \ref{prp:NonFiniteness} tells us that the elements + \[ + M_p = \left\langle\left(\frac{1}{1 - x}\right)^k : k \leq p\right\rangle_{\CC[y]_{\langle y\rangle}} + \] + form an infinite, ever increasing sequence of + $\CC[y]_{\langle y \rangle}$-submodules of $\CC[x]_{\langle x \rangle}$ and + that, hence, $\CC[x]_{\langle x \rangle}$ does not have a finite + $\CC[y]_{\langle y\rangle}$-module structure. +\end{example} +\printbibliography +\end{document}