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diff --git a/macros/latex/contrib/springer/svjour/amalog/example.tex b/macros/latex/contrib/springer/svjour/amalog/example.tex new file mode 100644 index 0000000000..50f6a31e35 --- /dev/null +++ b/macros/latex/contrib/springer/svjour/amalog/example.tex @@ -0,0 +1,331 @@ +\documentclass[amalog]{svjour} +\usepackage{amsmath} +\usepackage{amssymb} +% use Times for text fonts +%%%%%\usepackage{times} +% use Times also for mathematics +%%%%%\usepackage[LY1]{fontenc} +%%%%%\usepackage[LY1,mtbold]{mathtime} +%%%%%% coding for first page of a contribution +%%%%%\input{totalin.aml} +%%%%%\idline{40: 21--25}{21} % fuer Druckversion +%\idline{Probab. Theory Relat. Fields 116}{0} % fuer elektronische Fassung (ohne Seitennummern) +% +%%%%%\DOIyear{1999} % Jahr der Erstveroeffentlichung ("Online first") +%%%%%\DOI{123} % Manuskriptnummer +% +% Die Jahreszahl fuer die bibliographische Angabe, sowie die im +% Copyrightvermerk stammen aus dem Maschinendatum. Sie kann mittels +% \year=2022 % 2022 ist hier nur als Beispiel gewaehlt +% geaendert werden. +%%%%%\year=2000 +% +\def \R { {\bold R} } +\def \Q { {\bold Q} } +\def \A {{\bold A}} + +\def \F {{\bold F}} +\def \<{\langle} +\def \>{\rangle} +\def \ee{\preceq} +\def \tilde {\widetilde} +\def \N {\bold N} +\def \Z {{\bold Z}} +\def \E {\text E} +\def \L {\text L} + +\def\(( {(\!(} +\def\)) {)\!)} +\def\k{\text{\fontseries{b}\fontshape{it}\selectfont k}} + +\def \hat {\widehat} +\def \Supp {\operatorname{Supp}} +\def \Suppo {{\Supp}^* } +\def \iso{\cong} + +\def\Cal#1{{\mathcal #1}} +\def \m{\mathfrak{m}} +\def \Pos {\operatorname{Pos}} +\def \supp{\operatorname{supp}} +\def \bar{\overline} + +\spnewtheorem*{thm}{Theorem}{\bf}{\it} +\spnewtheorem*{prop}{Proposition}{\bf}{\it} +\spnewtheorem*{rem}{Remark}{\it}{\rm} + +\smartqed + +\begin{document} +\title{Riemannian geometry and Hilbert space applied to metamagical +game theory and the survival problem of Sch\"odinger's +cat\thanks{K.\,F.\,Gauss was supported by a grant of the Swiss +National Science Foundation. J.\,H.\,Poincar\'e was partially supported +by NSF grants DMS-98043192 and NCR-9850637 at Boston University}} + +\subtitle{I. Steps towards a theory of almost everything} + +\author{K.\,F.\,Gauss \and J.\,H.\,Poincar\'e } + +\titlerunning{Elementary theory of rings of Witt vectors} + +\institute{K. F. Gauss\at Department of Mathematics, University of +Illinois at +Urbana-Champaign,\\ Urbana, IL 61801, USA. +\email{gauss@math.uiuc.edu} +\and +J.\,H.\,Poincar\'e\at Equipe d'Analyse, Tour 46, Universit\'e de Paris +VI, 4 Place Jussieu,\\ F-75230 Paris Cedex 05, France. +\email{poincare@math.bu.edu}} + +\date{Received: 31 March 1999\,/\,Revised version: 14 October 1999\,/\\ +Published online: 20 December 1999} +%%%\def\year{2000} +\subclass{03A60, 12K05, 13L05} +\keywords{Riemannian geometry -- Hilbert space -- Game theory -- +Survival --\\ Schr\"odinger's cat} +\maketitle +\begin{abstract}We indicate a complete set of elementary invariants for +the ring of Witt vectors over a perfect field of prime characteristic, where this +ring is equipped with its unique multiplicative set of representatives for +the residue field. +\end{abstract} + +\noindent +Theorems of Ax, Kochen and Ersov tell us that the elementary theory of a +henselian valuation ring of equal +characteristic $0$ is completely determined +by the elementary theories of its value group and residue field, see +\cite{ref1,ref3,ref4}, and the references therein. This elementary +classification goes through even when +a predicate is added for a field of representatives of the residue field. + +Here we provide a mixed characteristic analogue of the latter when +the residue field is perfect of characteristic $p$ and the +maximal ideal is generated by $p$. For a complete discrete valuation ring +with these properties the analogue of `field of representatives' is +`multiplicative set of representatives for the residue field' and is due to +Witt. For proofs of this and related results mentioned below that we shall +use we refer to Serre \cite[Ch. II]{ref5}. +We now proceed to precise statements. + +Fix a prime number $p$. Let $A$ be a complete discrete valuation ring +with maximal ideal $\m=pA$ and perfect residue field $\k = A/\m$ (of +characteristic $p$). Let $\pi : A \to \k$ be the residue class map. +There is a unique multiplicatively closed set $S \subseteq A$ that is +mapped bijectively onto $\k$ by $\pi$. (Among the elements of $S$ are +$0$, $1$ and $-1$.) Each element $a\in A$ can be written uniquely as +$a=\sum_{i=0}^{\infty}s_ip^i$ with coefficients $s_i\in S$. By \cite[p. +413]{ref3} one can axiomatize $\operatorname{Th}(A)$ in terms of +$\operatorname{Th}(\k)$. We extend this to an axiomatization of +$\operatorname{Th}(A,S)$ in the theorem below. In its proof we shall use +the functor $W$ that assigns to $\k$ the corresponding ring $W(\k)$ of +``Witt vectors'' over $\k$. The rings $W(\k)$ and $A$ are isomorphic. + +As in \cite[p. 44]{ref5}, let $f : \k \to A$ denote +the system of multiplicative representatives, that is, $\pi(f(x)) = x$ and $f(xy) = f(x)f(y)$ +for $x,y\in \k$, and thus $f(\k) = S$. +The following easy result on $\Z$-linear relations among elements of $S$ is decisive. + +Let $k = (k_1,\dots,k_n)$ be an $n$-tuple of integers, and let $X=(X_1,\dots,X_n)$ be +an $n$-tuple of distinct indeterminates. Given an $n$-tuple $b= (b_1,\dots,b_n)$ of elements in an abelian +(additive) group $B$, put $ k\cdot b := k_1b_1 + \cdots + k_nb_n$. + +\begin{lemma}There are polynomials $R_1,\dots,R_N \in \F_p[X]$, +depending only on +$p$ and + $ k$ and not on $A$, such that for all $ x=(x_1,\dots,x_n) \in \k^n$: +$$ +k \cdot f(x)= 0 \Longleftrightarrow R_1( x) = \dots = R_N(x) = 0, +$$ +where $f(x) :=(f(x_1),\dots,f(x_n).$ +\end{lemma} +\begin{proof}By \cite[Prop. 9, p. 47]{ref5} we have for $x \in \k^n$: +$$ k \cdot f(x) = \sum_{i=0}^{\infty} f\bigl(P_i( x^{p^{-i}})\bigr)p^i$$ +where $P_i\in \F_p[X]$ depends only on $i$, $p$ and $k$. The ideal of $\F_p[X]$ +generated by the polynomials $P_i$, $i\in \N$, is generated by finitely many among them, say +$R_1,\dots,R_N$. Then $R_1,\dots,R_N$ have the property described in +the lemma.\qed\end{proof} + + +The $\Z$-linear relations together with the multiplicative relations $s=s_1s_2$ among the +elements of $S$ generate all polynomial relations over $\Z$ among elements of $S$: + + +\begin{lemma}Let $U$ and $V$ be multiplicatively closed subsets of +fields $E$ and $F$ +of characteristic $0$. Let $\lambda : U \to V$ be a bijection such that $\lambda(u_1u_2) = +\lambda(u_1)\lambda(u_2)$ for all $u_1,u_2\in U$, and such that for all +$ k \in \Z^n$ and all $u \in U^n$ we have: +$ k \cdot u = 0 \Longleftrightarrow k \cdot \lambda(u) = 0$. +Then $\lambda$ extends to an isomorphism from the subfield $\Q(U)$ of $E$ onto the subfield +$\Q(V)$ of $F$.\end{lemma} +\begin{proof}Let $P= \sum_i c_iX^i\in \Z[X]$ where the sum is over finitely +many $i\in \N^n$. Then, given $u\in U^n$, we have $P(u) = \sum_i c_iu^i= 0$ if and only if +$\sum_i c_i \lambda(u^i) = \sum_i c_i\lambda(u)^i = P(\lambda(u)) = 0$, by the hypothesis of +the lemma. The conclusion of the lemma follows easily.\qed\end{proof} + + For each $k\in \Z^n$ we fix a tuple $R = (R_1,\dots,R_N)\in \F_p[X]^N$ with the +property of Lemma 1. +Let $T$ be the theory in the language +$\{0,1,+,-,\cdot,{\bold S} \}$ (the language of rings with an extra unary predicate +${\bold S}$) whose models are the structures $(B,\Sigma)$ +such that \begin{description}[(2)] +\item[(1)] $B$ is a valuation ring with fraction field $E$ of +characteristic $0$. +\item[(2)] $\Sigma$ is a multiplicatively closed subset of $B$ that is +mapped bijectively +onto $B/\m(B)$ by the residue class map $b \mapsto \bar b : B \to B/\m(B)$. +\item[(3)] $\m(B) = pB$ and $B/\m(B)$ is a perfect field. +\item[(4)] The local ring $B$ is henselian. +\item[(5)] For each $k\in \Z^n$ we have: $k\cdot \sigma = 0 +\Longleftrightarrow R(\bar \sigma) = 0$, +for all $\sigma\in \Sigma^n$, where $R\in \F_p[X]^N$ is the tuple associated to $k$, and +$\bar \sigma := (\bar \sigma_1,\dots,\bar \sigma_n)$. +\end{description} + +\begin{thm} Two models $(B,\Sigma)$ and $(B',\Sigma')$ of $T$ are +elementarily +equivalent if and only if their residue fields $B/\m(B)$ and $B'/\m(B')$ are elementarily equivalent, +and their value groups $\Gamma$ and $\Gamma'$ are elementarily equivalent. +\end{thm} + +Here $\Gamma=v(E^{\times})$ is the value group of the valuation $v$ on the fraction field $E$ of $B$ with valuation ring $B$, and $\Gamma'$, $v'$ and $E'$ +are defined in the same way with $B'$ instead of $B$. These value groups are +considered as ordered abelian groups. + +Following Kochen \cite[pp. 407--408]{ref3}, the idea of the proof is to +pass to +sufficiently +saturated models where the valuation can be decomposed into a valuation of +equal characteristic $0$ and a complete discrete valuation. + +\begin{proof}One direction is obvious. For the other direction we assume that $B/\m(B) +\equiv B'/\m(B')$ and $\Gamma \equiv \Gamma'$. To show that then $(B,\Sigma) \equiv (B', \Sigma')$, we may assume these two models of $T$ are +$\aleph _1$-saturated. We focus on $(B,\Sigma)$, but the same analysis will apply +to $(B',\Sigma')$. We coarsen $v$ to the valuation $\tilde v$ +on $E$ with value group $\tilde \Gamma := \Gamma/\Z\cdot 1$ (where $1:= v(p)$ is the smallest +positive element of $\Gamma$) by setting $\tilde v(a) = v(a) + \Z\cdot 1$ for $a\in E^{\times}$. +The valuation ring of $\tilde v$ is +$$\tilde B := B[1/p] = \{a\in E: v(a) \ge -n\cdot 1 \text{ for some } n\}$$ with +maximal ideal $\tilde \m := \m(\tilde B) = \{a\in E: v(a) \ge n\cdot 1 \text{ for all } n\}$, and +residue field $K:= \tilde B/\tilde \m$ of characteristic $0$. Then $\tilde \m$ is also +a prime ideal of $B$, and $A:= B/\tilde \m$ is a valuation ring of $K$, with maximal ideal +$pA$. The residue class map $\lambda : \tilde B\to K$ maps $B$ onto $A$, and induces by passing to +quotients an isomorphism $B/pB \cong A/pA$ of the residue fields of $B$ and $A$. We put +$\k := B/pB = A/pA$ by identifying these residue fields +via this isomorphism. Thus $p=\pi \circ (\lambda|B)$ where $p: B \to \k$ and +$\pi : A \to \k$ are the residue class maps. Hence $S:= \lambda(\Sigma)$ is a multiplicatively +closed subset of $A$ that is mapped bijectively onto $\k$ by the residue class map $A \to \k$. +By $\aleph_1$-saturation $A$ is a complete discrete valuation ring, and therefore $(A,S)$ is +also a model of $T$, by Lemma 1. We now show how to ``lift'' the +quotient $(K,A,S)$ of +$(\tilde B,B,\Sigma)$ back to +$(\tilde B,B,\Sigma)$. The bijection $\sigma \mapsto \lambda(\sigma) : \Sigma \to S$ is +multiplicative, so by the second lemma $\lambda$ maps the ring $\Z[\Sigma]$ isomorphically +onto $\Z[S] \subseteq K$. Thus the fraction field $\Q(\Sigma)\subseteq E$ of $\Z[\Sigma]$ +is actually contained in $\tilde B$, and +$\lambda$ maps $\Q(\Sigma)$ isomorphically onto $\Q(S)$. Since $B$ is henselian, so is its +localization $\tilde B$. The residue field $K$ of $\tilde B$ being of characteristic $0$, +it follows that there is a field $L$ with $\Q(\Sigma) \subseteq L \subseteq \tilde B$ +such that $\lambda$ maps $L$ isomorphically onto all of $K$. Then $(L,B\cap L, \Sigma)$ is +the desired lifting of $(K,A,S)$, that is, $(L,B\cap L,\Sigma) \subseteq (\tilde B, B, \Sigma)$ +and $\lambda$ restricts to an isomorphism $(L,B\cap L,\Sigma)\cong (K,A,S)$. We now shift our +attention from $(B,\Sigma)$ (an expansion of the mixed characteristic valuation ring $B$) +to $(\tilde B, L, B\cap L, \Sigma)$ which we view as the equal characteristic valuation ring +$\tilde B$ equipped with a lifting of its expanded residue field $(K,A,S)$. Note that +$B$ is definable in $(\tilde B, L, B\cap L, \Sigma)$ as follows: $B=\{x\in \tilde B: x-y\in \tilde \m +\text{ for some } y\in B\cap L\}$. + +We now carry out the same construction with $(B',\Sigma')$, introducing +$\tilde v'$, $\tilde \Gamma'$, $\tilde B'$, $K'$, $\k'$, $A'$, $S'$ and $L'$ in the +same way we obtained the corresponding unaccented objects from $(B,\Sigma)$. As we indicated above it +now suffices to show that $(\tilde B, L, B\cap L, \Sigma)\equiv(\tilde B', L', B'\cap L', \Sigma')$. Consider the rings $W(\k)$ and $W(\k')$ of Witt vectors over +$\k$ and $\k'$, and for perfect subfields $F$ of $\k$ and $F'$ of $\k'$, consider the subrings +$W(F)$ and $W(F')$ of $W(\k)$ and $W(\k')$, as well as the corresponding multiplicatively +closed sets $S(F)\subseteq W(F)$ and $S(F')\subseteq W(F')$ that are mapped bijectively onto +$F$ and $F'$ by the canonical maps $W(F) \to F$ and $W(F') \to F'$. In particular we have +isomorphisms $(A,S) \cong (W(\k),S(\k))$ and $(A',S') \cong (W(\k'),S(\k'))$. Since +$\k$ and $\k'$ are elementarily equivalent and $\aleph_1$-saturated, the isomorphisms +$F\to F'$ between the countable $F\ee \k$ and $F'\ee \k'$ form a back-and-forth system +between $\k$ and $\k'$. Each isomorphism $F\to F'$ of this system induces an +isomorphism $(W(F),S(F))\to (W(F'),S(F'))$, thus giving rise to a back-and-forth system +between $(W(\k),S(F))$ and $(W(\k'),S(\k'))$. Hence $(W(\k),S(F))\equiv (W(\k'),S(\k'))$, and so + $(A,S)\equiv(A',S')$. Therefore $(K,A,S)\equiv (K',A',S')$, and thus $(L,B\cap L, \Sigma) +\equiv (L', B'\cap L', \Sigma')$. This allows us to apply Lemma 3 below to reach the desired +conclusion $(\tilde B, L, B\cap L, \Sigma)\equiv(\tilde B', L', B'\cap L', \Sigma')$. +This application also depends on the fact that $\Gamma \equiv \Gamma'$ implies $\tilde \Gamma \equiv +\tilde \Gamma'$.\qed\end{proof} + +The lemma appealed to at the end is a variant of the well-known results of Ax, Kochen and Ersov, and +can be proved in the same way, cf. \cite{ref1,ref3,ref4}. In this lemma +the value group $\Gamma$ of a valuation ring $\Cal O$ +refers to the value group of the valuation $v$ on the fraction field of $\Cal O$ such that +$v$ has $\Cal O$ as its valuation ring. This value group is considered as an ordered abelian group. + + +\begin{lemma}Let $\Cal O$ and $\Cal O'$ be henselian valuation rings of +equal characteristic +$0$ with value groups $\Gamma$ and $\Gamma'$, and let $L \subseteq \Cal O$ and +$L'\subseteq \Cal O'$ be fields that are mapped onto the +residue fields of $\Cal O$ and $\Cal O'$ by the residue class maps +$\Cal O \to \Cal O/\m(\Cal O)$ +and $\Cal O' \to \Cal O'/\m(\Cal O')$. Let $\Cal L$ be an extension of the +language of rings, and +let $L^*$ and ${L'}^*$ be expansions of the rings $L$ and $L'$ to $\Cal +L$-structures. Then +$$ (\Cal O, L^*) \equiv (\Cal O', {L'}^*) \Longleftrightarrow \Gamma +\equiv \Gamma' \text{ and } L^*\equiv {L'}^* .$$ +\end{lemma} + +The following variant of the theorem can be obtained in the same way, +by appealing to a corresponding variant of Lemma 3 (see \cite{ref4}). We +let $\k$ and $\k'$ denote the residue fields of the valuation rings $B$ and $B'$, +and let $\Gamma$ and $\Gamma'$ be their value groups as in the theorem. + +\begin{prop}Let $(B,\Sigma)$ and $(B',\Sigma')$ be models of $T$ +such that $(B,\Sigma)\subseteq (B',\Sigma')$ (so there are natural +inclusions $\k \subseteq \k'$ and $\Gamma \subseteq \Gamma')$. Then +$$(B,\Sigma)\preceq (B',\Sigma') \Longleftrightarrow \k\preceq \k' +\text{ and }\Gamma\preceq \Gamma'.$$ \end{prop} + +\begin{rem}In Lemma 1 we described the $\Z$-linear +relations among the elements of $S\subseteq A$. Another way to do this, in some respects more illuminating, is as follows. + + First, any root of unity in $A$ belongs to $S$ and any +tuple $\zeta = (\zeta_1,\dots,\zeta_n)$ ($n>0$) of roots of unity +$\zeta_i\in A$ satisfies non-trivial $\Z$-linear relations. These relations produce + in certain obvious ways further relations, for example, for any $s\in S\setminus \{0\}$ the tuple $s\zeta$ satisfies the same $\Z$-linear relations +as $\zeta$. + + Secondly, an element $a\in A$ belongs to $S$ if and only if $F(a)=a^p$, where $F$ is the canonical lifting of the Frobenius map to +an automorphism of $A$ (see \cite{ref5}). + +Using this last fact one can show, +following \cite{ref2}, that all $\Z$-linear relations among elements of +$S$ arise from the $\Z$-linear relations among the roots of unity in $A$. +This was pointed out to me by Hrushovski. +\end{rem} +\begin{acknowledgement}The authors wish to thank H.\,Minkowski and +D.\,Hilbert for stimulating discussions and encouragement. +\end{acknowledgement} + +\begin{thebibliography}{[KuPr89]} +\bibitem[A73]{ref1} Ax, J.: +A metamathematical approach to some problems in number theory. +AMS Symposium (1973) 161--190 +\bibitem[H]{ref2} Hrushovski, E.: +The Manin-Mumford conjecture and the model theory of difference fields. +Preprint +\bibitem[Ko75]{ref3} Kochen, S.: +The model theory of local fields. In: {\it Logic +Conference, Kiel 1974} (Proceedings), +Lecture Notes in Mathematics {\bf499}, Berlin 1975: Springer, pp. +384--425 +\bibitem[KuPr89]{ref4} Kuhlmann, F.-V. and Prestel, A.: +On places of algebraic function fields. +J. reine angew. Math. {\bf400}, 185--202 (1989) +\bibitem[S62]{ref5} Serre, J.-P.: +{\it Corps Locaux}. Paris: Hermann, 1962 +\end{thebibliography} + +\end{document} |