Approximation Theorems in Commutative Algebra: Classical and Categorical Methods

338 Pages·1992·27.8 MB·English

by  J. Alajbegović

Approximation Theorems in Commutative Algebra
Mathematics and Its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALYNICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland
H. KURKE, Humboldt University, Berlin, Germany
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOVÄSZ, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC, University of Belgrade, Yugoslavia
S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, University of Jena, Germany
Volume 59
Approximation
Theorems in
Commutative Algebra
Classical and Categorical Methods
by
J. Alajbegovic
Department of Mathematics,
RM1T, Melbourne,
Australia
and
J. Mockof
Department of Mathematics,
University of Ostrava,
Ostrava, Czechoslovakia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Alajbegovic , Jusuf H., 1951-
Approximation theorems i n commutative algebr a : classica l and
categorica l methods / Jusuf H. Alajbegovic , Jir i Mockor.
p. cm. — (Mathematics and it s applications . East European
serie s ; v. 59)
Includes bibliographica l reference s and indexes.
ISBN 978-94-010-5204-7 ISBNN 978-94-011-2716-5 (eBook)
DOII 10.1007/978-94-011-2716-5
1. Commutative rings . 2. Approximatio n theory . I . Mockor, Jiri .
II . Title . III . Series . Mathematics and it s application s (Kluwer
Academci Publishers) . East European serie s ; v. 59.
QA251.3.A38 1992
512.4—dc20 92-26603
All Rights Reserved
© 1992 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academci Publishers in 1992
as specified on appropriate pages within.
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanica,l
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To our parents
SERIES EDITOR'S PREFACE
ClDo _ IIIIIIIoaIIIics bu _ die
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Mathematics is a tool for dloogIrt. A bighly necessary tool in a world where both feedback and noolineari
ties abound. Similarly, all kinds of parts of IIIIIIhcmatiI:s serve as tools for odIcr parts and for ocher sci
eoccs.
Applying a simple rewriting rule to the quote on the right above one finds suc:h stalements as: 'One ser
vice topology has rcncIerM mathematical physics .. .'; 'One service logic has rendered computer science
.• .'; 'One service category theory has rmdcn:d mathematics ... '. All arguably true. And all statements
obrainable this way form part of the raison d'etm of this series.
This series, Mathmlatics tDIII Its Applications, saaned in 1977. Now that over one hundred volumcs have
appeared it seems opportune to reexamine its scope. AI. the time I wrote
"Growing spccialization and divenification have brought a host of monographs and textbooks
on incJeasingly specialized topics. However, the 'tree' of knowledge of JJJatbcmatics and
reIatcd ficIds docs not grow only by putting forth new bnDdIcs. It also happens, quite often in
fact, that brancbes which were thought to be comp1etcly disparate am suddenly seen to be
rdatcd. Furtbc:r, the kind and 1cvc1 of sopbisIication of mathematics applied in various sci
ences has changed drasIic:ally in nx:mt yean: _ theory is used (non-trivially) in
Rgional and tbcoretic:al ec:clIlOIDica; algelnic geomcuy inlClllCts with physics; the Minkowsky
1cmma, coding theory and the structure of water meet ODe another in pding and covering
theory; quantum fields, crystal clefects and f!!8thcmatical programming profit from homotopy
theory; Lie algebras am rdcvant to fikc:ring; and prediction and dccuil:al engineering can usc
Stein spaces. And in addition to this there am suc:h new emerging subdilciplincs as 'experi
mental mathematics', 'CPO', 'completdy integrable systems', '~synergetics and large
scale order', which am almost impossible to fit into the existing classificarioo schemes. They
draw upon widely different sections of mathematics ..
By and large, all this IIlill applies today. It is still true that at first sight mathematics seems rather frag
mented and that to find, sec, and exploit the cIecpc{ underlying iDteneIations more effort is nccdcd and so
am books that can bclp mathcmatic:ians and scicnIists do so. Accordingly MIA will continue to tty to make
suc:h boob available.
If anything, the dcac:ription I gave in 1977 is DOW an undcnIatcmcnL To the cumplcs of interaction
areas one sbouId add string theory where Riemaon smfacca, algebraic: gcomcay, modular functions, knots,
quantum field theory, Kac-Moody algebras, IDOIIIUOUI moonshine (and _) all come togcthc:r. And to
the cumples of things which can be usefully applied let me add the topic 'finite geometry'; a combination
of words which sounds like it might not cvm exist. let alone be applicable. And yet it is being applied: to
statistics via designs, to radar/sorw detection mays (via finite projective planes), and to bus connections
of VLSI chips (via diffcn:ucc sets). 1beIe seems to be DO part of (so-c:allt:d JlIIR') mathemarics that is not
in immediate danger of being applied. And, aa:onIingly, the applied matbcmatic:ian needs to be awam of
much more. Besides analysis and numerics, the traditional woddtorscs, he may nced~ kinds of combina
tories, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonIincar world and the extra
mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest
and a bit sad and depressing: proportional efforts and resuIts. It is in the noniinear world that infinitesimal
inputs may resuIt in macroscopic outputs (or vice vena). To appreciate what I am hinting at: if electronics
were linear we would have no fun with transistors and computers; we would have no TV; in fact you
would nOl be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, supcrspacc and
anticommuting integration, p-adic and ultnunctric space. All thRC have applications in both electrical
engineering and physics. Once, complex numbers were cquaIly outlandish, but they frequently proved the
monesl path bcIwcc:n 'rea!' results. Similarly, the tint IWO topics named have already provided a number
of •w ormhole , paths. There is no telling where all this is leading -fortunately.
Thus the original scope of the series, whiclI for various (sound) reasons now comprises five subscrics:
while (Japan), yellow (OJina), red (USSR), bluc (Eastern Europe), and green (everything else), still
applies. II has been enlarged a bit to include books treating of the tools from one subdiscipline which are
used in others. Thus the series still aims at books dealing with:
a cenlIal concept which plays an imponant role in scvCJal different mathematical and/or scientific
specialization areas;
new applications of the results and ideas from one area of scientific endeavour into another,
inftucnces which the results, problems and concepts of one field of enquiry have, and have had, on the
development of another.
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Busswn, 1992 Michicl Hazcwinkcl
Contents
Preface xiii
PART I CLASSICAL METHODS
1 APPROXIMATION THEOREMS FOR VALUATIONS
ON FIELDS 3
1.1 Introduction to Valuation Theory 3
Partially ordered groups · .. 3
Valuations on fields ...... 6
1.2 Approximation Theorems for Krull Valuations 11
Weak approximation theorems. . . . . . . 11
Approximation theorems .......... 15
Approximation theorems and upper classes. 18
Approximation theorems for Priifer domains 25
Approximation theorems for Priifer rings of Krull type 29
Extensions of approximation theorems 34
1.3 Applications in Topological Rings . . . . . . . . . 37
2 VALUATIONS ON COMMUTATIVE RINGS 49
2.1 Basic Properties of the Manis Valuation 50
Rings of quotients ........ 50
Large quotient rings ....... 51
Definition of the Manis valuation 52
Comparison of valuations 57
Some useful inequalities · . 64
2.2 R-Priifer Rings ......... 66
Definition of R-Priifer rings 66
R-Priifer valuation rings · . 69
2.3 Valuations with the Inverse Property 72
Definition and basic properties 72
Comparison of valuations with the inverse property 75
x CONTENTS
Inequalities for valuations with the inverse property . 77
The independence of valuations . . . . . 80
R-Priifer rings and the inverse property 85
2.4 Approximation Theorems .......... 89
Compatibility conditions . . . . . . . . . 90
Approximation theorem in the neighborhood of zero 92
General approximation theorem . . . . . 98
R-Priifer rings and families of valuations . . . . . . 110
3 ORDERED GROUPS AND HOMOMORPHISMS 127
3.1 Groups of Divisibility ... 127
Lattice-ordered groups ..... . 127
Groups of divisibility . . . . . . . 135
3.2 Groups with the Theory of Divisors . 146
4 APPROXIMATION THEOREMS FOR MULTI
STRUCTURES 159
4.1 Introduction to Multirings .. 160
Basic facts about m-rings 160
m-valuations ...... . 167
4.2 Approximation Theorem for Multirings 175
4.3 Introduction to d-Groups . ...... . 187
4.4 Approximation Theorems for d-Groups . 200
PART II CATEGORICAL METHODS
5 CATEGORICAL LOGIC 211
5.1 Topoi and Sheaves ......... . 211
5.2 Interpretation of Logic in Categories 230
The syntax of L. . . . . . . . 230
The semantic of L ..... . 231
Interpretations as subobjects 235
Interpretations as morphisms 237
Relations between interpretations 239
Canonical language and its interpretation 247
5.3 Axioms Valid in Interpretations of Logic in Categories 250
Example of completeness. . 250
Examples of valid sequents . . . . . . . . . . . . . . 253
CONTENTS Xl
6 APPROXIMATION THEOREMS IN CATEGORIES 263
6.1 Models of a Theory of Approximation Theorems. 263
Approximation theorems in algebras . 263
Approximation theorems in categories 268
6.2 Approximation Theorems and Sheaves 277
e-sheaves ... . . . . . . . . . . . . . 277
e-sheaves over cofinal subsets ..... 281
6.3 Relations between Approximation Theorems 288
Derived approximation theorems .... 288
Construction of derived approximation theorems . 301
Examples ..................... . 304
Bibliography 307
Index of Notation 317
Index 327