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Formal Methods in the Theories of Rings and Domains

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Formal Methods in the Theories of Rings and Domains Formal Methods in the Theories of Rings and Domains Davide Rinaldi Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München München, den 24. Juni 2014 Formal Methods in the Theories of Rings and Domains Davide Rinaldi Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München vorgelegt am 24. Juni 2014 von Davide Rinaldi aus Vittorio Veneto, Italien Erstgutachter: Dr. Peter Schuster Zweitgutachter: Prof. Dr. Helmut Schwichtenberg Externer Gutachter: Dr. Henri Lombardi, Besançon Tag der mündlichen Prüfung: 14. Juli 2014 Contents Contents v Introduction 1 1PreliminariesandResultsinFormalTopology 7 1.1 Foundation and Terminology .................... 7 1.2 Basic Pairs and Concrete Spaces .................. 9 1.2.1 Morphisms of Basic Pairs and Concrete Spaces ...... 14 1.3 Basic Topologies with Convergence Operation ........... 16 1.3.1 Morphisms of Basic Topologies with Operation ...... 20 1.4 Generation of Basic Topologies ................... 22 1.4.1 Finitary Basic Topologies .................. 25 1.4.2 Further Properties of the Convergence Operation ..... 28 1.5 Entailment Relations and Completeness Results .......... 31 1.6 Quotient and Localisation of a Basic Topology .......... 36 1.6.1 Krull Dimension of a Formal Topology ........... 39 2TheZariskiSpectrumasaBasicTopology 43 2.1 The Basic Zariski topology ..................... 44 2.1.1 The Product as Convergence Operation .......... 47 2.1.2 Ring Homomorphisms and Continuous Relations ..... 49 2.1.3 Quotients and Localisations of Zar(A) ........... 51 2.2 The Formal Zariski topology Zarf (A) ............... 54 2.2.1 Spatiality and Reducibility of Zarf (A) ........... 57 2.2.2 Connections with the Zariski Lattice ............ 65 3AConstructiveNotionofCodimension 67 3.1 Krull Dimension of the Zariski Formal Topology ......... 68 3.2 The ! k-topologies and Ring Codimension ............. 71 3.2.1 Nilregular Elements and Equidimensionality ........ 75 3.3 Krull’s Principal Ideal Theorem ................... 78 vi CONTENTS 4TheFormalProjectiveEisenbud-Evans-StorchTheorem 87 4.1 The Projective Spectrum as a Formal Topology .......... 88 4.2 The h-nilregular Element Property and Projective Dimension .. 91 4.3 The Projective Eisenbud-Evans-Storch Theorem ......... 95 5AUniversalKrull-LindenbaumTheorem 99 5.1 A Variant of Open Induction ....................100 5.2 A Universal Krull–Lindenbaum Theorem ..............103 5.3 Instantiations in Algebra and Logic .................111 6UnaryFormalTopologiesandFiniteDensity 123 6.1 The Scott Topology as a Basic Pair .................124 6.2 Unary Formal Topologies ......................129 6.3 Categorical Structure of UFTop∗ ..................137 6.4 The Unary Formal Topology Associated to an Algebra ......142 6.5 A Revisitation of the Classical Density Theorem .........145 6.5.1 A Finite Version of the Density Theorem .........151 6.6 An Abstract Notion of Totality ...................155 6.6.1 Application to Finitary Algebras ..............161 6.7 Comparing Classical and Abstract Totality ............162 Bibliography 167 Acknowledgements My first sincere thanks go to my advisor Dr. Peter Schuster, whose patience and support went much beyond his duties as an advisor. As a matter of fact, this thesis would have been neither started nor completed without his help and ideas within our mathematical interaction and his personal encouragement. This thesis does not do justice to the quality of his supervision. IamverygratefultoProf.HelmutSchwichtenbergwhogavemethe op- portunity of being, and welcomed me as, a PhD student in the Munich logic group. His presence, his comments and our mathematical discussions helped me to broaden my knowledge and to overcome some of my limits. IwouldliketoexpressmydeepgratitudetoProf.HenriLombardi, that accepted to be the external referee of this thesis. I regard this as an honour. Our mathematical discussions had a great influence in the development of this work. IwasfinanciallysupportedbyMALOA,anEU-fundedFP7project, includ- ing a generous travel allowance. Thanks to this, I had the chance to visit the University of Leeds for some months, take part to relevant conferences and meet leading scholars of my field. My adventure in Munich, as well as my office, has been shared withmycol- leagues Basil Karadais, Kenji Miyamoto and Iosif Petrakis. Despite the differ- ences in culture and research subject, we have been for sure a helpful and caring family of mathematicians. Temporary members of this family were also Daniel Bembé, Fredrik Forsberg, Florian Ranzi, Pedro Valencia and Albert Ziegler. IwanttothankMarcoBenini,UlrichBerger,ThierryCoquandand Giovanni Sambin for their encouragement and helpful discussions. Finally, I thank heartily my family and friends for their constant support. I am particularly grateful to my father, who passed away at the very beginning of my doctorate. He was my first teacher and, in his own unconventional fashion, the first example of a researcher I had. To him this thesis is dedicated. viii CONTENTS Abstract In recent years, Hilbert’s Programme has been resumed withintheframework of constructive mathematics. This undertaking has already shown its feasability for a considerable part of commutative algebra. In particular, point-free meth- ods have been playing a primary role, emerging as the appropriate language for expressing the interplay between real and ideal in mathematics. This dissertation is written within this tradition and has Sambin’s notion of formal topology at its core. We start by developing generaltools,inorder to make this notion more immediate for algebraic application. We revise the Zariski spectrum as an inductively generated basic topology, and we analyse the constructive status of the corresponding principles of spatiality and reducibility. Through a series of examples, we show how the principle of spatiality is recurrent in the mathematical practice. The tools developed before are applied to specific problems inconstructive algebra. In particular, we find an elementary characterization of the notion of codimension for ideals of a commutative ring, by means of which a constructive version of Krull’s principal ideal theorem can be stated and proved. We prove a formal version of the projective Eisenbud-Evans-Storch theorem. Finally, guided by the algebraic intuition, we present an application in constructive domain theory, by proving a finite version of Kleene-Kreisel densitytheoremfornon- flat information systems. x CONTENTS Zusammenfassung In den vergangenen Jahren wurde das Hilbertsche Programm im Rahmen der konstruktiven Mathematik wiederaufgenommen. Diese Unternehmung hat sich vor allem in der kommutativen Algebra als praktikabel erwiesen. Insbeson- dere spielen punktfreie Methoden eine wesentliche Rolle: sie haben sich als die angemessene Sprache herausgestellt, um das Zwischenspiel von “real” und “ideal” in der Mathematik auszudrücken. Die vorliegende Dissertation steht in dieser Tradition; zentral ist Sambins Begriffder formalen Topologie. Zunächst entwickeln wir ein allgemeines Instru- mentarium, das geeignet ist, diesen Begriffseinen algebraischen Anwendungen näherzubringen. Sodann arbeiten wir das Zariski-Spektrum in eine induktiv erzeugte “basic topology” um und analysieren den konstruktiven Status der ein- schlägigen Varianten von Spatialität und Reduzibilität. Durch Angabe einer Reihe von Instanzen zeigen wir, wie häufig das Prinzip der Spatialität in der mathematischen Praxis vorkommt. Die eigens entwickelten Werkzeuge werden schließlich auf spezifische Prob- leme aus der konstruktiven Algebra angewandt. Insbesonderegebenwireine elementare Charakterisierung der Kodimension eines Idealsineinemkommu- tativen Ring an, mit der eine konstruktive Fassung des Krullschen Hauptideal- satzes formuliert und bewiesen werden kann. Ferner beweisenwireineformale Fassung des Satzes von Eisenbud-Evans-Storch im projektiven Fall. Geleitet von der algebraischen Intuition stellen wir zuletzt eine Anwendung in der konstruk- tiven Bereichstheorie vor, indem wir eine finite Variante desDichtheitssatzes von Kleene und Kreisel für nicht-flache Informationssystemebeweisen. xii CONTENTS Publications Included in the Thesis This thesis contains material which has already been published or is submitted for publication. The chapter subdivision partially reflectsthefollowinglistof publications: 1. Giovanni Sambin, Davide Rinaldi and Peter Schuster. The Basic Zariski Topology.Submitted. 2. Davide Rinaldi. A constructive notion of codimension.JournalofAlgebra, 383:178-196, 2013. 3. Davide Rinaldi. A formal proof of the projective Eisenbud-Evans-Storch theorem,ArchivderMathematik,99:9–24,2012. 4. Davide Rinaldi and Peter Schuster. A Universal Krull-Lindenbaum The- orem.Submitted. xiv CONTENTS Introduction In the early 1920s, Hilbert collected his philosophical views in a foundational proposal, which became known as Hilbert’s Programme.Therealisationofsuch ambitious programme should have given the ultimate responsetothefounda- tional crisis of mathematics. In Hilbert’s view, there is a privileged part of mathematics, which we could call finitistic or “real”. He essentially identifies this part with elementary number theory, which stands beforelogicinasmuchas it leans on the intuition of “concrete signs”; on the other hand, there is a concep- tual and “ideal” side of mathematics, which has emerged with set theory, formal induction principles and the principle of excluded middle. The only method to provide
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