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Universl^ Micrdmlms International 300 N INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction. 1. The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity. 2. When an image on the film is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of ’ movement during exposure, or duplicate copy. 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In all cases we have filmed the best available copy. Universl^ MicrdMlms International 300 N. ZEEB RD„ ANN ARBOR. Ml 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8124267 F a s a n e l u , F l o r e n c e D o w d e l l THE CREATION OF SHEAF THEORY The American University PhJD. 1981 University Microfilms Internstionel 300 X . Zeeb Road. Ann Arbor. M I 48106 Copyright 1981 by Fasanelli, Florence Dowdell All Rights Reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE CREATION OF SHEAF THEORY BY Florence Dowdell Fasanelli Submitted to the Faculty of the Department of Mathematics, Statistics and Computer Science and the School of Education of The American University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Education Signatures of Committee: "1 -1 V J J ^ _____ Chair : ^ ;v? ^ Dean of the College^ n r i Member : Member : Member : 1981 The American University Washington, D.C. 20016 5 %'\ THE AMEHICiH ÜÎTIVEBSITY ÊIBRARY Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE CREATION OF SHEAF THEORY BY Florence Dowdell Fasanelli ABSTRACT A mathematical theory is often created by combining the resources of several branches of mathematics to solve a prob­ lem previously inaccessible in one (or more) branches. Establishing a combination as a theory is accomplished by 1 , the discovery of new axioms and theorems; 2, the putting together of new techniques for working between the branches (which however must remain autonomous); and 3, the systemati­ zation of an abstract structure of axioms, theorems and techniques by its extension to a general application. The creation of a new theory is often recognized by a change in the language employed to describe the concepts involved. This dissertation is an analysis of the creation of one such theory in the history of mathematics, recognizing 1953 as the year in which sheaf theory changed from an instrument of specific application used by Cartan to solve a long out­ standing problem (second Cousin problem) to an instrument of general application, extended by Serre's exposition beyond the original use (in analysis to algebra). The historical development of sheaf theory is discussed including the proliferation of its techniques in courses, texts, seminars Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and papers which were conducted, prepared and written by the pioneer mathematicians in this undertaking (1940-1972). The introduction contains the necessary definitions translated from Serre's "Faisceaux algébriques cohérents," called FAC, (1955). In Chap:er I the historical background in several complex variables begins with the work of Riemann, Weierstrass, Mittag-Leffier, Poincare, Appell, and Cousin. Recognition of Cousin's error by Gronwall is discussed, followed by a chronological interpretation of the work of Henri Cartan, Behnke, Thullen, and Oka and an introduction of Ideal by Ruckert, and faisceau by Leray. The solutions of the Cousin problems by Oka and Cartan are analyzed. In Chapter II the concern is the development of the terminology, primarily in the séminaire Cartan from 1948-1963. Abstraction of the theory is considered in Chapter III. In 1950 Leray published his paper on homology and cohomology of Lie algebras which gives applications of sheaves. By 1951 the definition of a sheaf had taken on a topological form (Lazard). In Stein's paper (1951) everything related back to convexity. In May 1952 general properties of coherent sheaves were developed (Serre and Cartan). Meanwhile at Princeton, Kodaira worked with Weyl on the Riemann-Roch theorem, and applications were developed in algebraic geometry using coherent sheaves. In 1953 Serre transferred the theory of sheaves to abstract varieties using the Zariski topology; as a consequence sheaves became a major instrument of general application. By 1954 Serre had developed an exposition of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the purely algebraic theory and gave the first systematic application of cohomological algebra to abstract algebraic geometry. The remainder of Chapter III deals with the work of Grothendieck, who developed the formal analogy between the theory of the cohomology of a space with coefficients in a sheaf and the theory of derived functors of modules, general­ ized by using categories. Thus the theory developed did not need the restriction hypothesis on the spaces which was required in FAC. Building on these ideas and that of adjoint functors, Godement published the first book which gave a complete exposition of sheaf theory (1958). Chapter IV contains a discussion of applications to rings, to categories and to topology. In Chapter V the educational implications of a historical study of theories are related to the creation of sheaf theory. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PREFACE The story of analysis of several complex variables from Bernard Riemann to Henri Cartan-Kiyoshi Oka is an exposition of the manner in which one discipline— analysis— has given rise to another, sheaf theory. The view taken here is that progress in mathematics frequently derives from attempts to deal with problems for which existing techniques are inade­ quate. Original quotations are employed extensively so that the contemporary judgments, when superimposed upon those of the preceding eras, do not obliterate the atmosphere in which the mathematics had developed. The essential story of the creation of sheaf theory takes place in three countries : France, Japan and the United States. The action takes place over 100 years, spanning two World Wars, which play an interesting part in the scenario. Several institutions are important: The séminaire Cartan from its inception in 1948 until its completion 16 years later; Bourbaki; the Institute for Advanced Study at Prince­ ton; the Mathematisches Institut at Munster. There mathema­ ticians were able to work in a productive atmosphere. The National Science Foundation made grants of at least $250,000 for papers discussed here. The Wars separated the Japanese from the West, yet during the Wars all countries produced significant mathematics even when the mathematicians were in confinement. As the IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. course of the solutions was developing new ideas came from unexpected sources outside the speciality and each one of these will be treated here with a briefer and less detailed history. In 1950 Jean Leray and Henri Cartan introduced the concept of a sheaf, using specific language to define and formalize ideas that had been developing since the early 1930s.. Sheaves play an important role in many branches of mathematics in 1981. This unifying concept has made it possible to state a problem in one area of mathematics but to solve it in another. Insights can be gained into the way mathematical ideas are promulgated, and the effect mathema­ ticians have on one another, by tracing the brief history of the development of sheaf theory. I hope to have given a flavor of the times, to have shown interchanges between math­ ematicians, to have noted the mathematics of the several disciplines and to have shown how what was originally a faisceau finally became a sheaf. It was in a seminar in Mary Gray's home that I first learned about sheaves. I would like to thank her for that introduction, her advice in my graduate studies, her patience, kindness and understanding. Judy Sunley and Richard Holzsager generously helped me to change the emphasis of my study. Ronald Maggiore and Daniel Antonoplos have welcomed me into mathematics education. For continuous encouragement, I want to thank Margaret and Peter Matelski, Antonia and James Fasanelli. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS PREFACE ...............................................ii INTRODUCTION ........................................
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