THE CREATION OF SHEAF THEORY
Creators: name: Fasanelli, Florence Dowdell
Publisher: name: American University
Year: 2023

description: A mathematical theory is often created by combining the resources of several branches of mathematics to solve a problem 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 systematization 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 outstanding 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 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 algebriques coherents," called FAC, (1955). In Chapter I the historical background in several complex variables begins with the work of Riemann, Weierstrass, Mittag-Leffler, 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 Seminaire 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 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, generalized 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.