Background Material

Abstract

This preliminary chapter provides some necessary background material and the most important tools to be used throughout the book. In order to make the book accessible to readers who are not experts in either the algebraic or analytic aspects, we have given the minimum of exposition of the various needed tools which are usually spread over several books. The chapter is divided into three parts: algebraic tools, analytical tools and elements of hyperfunctions theory. Under the first heading, we have collected the fundamental notions of commutative algebra that underlie both algebraic geometry and algebraic analysis, the basic notions of sheaf theory, and the foundations of the theory of Gröbner bases. Under the analytical heading, we have assembled the fundamental ideas on the space of distributions. The section on hyperfunctions provides the reader with the fundamental notions about these generalized function An appendix containing the basic definitions of category theory completes the chapter. Because of the purely instrumental nature of this chapter, we have made no pretense of completeness, and the reader might want to skip this material and refer to it only as the need occurs.