Homology of Analytic Sheaves and Duality Theorems

The homology of analytic sheaves is a natural apparatus in the theory of duality on complex spaces. The corresponding apparatus in algebraic geometry was developed by Grothendieck in the fifties. In complex ana- lytic geometry the apparatus of homology was missing until recently, and in its stead the hypercohomology of complex sheaves (the hyper-Ext func- tors) and the Aleksandrov-Cech homology with coefficients in co- presheaves were used. The homology of analytic sheaves, sheaves of germs of homology and homology groups of analytic sheaves, were intro- duced and studied in the mid-seventies in a number of papers by the author. The main goal of this book is to give a systematic and detailed account of the homology theory of analytic sheaves and some of its applications to duality theory on complex spaces and to the theory of hyperfunctions. In order to read this book one must be acquainted with the foundations of ho- mological algebra and the theory of topological vector spaces. Only the most elementary concepts and results from the theory of functions of sev- eral complex variables are assumed to be known. The information needed about sheaves and complex spaces is recounted briefly at the beginning of the fIrst chapter. v. D. Golovin v CONTENTS Chapter 1. ANALYTIC SHEA YES ...1 1. Prelirriinary Information ...1 2. Injectivity Test...16 3. Local Duality ...24 4. Injective and Global Dimension ...36 5. Properties of Fine Sheaves ...46 Chapter 2. HOMOLOGY THEORY ..." .. 63 1. Sheaves of Germs of Homology...63 ...