Etale Cohomology and the Weil Conjecture (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)

This book is concerned with one of the most important developments in algebraic geometry during the last decades. In 1949 Andr? Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. He himself proved his conjectures by means of an algebraic theory of Abelian varieties in the one-variable case. In 1960 appeared the first chapter of the "El?ments de G?ometrie Alg?braique" par A. Grothendieck (en collaboration avec J. Dieudonn?). In these "El?ments" Grothendieck evolved a new foundation of algebraic geometry with the declared aim to come to a proof of the Weil conjectures by means of a new algebraic cohomology theory. Deligne succeded in proving the Weil conjectures on the basis of Grothendiecks ideas. The aim of this "Ergebnisbericht" is to develop as self-contained as possible and as short as possible Grothendiecks 1-adic cohomology theory including Delignes monodromy theory and to present his original proof of the Weil conjectures. Preface......Page page000004.djvu Contents......Page page000006.djvu On the History of the Weil Conjectures (by J. A. Dieudonne)......Page page000008.djvu Introduction......Page page000018.djvu I. The Essentials of Etale Cohomology Theory......Page page000022.djvu 1. Etale Ring Extensions......Page page000024.djvu 2. Etale Cohomology of Schemes......Page page000035.djvu 3. Finite Morphisms......Page page000046.djvu 4. Finiteness Conditions on Sheaves and Compatibility of Cohomology with Limits......Page page000056.djvu 5. Calculation of Cohomology of Curves......Page page000070.djvu 6. The Base Change Theorem for Proper Morphisms......Page page000077.djvu 7. The Smooth Base Change Theorem......Page page000084.djvu 8. Cohomology with Compact Support; Applications of the Base Change Theorems......Page page000097.djvu 9. The Cohomological Dimension of Affine Algebraic Schemes......Page page000120.djvu 10. Purity Theorems......Page page000123.djvu 11. Comparison Theorems Between Etale Cohomology and Singular Cohomology......Page page000127.djvu 12. l-adic Sheaves......Page page000135.djvu II. Rationality of Weil ζ-Functions......Page page000149.djvu 1. Poincaré Duality......Page page000150.djvu 2. Cohomology Classes of Algebraic Cycles......Page page000167.djvu 3. The Fixed Point Formula of Grothendieck and Nielsen-Wecken for the Frobenius Homomorphism of Curve......Page page000174.djvu 4. Grothendieck Formula for L-Series of l-adic Sheaves......Page page000186.djvu 1. Lefschetz Pencils......Page page000192.djvu 2. Classification of Nondegenerate Double Points......Page page000198.djvu 3. The Monodromy Formalism......Page page000204.djvu 4. The Picard-Lefschetz Formulas......Page page000215.djvu 5. Computation of the Algebraic Monodromy Using Topological Monodromy......Page page000227.djvu 6. The Behavior of the Monodromy Mapping under Change of Base Ring......Page page000250.djvu 7. The Global Monodromy Theory of Lefschetz Pencils......Page page000262.djvu 1. Formulation of the Weil Conjecture......Page page000272.djvu 2. The Fundamental Estimate......Page page000276.djvu 3. A Rationality Proposition......Page page000281.djvu 4. Proof of Deligne's Theorem......Page page000274.djvu 5. Generalizations......Page page000290.djvu 6. Applications......Page page000292.djvu 7. The Weil Conjecture and the Standard Conjectures on Algebraic Cycles......Page page000297.djvu 1. The Fundamental Group......Page page000299.djvu 2. Derived Categories......Page page000309.djvu 3. Descent......Page page000321.djvu Bibliography......Page page000325.djvu Subject Index......Page page000332.djvu Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec­ tures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as self­ contained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very grateful to the staff of Springer-Verlag for their careful work. Eberhard Freitag, Reinhardt Kiehl ; With An Historical Introduction By J.a. Dieudonné ; [translated From The German Manuscript By Betty S. Waterhouse And William C. Waterhouse]. Includes Index. Bibliography: P. [308]-314.