Tale Cohomology is one of the most important methods in modern Algebraic Geometry and Number Theory. It has, in the last decades, brought fundamental new insights in arithmetic and algebraic geometric problems with many applications and many important results. The book gives a short and easy introduction into the world of Abelian Categories, Derived Functors, Grothendieck Topologies, Sheaves, General tale Cohomology, and tale Cohomology of Curves.

Inhalt

0. Preliminaries.- §1. Abelian Categories.- §2. Homological Algebra in Abelian Categories.- §3. Inductive Limits.- I. Topologies and Sheaves.- §1. Topologies.- §2. Abelian Presheaves on Topologies.- §3. Abelian,Sheaves on Topologies.- II. Étale Cohomology.- §1. The Étale Site of a Scheme.- §2. The Case X= spec(k).- §3. Examples of Étale Sheaves.- §4. The Theories of Artin-Schreier and of Kummer.- §5. Stalks of Étale Sheaves.- §6. Strict Localizations.- §7. The Artin Spectral Sequence.- §8. The Decomposition Theorem. Relative Cohomology.- §9. Torsion Sheaves, Locally Constant Sheaves, Constructible Sheaves.- §10. Étale Cohomology of Curves.- §11. General Theorems in Étale Cohomology Theory.