Introduction to tale Cohomology [Paperback]

A succinct introduction to etale cohomology. Well-presented and chosen this will be a most welcome addition to the algebraic geometrist's library.?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.0. Preliminaries.- ?1. Abelian Categories.- (1.1) Categories and Functors.- (1.2) Additive Categories.- (1.3) Abelian Categories.- (1.4) Injective Objects.- ?2. Homological Algebra in Abelian Categories.- (2.1) 3-Functors.- (2.2) Derived Functors.- (2.3) Spectral Sequences.- ?3. Inductive Limits.- (3.1) Limit Functors.- (3.2) Exactness of Inductive Limits.- (3.3) Final Subcategories.- I. Topologies and Sheaves.- ?1. Topologies.- (1.1) Preliminaries.- (1.2) Grothendiecks Notion of Topology.- (1.3) Examples.- ?2. Abelian Presheaves on Topologies.- (2.1) The Category of Abelian Presheaves.- (2.2) ?ech-Cohomology.- (2.3) The Functors fp and fp.- ?3. Abelian,Sheaves on Topologies.- (3.1) The Associated Sheaf of a Presheaf.- (3.2) The Category of Abelian Sheaves.- (3.3) Cohomology of Abelian Sheaves.- (3.4) The Spectral Sequences for ?ech Cohomology.- (3.5) Flabby Sheaves.- (3.6) The Functors fS and fs.- (3.7) The Leray Spectral Sequences.- (3.8) Localization.- (3.9) The Comparison Lemma.- (3.10) Noetherian Topologies.- (3.11) Commutation of the Functors Hq(U, ?) with Pseudofiltered Inductive Limits.- II. ?tale Cohomology.- ?1. The ?tale Site of a Scheme.- (1.1) ?tale Morphisms.- (1.2) The ?tale Site.- (1.3) The Relation between ?tale and Zariski Cohomology.- (1.4) The Functors f* and f*.- (1.5) The Restricted ?tale Site.- ?2. The Case X= spec(k).- ?3. Examples of ?tale Sheaves.- (3lsS