Introduction to Étale Cohomology

A succinct introduction to etale cohomology. Well-presented and chosen this will be a most welcome addition to the algebraic geometrist's library.

Inhalt

0. Preliminaries.- Abelian Categories.- (1.1) Categories and Functors.- (1.2) Additive Categories.- (1.3) Abelian Categories.- (1.4) Injective Objects.- Homological Algebra in Abelian Categories.- (2.1) 3-Functors.- (2.2) Derived Functors.- (2.3) Spectral Sequences.- Inductive Limits.- (3.1) Limit Functors.- (3.2) Exactness of Inductive Limits.- (3.3) Final Subcategories.- I. Topologies and Sheaves.- Topologies.- (1.1) Preliminaries.- (1.2) Grothendieck's Notion of Topology.- (1.3) Examples.- Abelian Presheaves on Topologies.- (2.1) The Category of Abelian Presheaves.- (2.2) ?ech-Cohomology.- (2.3) The Functors fp and fp.- 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. ale Cohomology.- The ale Site of a Scheme.- (1.1) ale Morphisms.- (1.2) The ale Site.- (1.3) The Relation between ale and Zariski Cohomology.- (1.4) The Functors f and f.- (1.5) The Restricted ale Site.- The Case X= spec(k).- Examples of ale Sheaves.- (3.1) Representable Sheaves.- (3.2) ale Sheaves of Ox -Modules.- (3.3) Appendix: The Big ale Site.- The Theories of Artin-Schreier and of Kummer.- (4.1) The Groups Hq(X,(Ga)x).- (4.2) The Artin-Schreier Sequence.- (4.3) The Groups Hq(X,(Gm)x).- (4.4) The Kummer Sequence.- (4.5) The Sheaf of Divisors on X.- Stalks of ale Sheaves.- Strict Localizations.- (6.1) Henselian Rings and Strictly Local Rings.- (6.2) Strict Localization of a Scheme.- (6.3) ale Cohomology on Projective Limits of Schemes.- (6.4) The Stalks of Rqf*(F).- The Artin Spectral Sequence.- The Decomposition Theorem. Relative Cohomology.- (8.1) The Decomposition Theorem.- (8.2) The functors j! and i!.- (8.3) Relative Cohomology.- Torsion Sheaves, Locally Constant Sheaves, Constructible Sheaves.- (9.1) Torsion Sheaves.- (9.2) Locally Constant Sheaves.- (9.3) Constructible Sheaves.- . ale Cohomology of Curves.- (10.1) Skyscraper Sheaves.- (10.2) The Cohomological Dimension of Algebraic Curves.- (10.3) The Groups Hq(X,(Gm)x) and Hq(X,(?n)x).- (10.4) The Finiteness Theorem for Constructible Sheaves.- . General Theorems in ale Cohomology Theory.- (11.1) The Comparison Theorem with Classical Cohomology.- (11.2) The Cohomological Dimension of Algebraic Schemes.- (11.3) The Base Change Theorem for Proper Morphisms.- (11.4) Finiteness Theorems.