Introduction to Étale Cohomology

ISBN-13: 9783540571162 / Angielski / Miękka / 1994 / 186 str.

Gunter Tamme; Gunter Tamme; M. Kolster

Opis

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

Spis treści

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) Grothendieck’s 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. Étale Cohomology.- §1. The Étale Site of a Scheme.- (1.1) Étale Morphisms.- (1.2) The Étale Site.- (1.3) The Relation between Étale and Zariski Cohomology.- (1.4) The Functors f* and f*.- (1.5) The Restricted Étale Site.- §2. The Case X= spec(k).- §3. Examples of Étale Sheaves.- (3.1) Representable Sheaves.- (3.2) Étale Sheaves of Ox -Modules.- (3.3) Appendix: The Big Étale Site.- §4. 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 Xét.- §5. Stalks of Étale Sheaves.- §6. Strict Localizations.- (6.1) Henselian Rings and Strictly Local Rings.- (6.2) Strict Localization of a Scheme.- (6.3) Étale Cohomology on Projective Limits of Schemes.- (6.4) The Stalks of Rqf*(F).- §7. The Artin Spectral Sequence.- §8. The Decomposition Theorem. Relative Cohomology.- (8.1) The Decomposition Theorem.- (8.2) The functors j! and i!.- (8.3) Relative Cohomology.- §9. Torsion Sheaves, Locally Constant Sheaves, Constructible Sheaves.- (9.1) Torsion Sheaves.- (9.2) Locally Constant Sheaves.- (9.3) Constructible Sheaves.- §10. Étale 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.- §11. General Theorems in Étale 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.

O autorze

Étale Cohomology is one of the most important methods in modern Algebraic Geometry and Number Theory. It is a large area with a large number of applications. The book gives a quick and easy introduction into Étale Cohomology. It is ample in its arguments but wisely restricted in the choice of material.