Introduction to Étale Cohomology

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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.

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


0. Preliminaries.- x1. Abelian Categories.- (1.1) Categories and Functors.- (1.2) Additive Categories.- (1.3) Abelian Categories.- (1.4) Injective Objects.- x2. Homological Algebra in Abelian Categories.- (2.1) 3-Functors.- (2.2) Derived Functors.- (2.3) Spectral Sequences.- x3. Inductive Limits.- (3.1) Limit Functors.- (3.2) Exactness of Inductive Limits.- (3.3) Final Subcategories.- I. Topologies and Sheaves.- x1. Topologies.- (1.1) Preliminaries.- (1.2) Grothendieck's Notion of Topology.- (1.3) Examples.- x2. Abelian Presheaves on Topologies.- (2.1) The Category of Abelian Presheaves.- (2.2) ?ech-Cohomology.- (2.3) The Functors fp and fp.- x3. 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. Etale Cohomology.- x1. The Etale Site of a Scheme.- (1.1) Etale Morphisms.- (1.2) The Etale Site.- (1.3) The Relation between Etale and Zariski Cohomology.- (1.4) The Functors f* and f*.- (1.5) The Restricted Etale Site.- x2. The Case X= spec(k).- x3. Examples of Etale Sheaves.- (3.1) Representable Sheaves.- (3.2) Etale Sheaves of Ox -Modules.- (3.3) Appendix: The Big Etale Site.- x4. 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 Xet.- x5. Stalks of Etale Sheaves.- x6. Strict Localizations.- (6.1) Henselian Rings and Strictly Local Rings.- (6.2) Strict Localization of a Scheme.- (6.3) Etale Cohomology on Projective Limits of Schemes.- (6.4) The Stalks of Rqf*(F).- x7. The Artin Spectral Sequence.- x8. The Decomposition Theorem. Relative Cohomology.- (8.1) The Decomposition Theorem.- (8.2) The functors j! and i!.- (8.3) Relative Cohomology.- x9. Torsion Sheaves, Locally Constant Sheaves, Constructible Sheaves.- (9.1) Torsion Sheaves.- (9.2) Locally Constant Sheaves.- (9.3) Constructible Sheaves.- x10. Etale 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.- x11. General Theorems in Etale 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.

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Introduction to Étale Cohomology
Kartonierter Einband
Springer, Berlin
Anzahl Seiten
H12mm x B235mm x T156mm
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