Introduction to Étale Cohomology by Günter Tamme (PDF)

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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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is aimed at readers who want to know etale cohomology–it spends little time on motivation. If you have not already heard of the SGA (Grothendieck’s Séminaire de Géométrie Algébrique) then you probably don’t want to read this. It is in effect an introduction to the SGA, especially SGA 4 (and 4 1/2). It is very helpful as a concise summary giving an overview in many fewer pages than the SGA do.A special virtue is the quick introduction to spectral sequences, and the many examples using them. Specialists will note these are all Grothendieck spectral sequences, relating the derived functors of a composite to the derived functors of the composed functors. So in one sense they are a special case more used in algebraic geometry than in topology perhaps. But a widely useful case.This book does not say a lot about the étale fundamental group of a scheme and in general it does not go far into algebraic geometry. It is a companion to other books that do that.It is a splendid short peek at Grothendieck’s view of cohomology.

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