Introduction to Etale Cohomology (Universitext Softcover reprint of the original 1st ed. 1994)

Introduction to Etale Cohomology (Universitext Softcover reprint of the original 1st ed. 1994)

By: Gunter Tamme (author), M. Kolster (translator)Paperback

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Description

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

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Contents

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. Etale Cohomology.- 1. 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.- 2. The Case X= spec(k).- 3. Examples of Etale Sheaves.- (3.1) Representable Sheaves.- (3.2) Etale Sheaves of Ox -Modules.- (3.3) Appendix: The Big Etale 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 Xet.- 5. Stalks of Etale Sheaves.- 6. 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).- 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. 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.- 11. 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.

Product Details

  • publication date: 28/09/1994
  • ISBN13: 9783540571162
  • Format: Paperback
  • Number Of Pages: 195
  • ID: 9783540571162
  • weight: 288
  • ISBN10: 3540571167
  • translations: German
  • edition: Softcover reprint of the original 1st ed. 1994

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