Cohomologie étale

Cohomologie étale
Name	Cohomologie étale
Caption	Diagram of étale morphisms and sheaves
Field	Algebraic geometry
Introduced	1960s
Contributors	Grothendieck, Artin, Verdier, Deligne, Milne

Cohomologie étale Cohomologie étale is a cohomology theory for schemes developed in the 1960s to extend topological techniques to algebraic varieties; it was introduced by Alexander Grothendieck and substantially advanced by Michael Artin, Jean-Louis Verdier, Pierre Deligne, and James Milne. It provides tools for relating arithmetic of schemes over fields like finite fields and number fields to invariants arising from sheaf cohomology, enabling proofs of central conjectures in arithmetic geometry such as parts of the Weil conjectures and results used in Fermat's Last Theorem via the Taniyama–Shimura conjecture. The theory interconnects with many institutions and results named for mathematicians and places, including the IHÉS, École Normale Supérieure, Institute for Advanced Study, and conferences where the ideas were developed.

Introduction

The subject originated in correspondence and seminars led by Alexander Grothendieck at the IHÉS and in notes circulated by Jean-Pierre Serre and John Tate, with major expositions in the "Séminaire de Géométrie Algébrique" and later texts by Pierre Deligne and James Milne. Étale cohomology uses the Grothendieck topology on schemes with the étale morphism notion to define sheaves whose cohomology groups capture arithmetic and geometric information about schemes such as those studied in Algebraic geometry, Diophantine geometry, and Arithmetic geometry. Key collaborators and users include Nick Katz, Alexander Beilinson, Vladimir Drinfeld, Maxim Kontsevich, and Gerd Faltings.

Definition and basic properties

Étale cohomology is defined by taking derived functors of the global sections functor on the category of sheaves on the étale site of a scheme, using the vocabulary of sheaves, derived categories, and derived functors developed by Jean-Louis Verdier and Alexander Grothendieck. For a scheme X and an abelian sheaf F on the étale site X_ét, one sets H^i(X_ét,F) using injective resolutions in the sense of Homological algebra, drawing on methods from Category theory and the work of Henri Cartan and Samuel Eilenberg. Fundamental properties include long exact sequences from short exact sequences of sheaves, cohomological dimension bounds proven by Michael Artin, Hochschild–Serre spectral sequences connecting cohomology of extensions named after Gerhard Hochschild and Jean-Pierre Serre, and proper base change and smooth base change theorems formulated by Alexander Grothendieck and proven with methods from Étale topology and Topos theory.

Étale cohomology with coefficients (ℓ-adic, torsion)

Coefficients play a central role: torsion coefficients such as Z/nZ and profinite coefficients lead to the development of ℓ-adic cohomology (for a prime ℓ invertible on the base), extensively used by Pierre Deligne in his proof of the Weil conjectures. ℓ-adic cohomology groups are inverse limits of cohomology with Z/ℓ^nZ coefficients, forming modules over the ℓ-adic integers and vector spaces over the ℓ-adic numbers. This construction interacts with Galois representations of the absolute Galois group of fields studied by John Tate and Serre, and with monodromy groups considered by Nicholas Katz and Pierre Deligne. Torsion phenomena were analyzed by Jean-Pierre Serre and Barry Mazur in arithmetic contexts such as elliptic curve torsion and Modular curve cohomology; related coefficient systems include constructible sheaves and perverse sheaves developed by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne.

Comparison theorems and functoriality

Comparison theorems relate étale cohomology to other cohomology theories: comparison with singular cohomology over Complex numbers is established by the étale-to-analytic comparison due to Artin and Grothendieck, while comparison with de Rham cohomology appears in the form of the de Rham–Witt complex and in p-adic Hodge theory developed by Jean-Marc Fontaine, Gerd Faltings, Kazuya Kato, and James Milne. Functoriality properties—covariance for proper maps, contravariance for arbitrary morphisms, cup products, and trace maps—are formalized using the six operations formalism pioneered by Grothendieck and refined by Pierre Deligne and Jean-Louis Verdier, and are essential in formulations by Alexander Beilinson and Joseph Bernstein.

Key theorems and applications (Weil conjectures, duality)

Étale cohomology was instrumental in the proof of the Weil conjectures: rationality and functional equation statements by Bernhard Dwork and the Riemann hypothesis analogue by Pierre Deligne used ℓ-adic methods, while Alexander Grothendieck provided the framework and reduction steps via the Lefschetz trace formula and the theory of weights. Poincaré duality and Verdier duality in the étale setting were established by Jean-Louis Verdier and applied by Michael Artin and Pierre Deligne in studying duality for schemes and local fields, with consequences for Tate duality studied by John Tate and arithmetic duality theorems proven by Jean-Michel Fontaine and Gerritzen. Applications span proofs and formulations in Diophantine geometry used by Gerd Faltings in his proof of the Mordell conjecture, and inputs to the study of L-functions and Modular forms relevant to the work of Andrew Wiles and Richard Taylor.

Computational techniques and examples

Concrete computations often involve cohomology of curves, surfaces, and abelian varieties: étale cohomology of smooth projective curves over finite fields uses the Weil conjectures machinery developed by Pierre Deligne and calculations by Nicholas Katz; examples include counting points on Fermat-type varieties studied by Bernhard Riemann-inspired techniques and explicit evaluations for Elliptic curve families used by Barry Mazur and John Tate. Computational tools include spectral sequences (Hochschild–Serre, Leray), the formalism of Cech cohomology adapted by Michael Artin, and methods from crystalline cohomology advanced by Alexander Berthelot and Pierre Berthelot for p-adic situations. Explicit algorithmic approaches appear in computational number theory contexts at institutions such as Max Planck Institute for Mathematics and software projects inspired by work at University of California, Berkeley and École Polytechnique.

Category:Algebraic geometry