étale cohomology

étale cohomology
Name	Étale cohomology
Field	Algebraic geometry
Introduced	1960s
Key people	Alexandre Grothendieck, Jean-Pierre Serre, Pierre Deligne, Michael Artin

étale cohomology is a cohomology theory for schemes developed to transfer topological methods into algebraic geometry, resolving obstacles posed by the failure of classical topology over fields such as finite fields and p-adic fields. It provides invariants compatible with operations of Galois theory, the Weil conjectures, and the theory of L-functions, and it underpins major achievements by Grothendieck, Serre, and Deligne.

History and motivation

The inception traces to work at the IHÉS and seminars in the 1960s where Grothendieck formulated new notions of topology for schemes, inspired by problems posed by Weil in the context of the Weil conjectures and by the need to construct cohomology theories compatible with Galois representations and étale fundamental group techniques introduced by Grothendieck and formalized by Artin and Serre. Early pioneers included Grothendieck, Serre, Artin, Deligne, and collaborators associated with seminars at CNRS institutions and École Normale Supérieure.

Foundations and definitions

Foundational ideas use the language of schemes, categories, and derived functors from homological algebra to define cohomology groups for sheaves on the étale site of a scheme, developed in the context of the Grothendieck topology formalism advocated by Grothendieck and systematized in the seminars at IHÉS. The construction employs the concepts of sheaves, injective resolutions from homological algebra, and derived categories as in the work of Verdier and Grothendieck to produce cohomology functors H^i(X, F) for a scheme X and an étale sheaf F.

Étale sheaves and Grothendieck topologies

Étale sheaves live on the étale site of a scheme and generalize classical sheaves from topology by respecting the Grothendieck topology axioms developed by Grothendieck, with coverings given by families of étale morphisms; this framework was elaborated in the seminars of Grothendieck and in expositions by Serre, Artin, and Deligne. Important classes of sheaves include constant finite ℓ-torsion sheaves related to ℓ-adic cohomology as treated by Deligne and constructible sheaves associated to stratifications used in the work of Beilinson and Bernstein.

Comparison theorems and basic properties

Comparison theorems relate étale cohomology with classical cohomology theories: for complex schemes there is the comparison with singular cohomology proved via the work of Serre and Grothendieck, and for proper smooth schemes there are comparison isomorphisms with de Rham cohomology in the context of Hodge theory and with Betti cohomology as treated by Deligne and Grothendieck. Étale cohomology satisfies finiteness and base change properties formalized by Artin, Serre, and Grothendieck, and it interacts with the étale fundamental group via monodromy representations studied by Grothendieck and Faltings.

Étale cohomology with coefficients and operations

Cohomology with coefficients in torsion sheaves, especially ℓ-adic coefficients for primes ℓ distinct from the residue characteristic, was organized by Deligne and Grothendieck to produce continuous representations of Galois groups and to define Weil cohomology theories compatible with Frobenius actions in the work culminating in proofs of instances of the Weil conjectures by Deligne. Operations such as cup product, pullback, pushforward, proper base change, and Verdier duality are available in derived categories following frameworks by Verdier, Grothendieck, and Artin.

Applications and major results

Étale cohomology underlies Deligne's proof of the last of the Weil conjectures and supports the construction of ℓ-adic representations central to modern arithmetic geometry, influencing the proofs of the Taniyama–Shimura conjecture and results of Wiles and Taylor on modularity via compatibility of Galois actions. It plays a central role in arithmetic geometry developments such as Faltings's theorem proven by Faltings, in the formulation of the Fontaine–Mazur conjecture studied by Serre and Mazur, and in the Langlands program pursued by Langlands, Deligne, and many others.

Technical tools and advanced topics

Advanced machinery includes ℓ-adic sheaves, perverse sheaves as introduced by Beilinson, Bernstein, and Deligne, the theory of weights developed by Deligne for the action of Frobenius, and purity and vanishing cycles techniques due to Grothendieck and Illusie. Further tools involve the étale topos, derived categories as in the work of Verdier, and p-adic Hodge theory connections developed by Fontaine, Faltings, and Skinner.

Category:Algebraic geometry