etale cohomology

etale cohomology
Name	Étale cohomology
Field	Algebraic geometry
Introduced	1960s
Introduced by	Alexandre Grothendieck, Jean-Pierre Serre
Uses	Arithmetic geometry, number theory, algebraic topology

etale cohomology

Étale cohomology is a cohomology theory for schemes developed to transfer topological and arithmetic information between algebraic varieties and arithmetic objects, introduced to solve problems in Weil conjectures and to provide tools for the study of algebraic cycles, Galois actions, and L-functions in contexts involving Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, John Tate, and contemporaries associated with the Institute des Hautes Études Scientifiques. It furnishes invariants that connect schemes over fields such as Number field, Finite field, and p-adic number fields to representations of absolute Galois groups like Galois group of a field, and it underpins deep results including proofs related to the Weil conjectures, compatibility with Hodge theory, and comparisons with Betti cohomology in the framework developed at institutions like École Normale Supérieure and Cambridge University.

History and motivation

The development of étale cohomology grew from efforts by Alexander Grothendieck and collaborators at the Séminaire de Géométrie Algébrique to resolve the Weil conjectures put forward by André Weil and to provide analogues of classical topological invariants for schemes lacking classical topologies, with major contributions from Jean-Pierre Serre, Pierre Deligne, Michael Artin, and John Tate. Early motivations included relating counting points over Finite field extensions to trace formulas informed by the Lefschetz fixed-point theorem and producing a cohomology theory satisfying Poincaré duality, Künneth formula, and finiteness theorems analogous to those known in Singular homology and De Rham cohomology. Deligne's proof of the last of the Weil conjectures employed étale cohomology together with notions drawn from Perverse sheaf theory and the formalism of the Grothendieck trace formula, developed in the milieu of research at Institut des Hautes Études Scientifiques and discussed in seminars at Harvard University and Institute for Advanced Study.

Étale topology and sites

The étale topology replaces the classical analytic topology by a Grothendieck topology on the category of schemes, introduced by Grothendieck at the Séminaire de Géométrie Algébrique and formalized in his work at the Institut des Hautes Études Scientifiques, building on categorical ideas associated with Alexander Grothendieck and influences from Paul Grothendieck's school; the notion of a site enables one to consider sheaves on a category of étale morphisms and to import sheaf cohomology machinery similar to that used in contexts like Zariski topology and Complex analytic space theory. Key constructions reference the notion of an étale morphism, fiber product, and coverings akin to those studied in Grothendieck topology, and the formalism aligns with earlier sheaf-theoretic frameworks developed at institutions such as Université Paris-Sud and University of California, Berkeley.

Étale sheaves and cohomology groups

Étale sheaves—defined on the étale site of a scheme—include constant torsion sheaves, constructible sheaves, and l-adic sheaves, and their cohomology groups H^i_{ét} carry actions of absolute Galois groups like Galois group of local and global fields, reflecting arithmetic phenomena studied by Emil Artin, Helmut Hasse, and Kurt Hensel. The formalism of derived functors, injective resolutions, and spectral sequences used in deriving these groups draws on categorical foundations advanced by Alexander Grothendieck and homological algebra developed by Jean-Louis Verdier and Henri Cartan; l-adic cohomology yields inverse systems connected to representations of Étale fundamental group introduced by Grothendieck and explored in relation to monodromy by Pierre Deligne and Nicholas Katz.

Comparison theorems and properties

Comparison theorems relate étale cohomology to classical theories: for complex varieties, comparison with Betti cohomology via the étale-to-topological comparison theorem connects to results associated with Élie Cartan and Henri Poincaré; p-adic comparison isomorphisms, crystalline and de Rham comparison theorems relate étale cohomology to Crystalline cohomology and de Rham cohomology in the work of Jean-Marc Fontaine, Gerd Faltings, and Aise Johan de Jong. Fundamental properties such as proper base change, smooth base change, and Poincaré duality were proven by Grothendieck and collaborators, and they underpin arithmetic applications used by researchers at Princeton University, Massachusetts Institute of Technology, and University of Cambridge in the study of zeta functions and trace formulas.

Applications in arithmetic geometry

Étale cohomology is central to proofs and formulations in arithmetic geometry: Deligne's proof of the Weil conjectures used étale cohomology and properties of l-adic sheaves; the theory informs the study of L-series and automorphic correspondences pursued by Robert Langlands in the context of the Langlands program and by Andrew Wiles and collaborators in the proof of Modularity theorem. It is instrumental in the study of rational points on varieties, influencing work by Faltings on the Mordell conjecture and by researchers at Institute for Advanced Study and University of Chicago on the arithmetic of elliptic curves, modular forms, and Galois representations central to modern number theory.

Technical tools and constructions

Technical tools include the étale fundamental group, l-adic sheaves and l-adic cohomology, the formalism of derived categories and perverse sheaves developed by Alexandre Grothendieck's school and formalized by Jean-Louis Verdier and Joseph Bernstein, and the use of spectral sequences, purity theorems, and cycle class maps studied by Spiro Karoubi and Gérard Laumon. Advanced comparisons—such as the Fontaine–Mazur conjectures, p-adic Hodge theory, and the study of vanishing cycles—connect étale cohomology with work by Jean-Marc Fontaine, Gerd Faltings, Kazuya Kato, and Bhargav Bhatt, and these constructions continue to be refined in research hubs like Harvard University, Princeton University, and Mathematical Sciences Research Institute.

Category:Cohomology theories