Etale cohomology
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| Etale cohomology | |
|---|---|
| Name | Étale cohomology |
| Field | Algebraic geometry, Number theory |
| Introduced | 1960s |
| Founders | Grothendieck |
| Main contributors | Grothendieck, Artin, Verdier, Deligne |
Etale cohomology is a cohomology theory for schemes developed to transfer topological and arithmetic methods into Algebraic geometry and Number theory. It was introduced to resolve specific problems such as the Weil conjectures and to provide Galois-equivariant invariants connecting schemes over finite and local fields with classical objects from Complex analysis, Topology, and Representation theory. Étale cohomology produces groups with actions of absolute Galois groups, enabling deep results linking Riemann hypothesis analogues, trace formulas, and L-functions in the tradition of Weil and Hasse.
History and motivation
The theory arose in the 1960s when Alexander Grothendieck sought a cohomology theory suitable for varieties over fields like finite fields and p-adic fields; Grothendieck, with collaborators such as Michael Artin and Jean-Louis Verdier, formulated the theory to attack the Weil conjectures, influenced by earlier work of André Weil and Helmut Hasse. Pierre Deligne completed the proof of the last of the Weil conjectures using étale cohomology and tools from Representation theory, drawing on ideas from Hodge theory and the topology of complex algebraic varieties as studied by Oscar Zariski and André Weil. The development interacted with institutions like the Institut des Hautes Études Scientifiques and the École Normale Supérieure, and it reshaped research at places including Harvard University and the Institute for Advanced Study.
Definitions and basic constructions
Étale cohomology is defined using the étale site of a scheme; Grothendieck's approach replaces classical topological sites with the category of étale morphisms and covers such as those studied by Michael Artin. For a scheme X one considers the category of sheaves on the étale site X_{ét} and defines cohomology groups H^i(X_{ét}, F) via derived functors, using injective resolutions in the spirit of Henri Cartan and Jean-Pierre Serre. The notion of étale morphism builds on concepts from Alexander Grothendieck's Éléments de géométrie algébrique and generalizes unramified coverings familiar from Riemann surface theory and the work of Bernhard Riemann and Oscar Zariski. Fundamental constructions include the étale fundamental group π_1^{ét}(X) introduced in Grothendieck's SGA for relating covers to Galois group actions, and cohomological tools like spectral sequences articulated by Jean Leray and Jean-Louis Verdier.
Étale cohomology with coefficients
Étale cohomology is commonly taken with torsion coefficients such as Z/nZ, or with ℓ-adic coefficients developed by Grothendieck and Pierre Deligne to study limits as ℓ-adic towers; these constructions connect to the Tate conjecture framework and to ℓ-adic representations of absolute Galois groups as studied by John Tate and Jean-Pierre Serre. For a prime ℓ different from the residue characteristics, the ℓ-adic cohomology groups H^i(X_{\bar{k}}, Q_ℓ) carry actions of the absolute Galois group Gal(\bar{k}/k), a key input in proofs by Deligne of weights and purity statements motivated by ideas from Alexander Grothendieck and André Weil. Coefficients in constructible sheaves, perverse sheaves shaped by work of Joseph Bernstein and Masaki Kashiwara, permit finer control and link to the decomposition theorem used by Rellich and Beilinson, while finite coefficient systems relate to étale analogues of local systems studied by Hermann Weyl and Élie Cartan.
Key properties and theorems
Étale cohomology satisfies many formal properties paralleling classical topological cohomology: long exact sequences, Mayer–Vietoris principles linked to Jean Leray, and proper base change theorem proved in Grothendieck's SGA with refinements by Michael Artin and Jean-Louis Verdier. The cohomological dimension bounds by Grothendieck and Artin control vanishing analogous to results in Singular cohomology studied by Henri Cartan. Poincaré duality and Verdier duality in the étale context were developed by Verdier and Deligne, enabling traces and Lefschetz fixed-point formulas akin to those of Solomon Lefschetz and John Tate. Other pivotal results include the finiteness theorems of Artin and Grothendieck, purity conjectures addressed by Alexander Grothendieck and later by Deligne and others, and the monodromy theorems connecting to work of Bernard Malgrange and Pierre Deligne.
Comparison theorems and relations to other cohomologies
Comparison theorems relate étale cohomology to classical theories: for complex varieties a comparison with singular cohomology and Hodge theory, building on work by Henri Poincaré and Hodge, shows compatibility between étale ℓ-adic cohomology and Betti cohomology via comparison isomorphisms. For p-adic geometry, comparisons with de Rham cohomology and crystalline cohomology were developed by Jean-Marc Fontaine, Alexander Grothendieck, and Pierre Berthelot, producing Fontaine's period rings and the Hyodo–Kato and Faltings comparison theorems—Faltings having deep connections to the Mordell conjecture proof by Gerd Faltings. These relations link to the Langlands program as envisioned by Robert Langlands and to modularity results involving Andrew Wiles and Richard Taylor.
Applications and arithmetic consequences
Étale cohomology underlies the proof of the Weil conjectures by Pierre Deligne and supports arithmetic applications such as the study of L-functions, local and global Galois representations central to work of John Tate, Jean-Pierre Serre, and Pierre Deligne, and modularity lifting techniques used by Andrew Wiles and Richard Taylor. It plays a role in the formulation and partial progress on the Tate conjecture, the Birch and Swinnerton-Dyer conjecture contextualized by Bryan Birch and Peter Swinnerton-Dyer, and in Iwasawa theory advanced by Kenkichi Iwasawa. Étale methods inform modern research at institutions like the Max Planck Institute for Mathematics and the Clay Mathematics Institute and continue to influence progress on the Langlands program, arithmetic geometry problems studied at the Institute for Advanced Study and the Princeton University mathematics community.