Weil cohomology

Weil cohomology
Name	Weil cohomology
Field	Algebraic geometry
Inventor	André Weil
Year	1960s
Related	Étale cohomology, de Rham cohomology, Betti cohomology

Weil cohomology is an axiomatic framework for assigning graded, finite-dimensional vector spaces with Poincaré duality and cycle maps to smooth projective varieties over fields, devised to provide the cohomological foundations needed for the Weil conjectures and subsequent developments in arithmetic geometry. It formalizes properties expected of cohomology theories such as comparison isomorphisms linking Hodge theory, \'etale cohomology, and crystalline cohomology, and underlies constructions in motivic cohomology and the theory of algebraic cycles.

Definition and axioms

A Weil cohomology theory is a contravariant functor from the category of smooth projective varieties over a base field to the category of graded finite-dimensional vector spaces over a field of coefficients, satisfying a list of axioms inspired by classical examples. Key axioms include functoriality with respect to morphisms of varieties, a Künneth formula reflecting product decompositions, Poincaré duality providing nondegenerate pairings, existence of a cycle class map from Chow groups, and a Lefschetz hard and weak theorem ensuring compatibility with hyperplane sections. These axioms were motivated by work of André Weil, refined through contributions of Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne, and are stated to facilitate proofs in the style of the proof of the Riemann hypothesis for varieties over finite fields.

Examples and standard theories

Standard examples of Weil cohomology include Betti cohomology for complex varieties producing Hodge structures, de Rham cohomology over fields of characteristic zero with connections to Gauss–Manin connection, and \'etale cohomology with ℓ-adic coefficients developed by Grothendieck and Michael Artin to handle arithmetic phenomena over finite fields. In characteristic p, crystalline cohomology introduced by Alexander Grothendieck and developed by Pierre Berthelot offers a Weil cohomology for p-adic coefficients in many cases. Other cohomology theories connected to the Weil framework include rigid cohomology by Amnon Berthelot and work by Bernard Dwork, as well as realizations emerging from the theory of mixed motives advanced by Yves André and Vladimir Voevodsky.

Properties and formal consequences

From the Weil axioms follow formal consequences such as the existence of a theory of Chern classes linking vector bundles to cycle classes, compatibility of cycle maps with products and pullbacks, and the Lefschetz trace formula connecting endomorphisms of varieties to traces on cohomology groups. These properties enabled the formulation and proof of the Weil conjectures by combining input from Pierre Deligne on weights with techniques from Grothendieck’s theory of \'etale cohomology and the monodromy methods of Jean-Pierre Serre and Nicholas Katz. The axioms also imply that correspondences act functorially on cohomology, giving rise to the definition of algebraic correspondences and their use in the construction of motivic Galois groups considered by Pierre Deligne and Alexander Beilinson.

Applications in algebraic geometry and number theory

Weil cohomology theories are instrumental in proving results about zeta functions of varieties over finite fields as in the Weil conjectures, in formulating the Tate conjecture about algebraic cycles, and in studying the Hasse–Weil L-function of varieties and motives. They provide the cohomological language used in the proof of the Riemann hypothesis for varieties over finite fields by Pierre Deligne and underpin approaches to the Birch and Swinnerton-Dyer conjecture via realization functors linking motives to classical invariants. Applications extend to the study of Shimura varieties influenced by Gérard Laumon and Michael Harris, the arithmetic of elliptic curves exemplified in work by John Tate and Jean-Pierre Serre, and to the analysis of Galois representations central to the Langlands program pursued by Robert Langlands and Pierre Deligne.

Construction techniques and realizations

Constructions of Weil cohomology theories use a variety of techniques: analytic comparison theorems connect Betti cohomology with de Rham cohomology via the Riemann–Hilbert correspondence developed by Kashiwara and Masaki Kashiwara’s collaborators; ℓ-adic realizations are built from inverse systems of finite étale covers using ideas of Grothendieck and Michael Artin; crystalline methods employ divided power structures and Witt vectors introduced by Ernst Witt and elaborated by Berthelot. Realization functors from hypothetical categories of mixed motives to concrete Weil cohomologies were advocated by Grothendieck and pursued by Uwe Jannsen, Alexander Beilinson, and Vladimir Voevodsky to give a unifying structure relating K-theory and regulators studied by Stephen Bloch and Alexander Goncharov.

Historical development and influence

The concept of an axiomatic Weil cohomology emerged from efforts by André Weil to understand zeta functions of varieties and from the foundational work of Alexander Grothendieck introducing étale topology during the 1960s. Subsequent milestones include the development of ℓ-adic cohomology by Grothendieck and collaborators, the formulation of the Weil conjectures, and the complete proof of the last conjecture by Pierre Deligne in the 1970s, which cemented the role of Weil cohomology in modern arithmetic geometry. The framework influenced the creation of the theory of motives, the formulation of deep conjectures by John Tate and Alexander Beilinson, and ongoing research by mathematicians such as Vladimir Voevodsky, Jacob Lurie, and Peter Scholze exploring new cohomological tools like prismatic cohomology and perfectoid methods developed by Peter Scholze that extend the landscape of Weil-type theories.

Category:Algebraic geometry