Weil cohomology

Weil cohomology. In algebraic geometry, a Weil cohomology theory is a system of cohomology groups for algebraic varieties over a field of characteristic zero, satisfying a specific set of axioms proposed by André Weil. These axioms formalize the properties expected of a "good" cohomology theory that could underpin a proof of the Weil conjectures. The search for and construction of such theories, most notably Étale cohomology and Crystalline cohomology, fundamentally transformed the field, leading to the resolution of the conjectures by Pierre Deligne.

Definition and axioms

A Weil cohomology theory assigns to every smooth projective variety $X$ over a field $k$ (of characteristic zero) a finite-dimensional graded vector space $H^*(X)$ over a coefficient field $K$ of characteristic zero. The theory must satisfy several key axioms. These include Poincaré duality, which relates cohomology groups in complementary dimensions, and the existence of a Künneth formula, describing the cohomology of a product variety. Crucially, it requires a cycle class map from the group of algebraic cycles to the cohomology ring, compatible with intersection products. The hard Lefschetz theorem must also hold, relating cohomology via cup product with the class of a hyperplane section. These structures mirror those of classical cohomology for complex manifolds, as formalized by Hodge theory.

Examples

The archetypal example is Betti cohomology for varieties over the complex numbers, where the coefficient field is $\backslash mathbb\{Q\}$. For varieties over other fields, several theories fulfilling the axioms were constructed. The most influential is Étale cohomology, developed by Alexander Grothendieck, Michael Artin, and Jean-Louis Verdier, with coefficients in $\backslash mathbb\{Q\}\_\backslash ell$ for a prime $\backslash ell$ different from the characteristic of the base field. Another major example is Crystalline cohomology, introduced by Pierre Berthelot and Alexander Grothendieck, which works in positive characteristic and has coefficients in the ring of Witt vectors. Algebraic de Rham cohomology, studied by Grothendieck and Friedrich Hirzebruch, provides another Weil theory for characteristic zero. The existence of these theories demonstrated the profound unity between different geometric contexts.

Properties and consequences

Any Weil cohomology theory yields powerful tools for studying algebraic cycles and the zeta functions of varieties. The axioms directly imply the rationality and functional equation for the zeta function, the first two of the Weil conjectures. The theory also provides a framework for formulating standard conjectures on algebraic cycles, such as those proposed by Grothendieck and Hodge. The Lefschetz trace formula in such a theory connects fixed points of morphisms to traces on cohomology, a cornerstone in proving the Riemann hypothesis for varieties over finite fields. These properties show that the axiomatic framework captures the essential homological features needed for deep arithmetic questions.

Comparison with other cohomology theories

Weil cohomologies are distinguished from more general cohomology theories by their stringent axioms, particularly the requirement of a cycle class map and the hard Lefschetz property. While sheaf cohomology with coherent coefficients is central to Serre's GAGA principles, it does not satisfy Poincaré duality with the correct coefficients. Chow groups provide a theory of cycles but lack the homological properties of a Weil theory. The theory of motives, envisioned by Grothendieck, aims to be a universal Weil cohomology, from which all others like Étale cohomology and Crystalline cohomology are realized via specific realization functors. This comparative landscape underscores the unique role of Weil theories as bridges between geometry, topology, and number theory.

History and development

The concept originated from the Weil conjectures, formulated by André Weil in 1949 based on analogies with the classical Riemann hypothesis and his work on curves over finite fields. Weil suggested that a cohomology theory for varieties over finite fields, akin to simplicial homology for topological spaces, would yield a proof. This challenge was taken up by Alexander Grothendieck and the Bourbaki school. The creation of Étale cohomology in the SGA 4 seminar, and later Crystalline cohomology, provided the necessary frameworks. The final proof of the third conjecture, the Riemann hypothesis, was achieved by Pierre Deligne in 1974 using Étale cohomology and sophisticated L-adic techniques. This saga cemented the central role of cohomological methods in modern arithmetic geometry.

Category:Algebraic geometry Category:Cohomology theories Category:Weil conjectures