rigid cohomology
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| rigid cohomology | |
|---|---|
| Name | Rigid cohomology |
| Field | Algebraic geometry; Number theory |
| Introduced | 1990s |
| Introduced by | Pierre Berthelot |
| Related | Crystalline cohomology; Monsky–Washnitzer cohomology; de Rham cohomology |
rigid cohomology is a p-adic cohomology theory for algebraic varieties over finite fields that yields well-behaved cohomological invariants with Frobenius actions. It was developed to provide a Weil cohomology for possibly singular or non-proper varieties, paralleling theories such as de Rham cohomology and étale cohomology while interacting with arithmetic objects like zeta functions and L-functions. Rigid cohomology fits into a web of comparison results connecting to crystalline cohomology, Monsky–Washnitzer cohomology, and p-adic Hodge theoretic frameworks linked to figures and institutions such as Pierre Berthelot, Jean-Pierre Serre, Alexander Grothendieck, and research at the IHÉS.
Introduction
Rigid cohomology is designed to assign finite-dimensional vector spaces over p-adic fields to algebraic varieties over finite fields, carrying linear operators coming from the Frobenius endomorphism and enabling the study of arithmetic invariants such as Weil conjectures-style eigenvalues, Hasse–Weil zeta function factors, and traces connected to counting points over finite extensions. The theory interfaces with works by Pierre Deligne, Jean-Marc Fontaine, and methods developed at institutions like École normale supérieure and CNRS, and it serves as a bridge linking the techniques of Monsky–Washnitzer cohomology used by researchers such as Bernard Dwork to more general geometric contexts treated by people including Luc Illusie.
Historical development and motivation
Motivated by the need to extend p-adic methods beyond smooth affine cases studied by Monsky and Washnitzer, and to supply a Weil cohomology compatible with Grothendieck’s vision from the Séminaire de Géométrie Algébrique, Pierre Berthelot introduced rigid cohomology in the 1990s to generalize crystalline cohomology of smooth proper varieties developed in the 1960s. The development responds to arithmetic problems studied by Emil Artin, Ernst Witt, and later addressed in the context of zeta function computations by Alan Lauder and explorations of p-adic differential equations by Kiran Kedlaya. Rigid cohomology also evolved alongside categorical and topos-theoretic efforts by Jean Giraud and subsequent formalizations at centers such as University of Paris and Harvard University.
Foundations and definitions
Foundationally, rigid cohomology uses rigid analytic geometry building on Tate’s theory pioneered at John Tate’s seminar and later formalized by researchers at the University of Cambridge and Princeton University. One constructs overconvergent de Rham complexes via rigid analytic tubes inside formal schemes linked to models over Witt vectors and uses notions related to admissible opens and Berkovich space ideas popularized by Vladimir Berkovich. The cohomology groups are vector spaces over fields like the fraction field of the Witt ring or suitable p-adic extensions, equipped with Frobenius and monodromy operators that echo structures introduced in the p-adic Hodge theory of Jean-Marc Fontaine and structural approaches from Alexander Grothendieck’s school.
Properties and comparison theorems
Rigid cohomology satisfies the expected finiteness, excision, and long exact sequence properties analogous to those in Évariste Galois-centred cohomological frameworks and enjoys comparison theorems with crystalline cohomology for proper smooth varieties and with Monsky–Washnitzer cohomology for smooth affine varieties. Comparison results were proved building on methods from P. Berthelot’s work and subsequent refinements by researchers at University of California, Berkeley and Institut des Hautes Études Scientifiques, often invoking techniques related to Dwork’s trace formula and the p-adic local monodromy theorem studied by Christophe Breuil and Kazuya Kato. Poincaré duality, Künneth formulae, and functoriality statements hold in suitable contexts, paralleling classical results associated with figures such as Henri Cartan and Jean Leray.
Computations and examples
Concrete computations in rigid cohomology have been carried out for curves, hypersurfaces, and complements of divisors, yielding explicit Frobenius matrices used to compute zeta functions and verify instances of the Weil conjectures. Algorithms by Alan Lauder, work by Kiran Kedlaya on slopes for zeta functions, and implementations in computational packages developed by researchers at University of Sydney and University of Warwick have produced examples for elliptic curves, hyperelliptic curves, and Calabi–Yau hypersurfaces studied in connection with mirror symmetry problems explored at institutions like Princeton University and Caltech. These computations often relate to point-counting applications originally motivated by the cryptographic community involving entities like National Institute of Standards and Technology.
Applications in arithmetic geometry
Rigid cohomology plays a role in proving rationality and functional equations for local zeta functions, analyzing slopes of Frobenius related to Newton polygons studied since the era of André Weil, and contributes to p-adic variants of the Hodge conjecture-inspired problems pursued by researchers at IHÉS and Université Paris-Sud. It informs the study of Galois representations arising from geometry as investigated by Jean-Pierre Serre and Pierre Deligne, and it is a tool in the investigation of L-functions in the tradition of work by Andrew Wiles and contemporaries. Applications extend to deformation problems, comparisons with étale cohomology in the style of Grothendieck–Deligne theory, and to explicit point-counting tasks useful in computational number theory centers such as Centre for Mathematics and its Applications.
Variants and extensions
Variants and extensions include overconvergent F-isocrystals, arithmetic D-module approaches advanced by researchers at Université de Rennes and École Polytechnique, and connections to rigid analytic period rings developed in p-adic Hodge theory by scholars like Jean-Marc Fontaine and Bhargav Bhatt. Further generalizations explore log-structures influenced by work of Kazuya Kato, interactions with Berkovich space techniques, and enhancements using derived algebraic geometry themes emerging from research groups at Massachusetts Institute of Technology and University of Oxford.