crystalline cohomology

crystalline cohomology
Name	Crystalline cohomology
Discipline	Algebraic geometry
Introduced	1960s–1970s
Major figures	Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, Nicholas Katz, Luc Illusie

crystalline cohomology is a p-adic cohomology theory developed to study algebraic varieties over fields of characteristic p and to lift their invariants to characteristic 0. It was introduced to address limitations of earlier approaches to the Weil conjectures and to give a robust cohomological tool compatible with deformation theory, Hodge theory, and the theory of schemes. The theory connects to foundational work by Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, Grothendieck–Messing, and Nicholas Katz and plays a central role in modern studies influenced by Luc Illusie and others.

Overview and Historical Context

Crystalline cohomology arose in the context of attempts to establish p-adic analogues of étale and de Rham theories after breakthroughs by Alexander Grothendieck and Jean-Pierre Serre on cohomological methods, alongside applications inspired by the Weil conjectures and the work of Pierre Deligne. Early motivators included lifting problems considered by Alexander Grothendieck in the Séminaire de Géométrie Algébrique and deformation insights related to Grothendieck–Messing theory; subsequent formalization and expansion involved contributions from Luc Illusie, Nicholas Katz, Pierre Berthelot, and contemporaries linked to the development of crystalline methods used in the proofs of results related to Hodge theory and arithmetic applications connected to Shimura varieties and the Langlands program.

Definitions and Basic Constructions

The basic definition uses a scheme X over a perfect field of characteristic p and a choice of a p-adic base such as the ring of Witt vectors W(k), a construction attributed to the work of Emil Artin and later formalized in contexts influenced by Évariste Galois-era algebraic structure. One constructs cohomology groups H^i_cris(X/W) as sheaf cohomology on a site built to encode infinitesimal thickenings; the approach parallels frameworks introduced by Alexander Grothendieck for the étale and fppf topologies and leverages formalism later employed by Pierre Deligne in mixed characteristic contexts. Fundamental operations—cup product, functoriality, and Frobenius action—are defined to interact with structures familiar from the work of Jean-Pierre Serre on local fields and John Tate on p-adic representations.

Crystalline Site and Crystals

The crystalline site is a category of thickenings of the scheme X equipped with divided power structures, an idea evolved from Grothendieck's crystalline intuition and influenced by notions appearing in Alexander Grothendieck's seminars. Objects are PD-thickenings and morphisms respect divided power structures analogous to constructions examined in the works of Grothendieck–Messing and A. Grothendieck collaborators. A crystal is a sheaf on this site satisfying rigidity under pullback, a concept that extends the notion of quasi-coherent sheaves in the Zariski setting and resonates with deformation theoretic viewpoints explored by Grothendieck and Pierre Deligne. The category of crystals yields modules with Frobenius and connection structures comparable to those studied by Nicholas Katz in his investigations of p-adic differential equations and in settings related to Dieudonné theory as developed by Jean Dieudonné and Alexander Grothendieck.

Comparison Theorems and Relations to Other Cohomologies

Crystalline cohomology admits comparison isomorphisms with de Rham cohomology for smooth liftable varieties, echoing compatibilities reminiscent of results by Pierre Deligne linking Hodge structures and algebraic de Rham theory. It relates to étale cohomology via p-adic Hodge theory results influenced by work of Jean-Marc Fontaine, Gerd Faltings, and Christophe Breuil, and it interfaces with rigid cohomology developed by Bernard Berthelot and furthered in contexts tied to Nicholas Katz. Comparison theorems situate crystalline groups within the tapestry connecting Hodge theory, the Langlands program, and arithmetic duality theorems explored by Alexander Grothendieck and John Tate.

Properties and Examples

For a smooth proper variety X over a perfect field of characteristic p, H^i_cris(X/W) is a finitely generated W(k)-module equipped with a semilinear Frobenius endomorphism; these features parallel structures appearing in the study of abelian varieties by Jean-Pierre Serre and in the classification of p-divisible groups in the work of Grothendieck–Messing and Jean Dieudonné. In concrete examples, crystalline cohomology computes classical invariants: for smooth projective curves it recovers information connected to Jacobians studied by André Weil and for K3 surfaces it yields lattices relevant to moduli considered by Igor Shafarevich and Pavel Shafarevich-related developments. Pathologies in singular or non-proper cases motivate variants such as rigid cohomology of Bernard Berthelot.

Applications in Arithmetic Geometry

Crystalline cohomology underpins many arithmetic applications: it contributes to proofs of instances of the Weil conjectures via p-adic methods, informs study of Galois representations akin to frameworks by Jean-Marc Fontaine and Gérard Laumon, and supports investigations into reductions of Shimura varieties and moduli of abelian varieties as in work related to Pierre Deligne and Nicholas Katz. It plays a role in formulating and proving p-adic comparison theorems used in modern approaches to the Langlands program and intersects with deformation theory central to the study of arithmetic schemes in research threads associated with Alexander Grothendieck, Gerd Faltings, and Pierre Berthelot.

Category:Algebraic geometry