de Rham–Witt complex

de Rham–Witt complex
Name	de Rham–Witt complex
Field	Algebraic geometry
Introduced	1970s
Authors	Pierre Deligne, Luc Illusie
Related	Crystalline cohomology, Hodge–Witt cohomology, p-adic Hodge theory

de Rham–Witt complex The de Rham–Witt complex is a tool in algebraic geometry that produces p-typical Witt vector-valued differential forms for schemes in characteristic p, linking arithmetic of Alexander Grothendieck-era cohomology theories with p-adic phenomena studied by Jean-Pierre Serre, John Tate, and Pierre Deligne. It was developed in the 1970s by Pierre Deligne and Luc Illusie to refine comparisons between Grothendieck's crystalline cohomology and classical Hodge theory, influencing work of Kenji Ueno, Jean-Louis Verdier, Mihalis Papanikolas, and later researchers such as Kazuya Kato and Gerd Faltings.

Introduction

The construction arises in the context of schemes over Évariste Galois-named finite fields like finite fields and more general bases with residue characteristic p studied by Alexander Grothendieck's school at IHÉS and Collège de France. It produces a pro-complex of sheaves on the small Zariski or étale site, connecting to theories developed by Jean-Pierre Serre, John Tate, Grothendieck, and later to p-adic Hodge comparisons by Jean-Marc Fontaine and Gerd Faltings. The de Rham–Witt complex carries operations named Frobenius and Verschiebung, reminiscent of structures considered by Ernst Witt and utilized by André Weil in Weil-type settings.

Construction and Definitions

For a scheme X over a perfect field of characteristic p, Illusie built the p-typical Witt vector functor W_n and organized sheaves W_nΩ^i_X with differential d, Frobenius F, and Verschiebung V; these maps echo structures studied by Ernst Witt and formal groups considered by Serge Lang and Michel Demazure. The complex is formed as an inverse system (W_nΩ^•_X)_{n≥1} with transition maps induced by V and projections, paralleling the Witt vector constructions of Teichmüller-type liftings used by Helmut Hasse and André Weil. Deligne and Illusie axiomatized the functorial behavior, enabling comparisons with crystalline complexes developed by Alexander Grothendieck's collaborators at IHÉS and in works of Arthur Ogus.

Properties and Functoriality

W_nΩ^•_X is contravariantly functorial for morphisms of schemes, paralleling formal properties in the work of Grothendieck and Jean-Pierre Serre; F and V satisfy relations F V = V F = p on appropriate degrees, evocative of identities in Ernst Witt's theory. For smooth proper schemes over perfect fields, Illusie proved finiteness and degeneration results used in comparisons by Deligne in the context of Weil-style cohomological arguments and by Kazuya Kato in logarithmic settings. The complexes admit filtrations analogous to those in Hodge theory developed by Phillip Griffiths and link to spectral sequences employed by Jean-Louis Verdier.

Relation to Crystalline and Hodge–Witt Cohomology

The hypercohomology of the de Rham–Witt complex computes Hodge–Witt cohomology, a p-adic analogue of Hodge cohomology studied by W. V. D. Hodge and integrated into arithmetic frameworks by Mazur and Grothendieck. Comparison isomorphisms relate de Rham–Witt hypercohomology to crystalline cohomology introduced by Grothendieck and developed by Pierre Berthelot; these comparisons underpin results of Gerd Faltings and Jean-Marc Fontaine in p-adic Hodge theory. The slope decompositions under Frobenius on crystalline cohomology reflect filtrations on de Rham–Witt cohomology studied in works of Luc Illusie and Kazuya Kato, with applications in the proofs of assertions originally posed in contexts by Alexander Grothendieck and Pierre Deligne.

Examples and Computations

For X = Spec k with k a perfect field of characteristic p, W_nΩ^0_X = W_n(k) reproduces Witt vectors of Ernst Witt and computations reduce to classical Witt theory used by André Weil and Jean-Pierre Serre. For smooth curves and abelian varieties studied by David Mumford and John Tate, Hodge–Witt groups can be computed explicitly using the de Rham–Witt complex, with applications to the study of Jacobians and étale cohomology phenomena investigated by Alexander Grothendieck and Pierre Deligne. Illusie's computations for surfaces and K3 surfaces, central in works by Shigeru Mukai and Phillip Griffiths, reveal relations to supersingular behavior examined by Takeshi Oda and Katsumi Nomizu.

Applications in Arithmetic Geometry

The de Rham–Witt complex is used in the study of zeta functions and L-functions of varieties over finite fields, topics central to André Weil and expanded by Pierre Deligne in his proof of the Weil conjectures. It underlies analyses of crystalline companions to ℓ-adic representations studied by Alexander Grothendieck and by contemporary researchers such as Jean-Marc Fontaine and Mark Kisin. Applications include results on reduction types of abelian varieties studied by Gerd Faltings and Jean-Pierre Serre, and contributions to the understanding of integral models in the work of Michael Harris and Richard Taylor.

Variants and Generalizations

Generalizations include logarithmic de Rham–Witt complexes developed by Kazuya Kato and Luc Illusie for schemes with singularities or normal crossings, echoing techniques from Alexander Grothendieck's log geometry program and later extensions by Pierre Deligne and Jean-Marc Fontaine. Recent developments relate de Rham–Witt-like constructions to prismatic cohomology introduced by Bhargav Bhatt and Peter Scholze, connecting to ideas in the work of Gerd Faltings and Jean-Marc Fontaine on integral p-adic Hodge theory and to frameworks explored by Mark Kisin and Matthew Emerton.

Category:Algebraic geometry