motivic cohomology

motivic cohomology
Name	motivic cohomology
Field	Algebraic geometry; Algebraic K-theory
Introduced	1990s
Contributors	Vladimir Voevodsky; Andrei Suslin; Spencer Bloch; Alexander Beilinson; Maxim Kontsevich

motivic cohomology

Motivic cohomology is a homological invariant arising in algebraic geometry that connects Algebraic K-theory, Étale cohomology, and regulators from Arakelov geometry to classical invariants studied by Grothendieck and contemporaries. Developed through work of Vladimir Voevodsky, Spencer Bloch, Andrei Suslin, and Alexander Beilinson, it underlies major results such as Voevodsky's proof of the Milnor conjecture and later progress on the Bloch–Kato conjecture and relations to the Beilinson conjectures. The theory is formulated using techniques inspired by the stable homotopy category and derived categories introduced by figures like Daniel Quillen and Jean-Louis Verdier.

Introduction

Motivic cohomology emerged from efforts by Bloch and Beilinson to produce a cohomology theory for algebraic varieties compatible with regulators studied by Ramanujan-era arithmetic (via Deligne cohomology) and conjectures proposed by Beilinson and Bloch; later formalization was achieved by Voevodsky with inputs from Suslin. It sits in a landscape with classical tools such as Hodge theory, de Rham cohomology, crystalline cohomology, and l-adic cohomology, and interacts with structures studied by Deligne, Grothendieck, Kontsevich, and Fulton.

Definitions and constructions

Several models define motivic cohomology. One approach uses the triangulated motivic stable homotopy category constructed by Voevodsky building on concepts from stable homotopy theory and the works of Adams, Boardman, and May. Another model is Bloch's higher Chow groups introduced by Bloch and related to cycle complexes studied by Suslin and Levine. The Beilinson regulator maps from K-theory developed by Quillen to real cohomology are mediated by motivic cohomology, following foundational ideas of Beilinson and Gillet and Soulé. Constructions employ simplicial sheaves on the Nisnevich topology and A^1-homotopy invariance echoing techniques of Morel and Voevodsky.

Basic properties and functoriality

Motivic cohomology satisfies localization, Mayer–Vietoris-type sequences, and A^1-homotopy invariance proved by Voevodsky and refined by Morel. It enjoys contravariant functoriality for morphisms of smooth varieties as in Grothendieck-style cohomology theories and covariant pushforwards compatible with Gysin maps studied by Fulton and Panin. There are long exact sequences linking motivic cohomology to higher Chow groups of pairs, echoing excision properties developed by Quillen and formalized via triangulated categories following Verdier. Étale sheafification yields comparison maps to étale cohomology linked to the work of Deligne and Gabber.

Examples and computations

Computations include classical cases: for fields, motivic cohomology groups recover Milnor K-theory by results of Milnor and proof of the Milnor conjecture by Voevodsky, with further generalizations addressed in the Bloch–Kato conjecture resolved by Voevodsky and Rost. For number fields, regulators connecting motivic cohomology to zeta values and the Beilinson conjectures were studied by Beilinson, Bloch, and Deligne. Calculations on projective spaces use projective bundle formula analogues found in works by Grothendieck and Quillen, and explicit descriptions for curves and surfaces were given by Bloch, Suslin, and Levine. Computational tools exploit spectral sequences such as the motivic-to-Künneth and Atiyah–Hirzebruch-type spectral sequences inspired by Atiyah and Hirzebruch, and methods of Voevodsky for slice filtrations akin to techniques of Sullivan.

Relation to other theories

Motivic cohomology interfaces with algebraic K-theory through Chern class maps inspired by Chern and Grothendieck–Riemann–Roch, and with Hodge theory via regulator maps to Deligne cohomology explored by Beilinson and Scholl. It links to l-adic cohomology used in proofs by Deligne of the Weil conjectures, and to crystalline cohomology studied by Berthelot and Illusie. The motivic stable homotopy category draws on concepts from Stable homotopy theory and reflects influences from Adams spectral sequence techniques and ideas of May used in classical homotopy theory.

Applications and conjectures

Applications include proofs and formulations of major conjectures: the Milnor conjecture proven by Voevodsky; the Bloch–Kato conjecture resolved by Voevodsky and Rost; and connections to the Beilinson conjectures on special values of L-functions studied by Beilinson, Bloch, and Deligne. Motivic cohomology contributes to arithmetic geometry problems concerning Shimura varieties and regulators in Iwasawa theory pursued by Iwasawa and Coates. Open conjectures involve relations proposed by Beilinson and Bloch, finer structural predictions by Voevodsky and Morel on slice filtrations, and interactions with categorical frameworks suggested by Kontsevich and Katzarkov.

Category:Algebraic geometry