motivic cohomology

The motivic cohomologyPlanetmathPlanetmath is “the algebraic version of singular cohomology, … an algebraic homology-like theory built out of the free abelian groupMathworldPlanetmath on the algebraic subvarietiesMathworldPlanetmath of [a regularPlanetmathPlanetmathPlanetmath algebro-geometric version of a manifoldMathworldPlanetmath] X, the algebraic cycles on X.” (Levine, 1997) This theory devised by Alexander Grothendieck and Enrico Bombieri to derive a conditional proof of the Weil conjectures employing algebraic geometryMathworldPlanetmathPlanetmath. Its inventors expected motivic cohomology to be a total generalizationPlanetmathPlanetmath of all homologyMathworldPlanetmathPlanetmath theories and Pierre Deligne pointed the way with his absolute Hodge cycles.

Since motivic cohomology is a type of cohomologyMathworldPlanetmathPlanetmath theory defined for schemes, in particular, algebraic varieties, there are a couple of different ways to define it. The Zariski topologyMathworldPlanetmath is so poor from an algebraic topology standpoint, alternative methods are required.

The first definition of motivic cohomolgy was only for rational coefficients. Recall that in topology, one has isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


given by the Chern character. Motivic cohomology is a theory which adapts this to the algebraic setting.

Actually, before this was done in topology, Grothendieck had done this for K0 of schemes where the ’cohomology theory’ was the Chow ring CH*(X), namely


Recall that the ring CHi(X) is the free abelain group of codimension i subvarieties of X modulo rational equivalence.

Motivic cohomology was motivated by these two facts. Namely, weight zero motivic cohomology are the Chow groups, and rationally is the graded pieces (with respect to the gamma filtrationMathworldPlanetmathPlanetmath) of the group K0(X). Motivic cohomology in higher weights then corresponds to Bloch’s higher Chow groups, and rationally coincides with graded pieces of higher algebraic K-theory Ki(X).

If Δn denotes the algebraic n-simplex given by the single equation t0++tn=1, then let zq(X,n) denote the group of algebraic cycles on X×Δn of codimension q which intersect each face of Δn properly. This intersectionMathworldPlanetmath condition allows one to turn zq(X,n) into a simplicial abelian group in the index q, and hence a chain complexMathworldPlanetmath. The cohomology groupsPlanetmathPlanetmath of this chain complex (or equivalently, the homotopy group of the simplicial abelian group) are denoted CHq(X,n) are are called higher Chow groups, and were introduced by Bloch. They provide one of the possible equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definitions of motivic cohomology, and one has


Another definition comes from the work of Suslin and Voevodsky, which is more technical. For a smooth scheme of finite type over a field k, let tr(X)(Y) denote the free abelain group of closed integral subschemes of X×Y whose supportMathworldPlanetmathPlanetmath is finite and surjectivePlanetmathPlanetmath over a component of X. Then tr(X) becomes a presheafMathworldPlanetmathPlanetmathPlanetmath, and is actually a sheaf in the Zariski, Nisnevich, and étale topologies. Then tr(𝔾mq) is defined to be Z(q)=tr(𝔾m×q) mod out the images of 𝔾m×q-1 via the q embeddingsPlanetmathPlanetmath with one coordinate equal to 1.

For any presheaf F, let C*(F) denote the chain complex, where in weight n we have Cn(F)(X)=F(Δn×X). Then the motivic complex is given by C*tr(𝔾mq)[-q], where the [-q] means shift by q. The motivic cohomology groups are then defined to be the hypercohomology of this complexes of sheaves, taken in either the Zariski or Nisnevich topologies:


Suslin has shown that the above two deifnitions of motivic cohomology agree. There are other defintions by Voevodsky, who constructed a triangulated category of motives, and a motivic homotopy category, in which motivic cohomology theory (among other theories) are representable.


  • 1 Marc Levine, “Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky”, Bull. Amer. Math. Soc. 34 (1997): 297
  • 2 Anton Suslin & Viktor Voevodsky, “Bloch-Kato conjecture and motivic cohomology with finite coefficients” NATO ASI Series C Mathematical and Physical Sciences 548 (2000): 117 - 192
  • 3 Viktor Voevodsky, “Motivic cohomology with Z/2-coefficients”. Publications mathématiques Institut des hautes études scientifiques (0073-8301), 98 1 (2003): 59 - 73
Title motivic cohomology
Canonical name MotivicCohomology
Date of creation 2013-03-22 16:43:34
Last modified on 2013-03-22 16:43:34
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Definition
Classification msc 57M07