motivic cohomology

motivic cohomology

The

motivic cohomology

is “the

algebraic

version of singular cohomology, … an algebraic homology-like theory built out of the

free abelian group

on the algebraic

subvarieties

of [a

regular

algebro-geometric version of a

manifold

]

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 geometry

. Its inventors expected motivic cohomology to be a total

generalization

of all

homology

theories and Pierre Deligne pointed the way with his absolute

Hodge cycles

.

Since motivic cohomology is a type of

cohomology

theory defined for schemes, in particular,

algebraic varieties

, there are a couple of different ways to define it. The

Zariski topology

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

isomorphisms

K

*

⁢

(

X

)

ℚ

≅

⊕

i

H

i

⁢

(

X

,

ℚ

)

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

K

0

of schemes where the ’cohomology theory’ was the

Chow ring

C

⁢

H

*

⁢

(

X

)

, namely

K

0

⁢

(

X

)

ℚ

≅

⊕

i

C

⁢

H

i

⁢

(

X

)

ℚ

Recall that the ring

C

⁢

H

i

⁢

(

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

filtration

) of the group

K

0

⁢

(

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

K

i

⁢

(

X

)

.

If

Δ

n

denotes the algebraic

n

-simplex given by the single

equation

t

0

+

…

+

t

n

=

1

, then let

z

q

⁢

(

X

,

n

)

denote the group of algebraic cycles on

X

×

Δ

n

of codimension

q

which

intersect

each face of

Δ

n

properly. This

intersection

condition allows one to turn

z

q

⁢

(

X

,

n

)

into a simplicial abelian group in the index

q

, and hence a

chain complex

. The

cohomology groups

of this chain complex (or equivalently, the

homotopy group

of the simplicial abelian group) are denoted

C

⁢

H

q

⁢

(

X

,

n

)

are are called higher Chow groups, and were introduced by Bloch. They provide one of the possible

equivalent

definitions of motivic cohomology, and one has

K

i

⁢

(

X

)

ℚ

≅

⊕

j

C

⁢

H

j

⁢

(

X

,

i

)

ℚ

.

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

ℤ

t

⁢

r

⁢

(

X

)

⁢

(

Y

)

denote the free abelain group of closed integral subschemes of

X

×

Y

whose

support

is finite and

surjective

over a

component

of

X

. Then

ℤ

t

⁢

r

⁢

(

X

)

becomes a

presheaf

, and is actually a sheaf in the Zariski, Nisnevich, and étale topologies. Then

ℤ

t

⁢

r

⁢

(

𝔾

m

∧

q

)

is defined to be

Z

⁢

(

q

)

=

ℤ

t

⁢

r

⁢

(

𝔾

m

×

q

)

mod out the images of

𝔾

m

×

q

-

1

via the

q

embeddings

with one coordinate equal to

1

.

For any presheaf

F

, let

C

*

⁢

(

F

)

denote the chain complex, where in weight

n

we have

C

n

⁢

(

F

)

⁢

(

X

)

=

F

⁢

(

Δ

n

×

X

)

. Then the motivic complex is given by

C

*

⁢

ℤ

t

⁢

r

⁢

(

𝔾

m

∧

q

)

⁢

[

-

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:

H

p

⁢

(

X

,

ℤ

⁢

(

q

)

)

=

ℍ

Z

⁢

a

⁢

r

p

⁢

(

X

,

ℤ

⁢

(

q

)

)

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

.

References

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 mathÃÂ©matiques Institut des hautes ÃÂ©tudes 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

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