Motivic cohomology and infinitesimal group schemes

Publication date: 14 December 2020

Abstract: For

k

a perfect field of characteristic

p > 0

and

G / k

a split reductive group with

p

a non-torsion prime for

G,

we compute the mod

p

motivic cohomology of the geometric classifying space

B G_{(r)}

, where

G_{(r)}

is the

r

th Frobenius kernel of

G .

Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme

G / k

of finite type, we define a cycle class map from the mod

p

motivic cohomology of the classifying space

B G

to the mod

p

'etale motivic cohomology of the classifying stack

m a t h c a l B G .

This also gives a cycle class map into the Hodge cohomology of

m a t h c a l B G .

We study the cycle class map for some examples, including Frobenius kernels.