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Motivic cohomology and infinitesimal group schemes

Eric Primozic

Vol. 7 (2022), No. 3, 441–466
Abstract

For k a perfect field of characteristic p > 0 and Gk a split reductive group with p a nontorsion prime for G, we compute the mod p motivic cohomology of the geometric classifying space BG(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 an algebraic group Gk, we define a cycle class map from the mod p motivic cohomology of the classifying space BG to the mod p étale motivic cohomology of the classifying stack G. This also gives a cycle class map into the Hodge cohomology of G. We study the cycle class map for some examples, including Frobenius kernels.

Keywords
motivic cohomology, Frobenius, infinitesimal group schemes
Mathematical Subject Classification
Primary: 14F42
Milestones
Received: 14 January 2021
Revised: 26 October 2021
Accepted: 27 April 2022
Published: 19 December 2022
Authors
Eric Primozic
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, AB
Canada