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Motivic cohomology and infinitesimal group schemes

Eric Primozic

Vol. 7 (2022), No. 3, 441–466

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.

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