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Prorepresentability of $K^M$-cohomology in weight 3 generalizing a result of Bloch

Eoin Mackall

Vol. 8 (2023), No. 1, 127–139
Abstract

We generalize a result due to Bloch on the prorepresentability of Milnor K-cohomology groups at the identity. In particular, we prove, for X a smooth, proper, and geometrically connected variety defined over an algebraic field extension  k, that the functor

𝒯Xi,3(A) = ker (H i(X A,𝒦3,XAM) H i(X,𝒦 3,XM)),

defined on Artin local k-algebras (A,𝔪A) with A𝔪Ak, is prorepresentable provided that certain Hodge numbers of X vanish.

Keywords
$K\mkern-2mu$-cohomology, prorepresentability
Mathematical Subject Classification
Primary: 19E08
Secondary: 14D15
Milestones
Received: 18 July 2022
Revised: 27 December 2022
Accepted: 19 January 2023
Published: 1 May 2023
Authors
Eoin Mackall
University of Maryland
College Park, MD
United States