Prorepresentability of KM-cohomology in weight 3 generalizing a result of Bloch

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 $k ∕ ℚ$ , that the functor

𝒯_{X}^{i, 3} (A) = \ker (H^{i} (X_{A}, 𝒦_{3, X_{A}}^{M}) \to H^{i} (X, 𝒦_{3, X}^{M})),

defined on Artin local $k$ -algebras $(A, 𝔪_{A})$ with $A ∕ 𝔪_{A} ≅ k$ , is prorepresentable provided that certain Hodge numbers of $X$ vanish.

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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

Vol. 8, No. 1, 2023

Eoin Mackall

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors