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Strongly A^1-invariant sheaves (after F. Morel)

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arxiv 2406.11526 v1 pith:LHCDTUZV submitted 2024-06-17 math.AG math.ATmath.KT

Tom Bachmann This is my paper · ORCID
classification math.AGmath.ATmath.KT
keywords a1-invariantgroupsnisnevichsheavesstronglyabeliancohomologyfield
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Strongly (respectively strictly) A1-invariant sheaves are foundational for motivic homotopy theory over fields. They are sheaves of (abelian) groups on the Nisnevich site of smooth varieties over a field k, with the property that their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich cohomology groups) are invariant under replacing a variety X by the affine line over X. A celebrated theorem of Fabien Morel states that if the base field k is perfect, then any strongly A1-invariant sheaf of abelian groups is automatically strictly A1-invariant. The aim of these lecture notes is twofold: (1) provide a complete proof if this result, and (2) outline some of its applications.

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  1. On The Morel Structure Conjecture

    math.KT 2026-07 accept novelty 6.0 of 10

    Witt K-theory is effective over any perfect field of characteristic 2, finishing the proof of the Morel structure conjecture.

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