Motivic spheres and the image of the Suslin–Hurewicz map

We show that an old conjecture of A. A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable A 1 -homotopy sheaves of the general linear group. Using this identification, we establish Suslin’s...

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Bibliographic Details
Published in:Inventiones mathematicae 2020-01, Vol.219 (1), p.39-73
Main Authors: Asok, Aravind, Fasel, Jean, Williams, Ben
Format: Article
Language:English
Subjects:
Algebraic Geometry
Algebraic Topology
Commutative Algebra
Mathematics
Mathematics and Statistics
Sheaves
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Summary:We show that an old conjecture of A. A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable A 1 -homotopy sheaves of the general linear group. Using this identification, we establish Suslin’s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3. We do this by linking the relevant unstable A 1 -homotopy sheaf of the general linear group to the stable A 1 -homotopy of motivic spheres.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-019-00907-z