Making the motivic group structure on the endomorphisms of the projective line explicit

We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular,...

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Bibliographic Details
Published in:arXiv.org 2024-12
Main Authors: Viktor Balch Barth, Hornslien, William, Quick, Gereon, Wilson, Glen Matthew
Format: Article
Language:English
Subjects:
Monoids
Multiplication
Subgroups
Online Access:Get full text
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Summary:We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave's monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are simple to describe.
ISSN:2331-8422