Making the motivic group structure on the endomorphisms of the projective line explicit
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916507794014208 |
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| author | Barth, Viktor Balch Hornslien, William Quick, Gereon Wilson, Glen Matthew |
| author_facet | Barth, Viktor Balch Hornslien, William Quick, Gereon Wilson, Glen Matthew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Making the motivic group structure on the endomorphisms of the projective line explicit Barth, Viktor Balch Hornslien, William Quick, Gereon Wilson, Glen Matthew Algebraic Geometry Algebraic Topology K-Theory and Homology 14F42, 55P99, 14F35 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. |
| title | Making the motivic group structure on the endomorphisms of the projective line explicit |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology 14F42, 55P99, 14F35 |
| url | https://arxiv.org/abs/2306.00628 |