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

We construct a group structure on the set of pointed naive\nhomotopy classes of scheme morphisms from the Jouanolou\ndevice to the projective line. The group operation is defined\nvia matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular,\nit is completely independent of the construction of the motivic\nhomotopy 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\nfrom Cazanave’s monoid to this group. Moreover, we show\nthat this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou\ndevice to the projective line. This subgroup is generated by a\nset of morphisms that are very simple to describe.