Affine quadrics and the Picard group of the motivic category

Abstract: In this article we study the subgroup of the Picard group of Voevodsky's category of geometric motives generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann, but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the Criterion of motivic equivalence of projective quadrics.