Affine quadrics and the Picard group of the motivic category

TL;DR

This paper investigates the subgroup of the Picard group in Voevodsky's geometric motives generated by affine quadrics, providing new descriptions and relations using functors and indecomposable summands.

Contribution

It introduces a novel description of the subgroup generated by affine quadrics in the Picard group and extends the criterion for motivic equivalence of projective quadrics.

Findings

The subgroup can be described via indecomposable summands of projective quadrics.

All relations among reduced motives of affine quadrics are characterized.

The motivic equivalence criterion for projective quadrics is extended.

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.