On the Invertibility of Motives of Affine Quadrics

Abstract

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives $\mathbf{DM}(k,\mathbb{Z}[1/e])$ (where $k$ is a perfect field of exponential characteristic $e$). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) $\mathbf{DM}(k,\mathbb{Z}[1/e])$, the construction of which constitutes the main part of this work.