On the Invertibility of Motives of Affine Quadrics

TL;DR

This paper proves the invertibility of motives of smooth affine quadrics in a specific motivic category and establishes a motivic version of conjectures related to products of affine Pfister quadrics, using a new conservative functor.

Contribution

It introduces a novel conservative functor on the category of motives and applies it to prove invertibility and conjecture results for affine quadrics.

Findings

Reduced motives of smooth affine quadrics are invertible in DM(k, ZZ[1/e])

A motivic version of Po Hu's conjectures on affine Pfister quadrics is established

The new conservative functor is key to these results

Abstract

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, ZZ[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) DM(k, ZZ[1/e]), the construction of which constitutes the main part of this work.