Lifting of coefficients for Chow motives of quadrics

TL;DR

The work proves that the natural functor from the category of Chow motives of smooth projective quadrics with integral coefficients to the modulo $2$ coefficient category induces a bijection on isomorphism classes of objects. It combines Rost’s nilpotence for quadrics with a detailed analysis of low-rank Chow groups under splitting fields and a Galois-action framework to lift modulo $2$ data to integral coefficients. The surjectivity and injectivity arguments hinge on constructing and lifting degree-0 projectors and using discriminant and indecomposability considerations to manage special cases. By providing a self-contained, direct proof, the paper recovers Vishik’s integral description from the modulo $2$ theory (EKM) and clarifies the relationship between integral and modulo $2$ Chow motives for quadrics.

Abstract

We prove that the natural functor from the category of Chow motives of smooth projective quadrics with integral coefficients to the category with coefficients modulo 2 induces a bijection on the isomorphism classes of objects.

Theorems & Definitions (19)

• Theorem 1
• Proposition 2
• proof
• Theorem 3: Rost's nilpotence for quadrics
• Corollary 4: EKM Corollary 92.5
• Corollary 5: EKM Corollary 92.7
• Proposition 6
• proof
• Proposition 7
• proof