A-upper motives of reductive groups
Charles De Clercq, Nikita Karpenko, Anne Quéguiner-Mathieu
TL;DR
This work extends motivic theory for projective homogeneous varieties to reductive groups that are $p'$-inner and $p$-consistent. It introduces $A$-upper motives and an exact retraction to Artin motives, enabling a decomposition of motives into Tate shifts of $A$-upper pieces. The authors prove that indecomposable summands are governed by higher Artin-Tate traces and that isomorphism of motives is detected by these traces, culminating in a criterion for motivic equivalence via higher Tits $p$-indexes. The framework unifies inner and non-inner types and provides practical criteria for motivic classifications across field extensions. These results generalize TateTraces and yield a comprehensive picture of how higher isotropy data controls motives of projective homogeneous varieties.
Abstract
Given a prime number $p$, we perform the study of Chow motives and motivic decompositions, with coefficients in $\mathbb{Z}/p\mathbb{Z}$, of projective homogeneous varieties for $p'$-inner $p$-consistent reductive algebraic groups. Assorted with the known case of $p$-inner reductive groups, our results cover all absolutely simple groups of type not $^3\!D_4$ or $^6\!D_4$, among other examples. First, we define the A-upper motives of such a reductive group $G$; they are indecomposable motives, naturally related to Artin motives built out of spectra of subextensions of a minimal extension over which $G$ become of inner type. With this in hand, we carry on the qualitative study of motivic decompositions for projective $G$-homogeneous varieties. Providing geometric isomorphism criteria for A-upper motives, we obtain a classification of motives of projective $G$-homogeneous varieties, by means of their higher Artin-Tate traces. We also show that the higher Tits $p$-indexes of the group $G$ determine its motivic equivalence class.