A-upper motives

2024/03/16 by Charles De Clercq, De Clercq, Charles, Nikita A. Karpenko +3
Computer Science · Psychology · #Algebraic Geometry (math.AG) #Cognitive Science and Mapping #FOS: Mathematics #Psychological Testing and Assessment

paper · pdf · doi:10.48550/arxiv.2403.11030

openalex publication_date 2024/03/16 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28

Abstract

Let E/F be a finite field extension with every intermediate field being Galois over F. Given a reductive group G over F such that G_E is of inner type, we define its A-upper motives. These motives are indecomposable and naturally related with the indecomposable summands in the motives of spectra of intermediate fields in E/F. We show that motives of projective homogeneous varieties under G are isomorphic to direct sums of Tate shifts of A-upper motives. Based on that, we get a classification of the motives of projective homogeneous varieties under absolutely simple groups of type not ⁶ D₄, by means of their higher Artin-Tate traces. We also show how Tits indexes over suitable field extensions determine motivic equivalence classes for these algebraic groups.

Related

• Higher Tate traces of Chow motives
• Upper motives of outer algebraic groups
• Upper motives of products of projective linear groups
• Motivic equivalence of algebraic groups
• Motivic decompositions of projective homogeneous varieties and change of coefficients