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Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Research paper by V. Chernousov, A. Merkurjev

Indexed on: 01 Sep '06Published on: 01 Sep '06Published in: Transformation Groups



Abstract

Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory \({\mathcal M}(G,R)\) of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that \({\mathcal M}(G,R)\) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in \({\mathcal M}(G,R).\) We prove that the Krull--Schmidt theorem holds in many cases.