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

Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory 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 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 M(G; R). We prove that the Krull-Schmidt theorem holds in many cases.