# 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.