Motivic decompositions of projective homogeneous varieties and change of coefficients

Charles de Clercq (1)

1 UPMC - Université Pierre et Marie Curie - Paris 6

We prove that under some assumptions on an algebraic group $G$, indecomposable direct summands of the motive of a projective $G$-homogeneous variety with coefficients in $\mathbb{F}_p$ remain indecomposable if the ring of coefficients is any field of characteristic $p$. In particular for any projective $G$-homogeneous variety $X$, the decomposition of the motive of $X$ in a direct sum of indecomposable motives with coefficients in any finite field of characteristic $p$ corresponds to the decomposition of the motive of $X$ with coefficients in $\mathbb{F}_p$. We also construct a counterexample to this result in the case where $G$ is arbitrary.

Mots clés: Projective homogeneous varieties; Grothendieck Chow motives; Motivic decomposition

Domaines: Géométrie algébrique [math.AG]

Charles de Clercq. Motivic decompositions of projective homogeneous varieties and change of coefficients. 2010. ⟨hal-00475569v2⟩