Motivic decompositions of projective homogeneous varieties and change of coefficients

TL;DR

This paper investigates how motivic decompositions of projective homogeneous varieties behave under change of coefficients, proving indecomposability preservation under certain conditions and providing counterexamples.

Contribution

It establishes conditions under which indecomposable motives remain indecomposable when coefficients change, and constructs counterexamples for arbitrary groups.

Findings

Indecomposable motives remain indecomposable over any field of characteristic p under certain assumptions.

Decomposition of motives with coefficients in finite fields of characteristic p matches that with coefficients in _p.

Counterexamples show the limits of these results for arbitrary algebraic groups.

Abstract

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.