DOCUMENTA MATHEMATICA, Vol. 17 (2012), 151-165

Maksim Zhykhovich

Decompositions of Motives of Generalized Severi-Brauer Varieties

Let $p$ be a positive prime number and $X$ be a Severi-Brauer variety of a central division algebra $D$ of degree $p^n$, with $n\geq 1$. We describe all shifts of the motive of $X$ in the complete motivic decomposition of a variety $Y$, which splits over the function field of $X$ and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties $X(p^m,D)$ of right ideals in $D$ of reduced dimension $p^m$, $m=0,1,ldots,n-1$, except the cases $p=2$, $m=1$ and $m=0$ (for any prime $p$), where motivic indecomposability was proven by Nikita Karpenko.

2010 Mathematics Subject Classification: 14L17; 14C25

Keywords and Phrases: Central simple algebras, generalized Severi-Brauer varieties, Chow groups and motives.

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