Motivic decomposability of generalized Severi-Brauer varieties (Q710845)

Let \(F\) be a field, let \(p\) be a positive prime number and let \(D\) be a central division \(F\)-algebra of degree \(p^n\), with \(n\geq 1\). According to \textit{N. A. Karpenko} [St. Petersbg. Math. J. 7, No. 4, 649--661 (1996); translation from Algebra Anal. 7, No. 4, 196--213 (1995; Zbl 0866.14006); J. Reine Angew. Math. 677, 179--198 (2013; Zbl 1267.14009)], the Chow motive \(M(SB(p^m, D))\) with coefficients in \(\mathbb{F}_p\) of the generalized Severi--Brauer variety \(SB(p^m, D)\) of right ideals in \(D\) of reduced dimension \(p^m\) for \(m=0, 1,\dots, n-1\) is indecomposable for any \(p\) and \(m=0\) and for \(p=2\), \(m=1\). In this note decomposability of \(M(SB(p^m, D))\) in all the other cases is proved.