Motivic decomposability of generalized Severi–Brauer varieties

Abstract

Let F be an arbitrary field. Let p be a positive prime number and D a central division F-algebra of degree p n , with n ⩾ 1 . We write SB ( p m , D ) for the generalized Severi–Brauer variety of right ideals in D of reduced dimension p m for m = 0 , 1 , … , n − 1 . We note by M ( SB ( p m , D ) ) the Chow motive with coefficients in F p of the variety SB ( p m , D ) . It was proven by Nikita Karpenko that this motive is indecomposable for any prime p and m = 0 and for p = 2 , m = 1 . We prove decomposability of M ( SB ( p m , D ) ) in all the other cases.