Grothendieck Chow-motives of Severi-Brauer varieties

For any central simple algebra, the Grothendieck Chow-motive of the corresponding Severi-Brauer variety is decomposed in a direct sum where each summand is a twisted motive of the Severi-Brauer variety corresponding to the underlying division algebra. It leads to decompositions in other theories (for instance, of K-cohomologies) because of the universal property of the Chow-motives. In the second part, it is shown that the Chow-motive of a Severi-Brauer variety corresponding to a division algebra is indecomposable as an object in the category of motives. We fix a basefield F , a central simple algebra D over F , put r = degD and X be the Severi-Brauer variety SB (D) corresponding to D [1]. Moreover, putX = SB (Mn(D)) whereMn(D) is the F -algebra of n×n-matrices over D. In the first part, we decompose the Grothendieck Chow-motive X̃n of the variety X [5] in the direct sum n−1 ⊕ i=0 X̃(ir) of twisted motives of X (1.3.2) (note that in the trivial case D = F it is the well-known decomposition of the motive of the projective space [5]). Hence in any “geometrical” cohomology theory H we have: