On motives of algebraic groups associated to a divsion algebra

In this paper we consider motives and motivic cohomology of algebraic groups GL1(A) and SL1(A) for a central simple algebra A of prime degree n over a field F . Motivation to study these groups comes from the problems arising in algebraic K-theory, in particular non-triviality of SK1(A) [16], [9]. It is proved by Biglari [4] that motives of split reductive algebraic groups such as GLn(F ) and SLn(F ) are Tate motives. Furthermore, using higher Chern classes in motivic cohomology constructed by Pushin [11] one can write down explicit direct sum decompositions for the motives of these two groups with integral coefficients. Non-split algebraic groups such as GL1(A) and SL1(A) are more intricate. We note however that all the complications lie in n-torsion effects (n is the degree of A): the situation becomes trivial if we make n invertible. For GL1(A) we follow an idea of Suslin to split the motive M(GL1(A)) into two pieces: the first piece is a very simple Tate motive, whereas the second piece is a twisted Tate motive M over X , where X is the Voevodsky-Chech simplicial scheme associated to the Severi-Brauer variety SB(A) (Theorem 4.6). We investigate the structure of the latter motive M using the twisted slice filtration, and compute the second differential in the arising spectral sequence (Theorem 4.7). Using the spectral sequence we compute some lower weight motivic cohomology groups of GL1(A) (Corollary 4.9). We also consider the case of degree 2 algebra where one can write explicit decomposition for M(GL1(A)) (Proposition 4.4). For SL1(A) we only consider algebras A of degree either 2 or 3. In both of these cases SL1(A) admits an explicit smooth compactification XA given as a hyperplane section of a generalized Severi-Brauer variety (Proposition 5.1). In the degree 2 case XA is a 3-dimensional quadric, whose motive can be computed explicitly (Proposition 5.4). In the degree 3 case XA is a hyperplane section of the twisted form of the Grassmannian Gr(3, 6). The motives of such hyperplane sections with coefficients in Z/3 were already considered in [13]. We give a slightly different proof of the decomposition we need with integral (or more precisely, with Z[ 12 ]) coefficients (Proposition 5.8), using a version of Rost nilpotence theorem that we prove along the way (Proposition 3.9). The author expresses his gratitute to A.Suslin for numerous conversations on matters discussed in the paper. We fix the notation we need. • F is a perfect field. We assume char(F ) 6= 2 whenever we speak of quaternion algebars and char(F ) = 0 in Section 5.3. • A is a central simple algebra over F of degree n. We assume n to be prime in Section 4.