Algebraic K-theory and higher Chow groups of linear varieties

The main focus in this pager is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-l(v) higher Chow groups to mod-l(v) etale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p greater than or equal to 0 with l double dagger p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-l(v) motivic cohomology and mod-l(v) algebraic K-theory of projective nonsingular linear varieties to bt: free Z/l(v)-modules.