Note on the filtrations of the K-theory
Let X be a (colimit of) smooth algebraic variety over a subfield of C. Let K alg (X) (resp. K top(X(C))) be the algebraic (resp. topological) K-theory of k (resp. complex) vector bundles over X ( resp. X(C))). When K alg (X) = K top(X(C)), we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider the cases X = BG for an algebraically closed field k, and X = Gk/Tk the twisted form of flag varieties G/T for non-algebraically closed field k.