Note on the filtrations of the K-theory

Abstract

Let $X$ be a (colimit of) smooth algebraic variety over a subfield $k$ of $\mathbf C$. Let $K_{alg}^0(X)$ (resp. $K_{top}^0(X(\mathbf C)))$ be the algebraic (resp. topological) $K$-theory of $k$ (resp. complex) vector bundles over $X$ (resp. $X(\mathbf C)))$. When $K_{alg}^0(X) \cong K_{top}^0(X(\mathbf C))$, we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider in the cases $X = BG$ for algebraic group $G$ over algebraically closed fields $k$, and $X = \mathbf G_k/T_k$ the twisted form of flag varieties $G/T$ for non-algebraically closed field $k$.