  We consider the two main classification methods of D-brane charges via
K-theory, in type II superstring theory with vanishing B-field: the Gysin map
approach and the one based on the Atiyah-Hirzebruch spectral sequence. Then, we
find out an explicit link between these two approaches: the Gysin map provides
a representative element of the equivalence class obtained via the spectral
sequence. We also briefly discuss the case of rational coefficients,
characterized by a complete equivalence between the two classification methods.
