  In this article, we study how the Grothendieck group of coherent sheaves can
be used to describe D-branes. We show how global bound state construction in
topological $K$-theory can be adapted to our context, showing that D-branes
wrapping a subvariety are holomorphically classified by a relative $K$-group.
By taking the duality between the relative $K$-groups and the $K$-homologies,
we show that D-brane charge of type IIB superstrings is properly classified by
the $K$-homology.
