  We introduce the notion of a Thom class of a current and define the localized
intersection of currents. In particular we consider the situation where we have
a smooth map of manifolds and study localized intersections of the source
manifold and currents on the target manifold. We then obtain a residue theorem
on the source manifold and give explicit formulas for the residues in some
cases. These are applied to the problem of coincidence points of two maps. We
define the global and local coincidence homology classes and indices. A
representation of the Thom class of the graph as a Cech-de~Rham cocycle
immediately gives us an explicit expression of the index at an isolated
coincidence point, which in turn gives explicit coincidence classes in some
non-isolated components. Combining these, we have a general coincidence point
theorem including the one by S. Lefschetz.
