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

Localized intersection of currents and the Lefschetz coincidence point theorem