Equivalences in the derived limits

In this article, we study the derived limits on the category of inverse systems of free abelian groups, and construct a long exact sequence for the derived limits of inverse systems, and then establish a certain kind of cochain equivalences between cochain complexes induced by maps of systems based on inverse systems. We can apply this result to the inverse systems of singular (or simplicial) cohomology groups induced by the direct systems of topological spaces.