  We study the cohomology theory of sheaf complexes for open embeddings of
topological spaces and related subjects. The theory is situated in the
intersection of the general Cech theory and the theory of derived categories.
That is to say, on the one hand the cohomology is described as the relative
cohomology of the sections of the sheaf complex, which appears naturally in the
theory of Cech cohomology of sheaf complexes. On the other hand it is
interpreted as the cohomology of a complex dual to the mapping cone of a
certain morphism of complexes in the theory of derived categories. We prove a
"relative de Rham type theorem" from the above two viewpoints. It says that, in
the case the complex is a soft or fine resolution of a certain sheaf, the
cohomology is canonically isomorphic with the relative cohomology of the sheaf.
Thus the former provides a handy way of representing the latter. Along the way
we develop various theories and establishes canonical isomorphisms among the
cohomologies that appear therein. The second viewpoint leads to a
generalization of the theory to the case of cohomology of sheaf morphisms. Some
special cases together with applications are also indicated.
