Affiliation:
1. Department of Mathematics , Hokkaido University , Sapporo 060-0810 , Japan
Abstract
Abstract
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 Čech 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 Čech 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.
Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction
theory and other problems in algebraic analysis.
Funder
Japan Society for the Promotion of Science
Subject
Applied Mathematics,General Mathematics