  The present paper is a continuation of our work on curved finitary spacetime
sheaves of incidence algebras and treats the latter along Cech cohomological
lines. In particular, we entertain the possibility of constructing a
non-trivial de Rham complex on these finite dimensional algebra sheaves along
the lines of the first author's axiomatic approach to differential geometry via
the theory of vector and algebra sheaves. The upshot of this study is that
important `classical' differential geometric constructions and results usually
thought of as being intimately associated with smooth manifolds carry through,
virtually unaltered, to the finitary-algebraic regime with the help of some
quite universal, because abstract, ideas taken mainly from sheaf-cohomology as
developed in the first author's Abstract Differential Geometry theory. At the
end of the paper, and due to the fact that the incidence algebras involved have
been previously interpreted as quantum causal sets, we discuss how these ideas
may be used in certain aspects of current research on discrete Lorentzian
quantum gravity.
