The present paper is a continuation of our work on curved finitary spacetimesheaves of incidence algebras and treats the latter along Cech cohomologicallines. In particular, we entertain the possibility of constructing anon-trivial de Rham complex on these finite dimensional algebra sheaves alongthe lines of the first author's axiomatic approach to differential geometry viathe theory of vector and algebra sheaves. The upshot of this study is thatimportant `classical' differential geometric constructions and results usuallythought of as being intimately associated with smooth manifolds carry through,virtually unaltered, to the finitary-algebraic regime with the help of somequite universal, because abstract, ideas taken mainly from sheaf-cohomology asdeveloped in the first author's Abstract Differential Geometry theory. At theend of the paper, and due to the fact that the incidence algebras involved havebeen previously interpreted as quantum causal sets, we discuss how these ideasmay be used in certain aspects of current research on discrete Lorentzianquantum gravity.
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