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Finitary Čech-de Rham cohomology. (English) Zbl 1015.83013
Summary: The present paper continues [{\it A. Mallios} and {\it I. Raptis}, Int. J. Theor. Phys. 40, 1885-1928 (2001; Zbl 0987.83003)] and studies the curved finitary spacetime sheaves of incidence algebras presented therein from a Čech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial 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 [{\it A. Mallios}, Geometry of vector sheaves: An axiomatic approach to differential geometry, Vols. 1-2, Kluwer, Dordrecht (1998; Zbl 0904.18001, Zbl 0904.18002); Math. Jap. 48, 93-180 (1998; Zbl 0910.53013)]. The upshot of this study is that important “classical” differential geometric constructions and results usually thought of as being intimately associated with $\cal C^{\infty}$-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 Mallios (loc. cit.). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets [{\it I. Raptis} and {\it R. R. Zapatrin}, Int. J. Theor. Phys. 39, 1-13 (2000; Zbl 0974.83014); Mallios-Raptis (loc. cit.)], we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.

83C45Quantization of the gravitational field
58A12de Rham theory (global analysis)
53C80Applications of global differential geometry to physics
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