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Finitary Čech-de Rham cohomology. (English) Zbl 1015.83013

Summary: The present paper continues [A. Mallios and 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 [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 𝒞 -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 [I. Raptis and 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.


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