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Finitary Čech-de Rham Cohomology

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Abstract

The present paper continues (Mallios & Raptis, International Journal of Theoretical Physics, 2001, 40, 1885) 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 (Mallios, Geometry of Vector Sheaves: An Axiomtic Approach to Differential Geometry, Vols. 1–2, Kluwer, Dordrecht, 1998a; Mathematica Japonica (International Plaza), 1998b, 48, 93). 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 (1998a,b). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets (Raptis, International Journal of Theoretical Physics, 2000, 39, 1233; Mallios & Raptis, 2001), we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.

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Authors and Affiliations

  1. Algebra and Geometry Section, Department of Mathematics, University of Athens, Athens, Greece

    Anastasios Mallios

  2. Theoretical Physics Group, Blackett Laboratory of Physics, Imperial College of Science, Technology and Medicine, London, UK

    Ioannis Raptis

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  1. Anastasios Mallios
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  2. Ioannis Raptis
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Mallios, A., Raptis, I. Finitary Čech-de Rham Cohomology. International Journal of Theoretical Physics 41, 1857–1902 (2002). https://doi.org/10.1023/A:1021000806312

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