*(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 ${\mathcal{C}}^{\infty}$-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 [*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:

83C45 | Quantization of the gravitational field |

58A12 | de Rham theory (global analysis) |

53C80 | Applications of global differential geometry to physics |