The Homological Kähler-De Rham Differential Mechanism part I: Application in General Theory of Relativity

The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalarsdefined together with the associated de Rham complex. In this communication,we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler-De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, isrelated with the absolute representability of the theory in thefield of real numbers, a byproduct of which is the fixed backgroundmanifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamicsin quantum theory, where the adherence to a fixed manifold substratum isproblematic due to singularities or other topological defects