Until now, the correspondence between the Alexander-Kolmogorov Complex, and the De Rham one, by means of a small scale parameter, has not gone that far as passing to the limit of the resolvent of the associated Laplacian, when the small parameter tends towards zero. In this line, a result proving a complete Hodge decomposition was missing. We bridge this gap by means of our own rescaled h-cohomology, h being a very small parameter. Passing to the limit of the resolvent enables us to consider the extension to singular spaces, in particular, our h-differential operators also enable us to also make the connection with those of analysis on fractals, as introduced by Jun Kigami, and taken up by Robert S. Strichartz.