It is well understood that the exactness of the de Rham complex (firstly obtained for finite-dimensional manifolds and next extended to Banach manifolds, Fréchet manifolds, locally convex spaces) is of great importance because it describes the singular cohomology. The aim of this paper is to give conditions under which a de Rham complex is constructed and is exact, within the framework of abstract sheaves of differentiable functions taking values in an arbitrary topological algebra \({\mathcal A}\) instead of the field \(\mathbb{R}\) of real numbers.