We study the dg-algebra Ω*_A|R of algebraic de Rham forms of a real soft function algebra A, i.e., the algebra of global sections of a soft subsheaf of C_X, the sheaf of continuous functions on a space X. We obtain a canonical splitting Hⁿ(Ω*_A|R) ≅ Hⁿ(X,R)⊕ V, where V is some vector space. In particular, we consider the cases A=C(X) for X a compact Hausdorff space and A = C^∞(X) for X a compact smooth manifold. For the algebra PPol_K(|K|) of piecewise polynomial functions on a polyhedron K the above splitting reduces to a canonical isomorphism H*(Ω*_{PPolK (|K|)|R}) ≅ H*(|K|,R). We also prove that the algebraic de Rham cohomology Hⁿ(Ω*_C(X)|R) is nontrivial for each n>0 if X is an infinite compact Hausdorff space.

This research was supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2019-1620 date 08/11/2019 and 075-15-2022-289 date 06/04/2022.