The de Rham cohomology of soft function algebras

We study the dg-algebra $\Omega ^\bullet_{A|\mathbb{R}}$ 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 $\mathrm H ^n (\Omega ^\bullet_{A|\mathbb{R}}) \cong \mathrm H ^n (X,\mathbb{R})\oplus V$, where $V$ is some vector space. In particular, we consider the cases $A=C(X)$ for $X$ a compact Hausdorff space and $A = C^\infty (X)$ for $X$ a compact smooth manifold. For the algebra $\mathrm{PPol}_K (|K|)$ of piecewise polynomial functions on a polyhedron $K$ the above splitting reduces to a canonical isomorphism $\mathrm H ^* (\Omega ^\bullet_{\mathrm{PPol}_K (|K|)|\mathbb{R}}) \cong \mathrm H ^* (|K|,\mathbb{R})$. We also prove that the algebraic de Rham cohomology $\mathrm H ^n (\Omega ^\bullet_{C(X)|\mathbb{R}})$ is nontrivial for each $n\geq 1$ if $X$ is an infinite compact Hausdorff space.