The de Rham cohomology of soft function algebras

Abstract

We study the dg-algebra ${\Omega }_{A|\mathbb{R}}^{\bullet }$ 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 ${\mathrm{H}}^n({\Omega }_{A|\mathbb{R}}^{\bullet })\cong {\mathrm{H}}^n(X,\mathbb{R})\oplus V$, 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 ${\mathrm{H}}^{\ast }({\Omega }_{{\mathrm{PPol}}_K(|K|)|\mathbb{R}}^{\bullet })\cong {\mathrm{H}}^{\ast }(|K|,\mathbb{R})$. We also prove that the algebraic de Rham cohomology ${\mathrm{H}}^n({\Omega }_{C(X)|\mathbb{R}}^{\bullet })$ is nontrivial for each $n\ge 1$ if $X$ is an infinite compact Hausdorff space.

32 pages