The de Rham cohomology of the algebra of polynomial functions on a simplicial complex

Abstract

We consider the algebra $A^0 (X)$ of polynomial functions on a simplicial complex $X$. The algebra $A^0 (X)$ is the $0$th component of Sullivan's dg-algebra $A^\bullet (X)$ of polynomial forms on $X$. Our main interest lies in computing the de Rham cohomology of the algebra $A^0(X)$, that is, the cohomology of the universal dg-algebra $\Omega ^\bullet _{A^0(X)}$. There is a canonical morphism of dg-algebras $P:\Omega ^\bullet _{A^0(X)} \to A^\bullet (X)$. We prove that $P$ is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^0 (X)$ is canonically isomorphic to the cohomology of the simplicial complex $X$ with coefficients in $k$. Moreover, for $k=\mathbb{Q}$ the dg-algebra $\Omega ^\bullet _{A^0 (X)}$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.