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 ${\Omega }_{A^0(X)}^{\bullet }$. There is a canonical morphism of dg-algebras $P:{\Omega }_{A^0(X)}^{\bullet }\to A^{\bullet }(X)$. 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 ${\Omega }_{A^0(X)}^{\bullet }$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.

8 pages