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

I. S. Baskov

Čebyševskij sbornik, Tome 24 (2023) no. 1, pp. 203-212

Résumé

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 $X$. All algebras are over an arbitrary field $k$ of characteristic $0$. 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. Our result shows that for the algebra $A^0 (X)$ the statement of Grothendieck's comparison theorem holds (proved by him for smooth algebras). In order to prove the statement we consider Čech resolution associated to the cover of the simplicial complex by the stars of the vertices. Earlier, Kan–Miller proved that the morphism $P$ is surjective and gave a description of its kernel. Another description of the kernel was given by Sullivan and Félix–Jessup–Parent.

Détail

Keywords: algebraic de Rham cohomology, universal dg-algebra, algebra of polynomial functions, dg-algebra of polynomial forms, rational homotopy theory.