The splitting of the de Rham cohomology of soft function algebras is multiplicative

TL;DR

This paper proves that the canonical splitting maps in the de Rham cohomology of real soft function algebras are multiplicative, enhancing the understanding of their algebraic structure.

Contribution

It demonstrates that the canonical splitting maps in the de Rham cohomology of soft function algebras are multiplicative, building on prior splitting results.

Findings

Splitting maps are multiplicative.

Enhances understanding of cohomology algebra structure.

Builds on previous splitting results.

Abstract

Let $A$ be a real soft function algebra. In arXiv:2208.11431 we have obtained a canonical splitting $\mathrm{H}^* (\Omega ^\bullet _{A|\mathrm{R}}) \cong \mathrm{H} ^* (X,\mathrm{R})\oplus \text{(something)}$ via the canonical maps $\Lambda_A:\mathrm{H} ^* (X,\mathrm{R})\to\mathrm{H} ^* (\Omega ^\bullet _{A|\mathrm{R}})$ and $\Psi_A:\mathrm{H} ^* (\Omega ^\bullet _{A|\mathrm{R}})\to\mathrm{H} ^* (X,\mathrm{R})$. In this paper we prove that these maps are multiplicative.