The de Rham cohomology of soft function algebras

For a commutative $k$-algebra $A$ one defines the de Rham cohomology $H^*_{DR} (A)$. The Grothendieck's comparison theorem states that the de Rham cohomology groups of the algebra of the regular functions on a smooth affine algebraic variety $X$ are isomorphic to the singular cohomology groups of the analytification of $X$.
If one considers an arbitrary soft function $R$-algebra $A$ on a space $X$ then there is no such isomorphism, but the singular cohomology groups canonically split off of the de Rham cohomology groups of $A$.
The splitting is obtained via the canonical maps
$\Lambda : H^*_{sing} (X, R) \to H^*_{DR} (A)$
and $\Psi : H^*_{DR} (A) \to H^*_{sing} (X, R)$
During the talk we will briefly discuss the construction of these maps.
We will consider these maps for various function algebras:
- an algebra of continuous functions on a topological space,
- an algebra of smooth functions on a smooth manifold,
- an algebra of piecewise polynomial functions on a polyhedron,
- an algebra of polynomial functions on a polyhedron.
We will also discuss when the maps $\Lambda$ and $\Psi$ are isomorphisms.
The talk is based on the papers https://arxiv.org/abs/2208.11431 and https://arxiv.org/abs/2208.11431.