Coincidence of the homological dimensions of the Fréchet algebra of smooth functions on a manifold with the dimension of the manifold (Q580668)

The principal result of this paper is the following: ds \(C^{\infty}(M)=dg C^{\infty}(M)=db C^{\infty}(M)=m\); here M is a smooth real m-dimensional manifold, \(C^{\infty}(M)\) is the topological algebra of \(C^{\infty}\) functions on M and ds A, dg A, db A denote the cohomological dimensions of a topological algebra A in the sense of \textit{A. Ya. Khelemskij} [Homology in Banach and topological algebras (in Russian) (1986; Zbl 0608.46046)].