Detailed proof of a theorem on coincidence of homological dimensions of Fr\'echet algebras of smooth functions on a manifold with the dimension of the manifold

Abstract

Given work contains the full text of the proof of the following assertion: For the topological algebra $C^{\infty}(\mathcal{M})$ of smooth functions on a smooth $m$-dimensional real manifold $\mathcal{M}$ the small global dimension $(\mathop{\mathrm{ds}} C^\infty (\mathcal{M}))$, the global homological dimension $(\mathop{\mathrm{dg}} C^\infty (\mathcal{M}))$ and the bidimension $(\mathop{\mathrm{db}} C^\infty(\mathcal{M}))$ are equal to $m$ (all dimensions are understood in the sense of the homology of topological (locally convex) algebras).