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

TL;DR

This paper provides a detailed proof that for the algebra of smooth functions on an m-dimensional manifold, various homological dimensions all equal m, linking algebraic properties directly to the manifold's dimension.

Contribution

It offers a complete proof that the small global, global, and bidimensions of the algebra of smooth functions on a manifold all equal its dimension.

Findings

Homological dimensions of $C^{ abla}( abla)$ equal the manifold's dimension

Unified understanding of different homological dimensions for smooth function algebras

Explicit proof connecting algebraic dimensions with geometric dimension

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).