  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).
