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Let $M$ be a $C^\infty$ manifold, $C^\infty(M)$ be its function ring and $X=\text{Spec}(C^\infty(M))$ be the corresponding affine scheme.

Both $M$ and $X$ are locally ringed spaces. Are they isomorphic? If not, is it possible to reconstruct $M$ from $X$?

  • $X$ and $C^\infty(M)$ have exactly the same information, so you can take a look to http://math.stackexchange.com/questions/1764947/can-we-recover-a-compact-smooth-manifold-from-its-ring-of-smooth-functions for answers to your second question. – PseudoNeo Mar 16 '17 at 07:41
  • And the comment by Eric Wofsey to Mike Miller's answer (in the link) will answer your 1st question. – Moishe Kohan Mar 16 '17 at 13:14

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