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This paper gives a proof that the underlying topological spaces of affine schemes are precisely the spectral spaces (compact, T0, sober, and the compact open subsets are closed under finite intersection and generate the topology; equivalently, they are the inverse limits of families of finite T0 spaces).

Instead of starting with a bare topological space, suppose we have a locally ringed space. If it is an affine scheme, then the underlying space is spectral, the supports of sections generate the topology, every restriction map of the sheaf is a localization of rings, etc.

Does anyone know whether we can impose restrictions (such as the ones I have just listed, and probably together with others) to guarantee that a given locally ringed space is actually an affine scheme?

This poster asked something similar - whether locally ringed spaces had been classified in some way. This is not exactly what I'm looking for, but it would be interesting to know in any case.

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Would this be better attempted at MathOverflow? – Xander Flood Dec 9 '12 at 19:17

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