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This problem has Hartshorne Height 1.
- Define an inverse map:
- As topological spaces: use and the restriction map to determine what prime of the point should map to; call the map m.
- Should find , prove a lemma (or do II.2.16a) to show the latter subset is open; this shows continuity.
- Define the map of sheaves on distinguished affines:
- Use and the universal property of to get a map
- Check these maps behave well with restrictions, i.e. if then is the same as
- Check that is a local homomorphism.
- Conclude there is a well defined map of sheaves
- Check that the two maps defined between compose to give the identity on both sets.
- For one way to prove it is to say you know what the map of sheaves does on distinguished affines and note that .
- the other direction: requires more work:
- The notation: , , .
- Show (this requires some thought!) and probably a diagram involving and some arguments of uniqueness of certain maps.
- Argue that the map of sheaves agree on distinguished affines (basically the restriction maps from the have to agree with the unique maps that come from localization ), hence the maps of schemes is the same.