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This problem has Hartshorne Height 1.
Prop-Def. II.1.2: For any presheaf there is a sheaf and a morphism such that any sheaf and morphism factors as for a unique
- Straightforward application of the definition stalk and thinking of elements of the stalk as equivalences classes etc.
- First show is a locally ringed space. One way of doing this is showing the following:
- is a local ring.
- , i.e.
- one way to prove the above is to show one side has the universal property of the other, this involved using part a)
- Next, show for that (as locally ringed spaces!) to conclude that is a scheme.
- this consists of a map on topological spaces (the identity) and a sheaf map .
- define the sheaf map on distinguished affines and show its and isomorphism i.e. .
- Define a map of schemes
- Map on topological spaces is the identity.
- For use the canonical projection and the presheaf to sheaf map to get the desired sheaf map.
- Check the map on local rings is and hence a local homomorphism.
- The desired factorization on topological spaces is clear.
- Show the desired factorization on the level of presheaves:
- Use that is reduced and the universal property of kernels to show there is a factorization .
- Argue this gives a factorization as sheaves, giving a morphism of schemes
- Use Prop-Def II.1.2 to show there is the desired factorization on the level of sheaves and hence a factorization as locally ringed spaces.
- The above factorization gives a factorization into
- Use that is local (and surjective!) and that is local to conclude that is also local.
- Uniqueness follows essentially from the uniqueness of of the map in Prop-Def II.1.2.