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II.2.3

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This problem has Hartshorne Height 1.


Relevant Results

Prop-Def. II.1.2: For any presheaf F there is a sheaf F^+ and a morphism F \to F^+ such that any sheaf G and morphism  F \to G factors as  F \to F^+ \to G for a unique F^+ \to G

HAPPY

Part a)

  • Straightforward application of the definition stalk and thinking of elements of the stalk as equivalences classes \langle s, U \rangle etc.

Part b)

  • First show X_{red} is a locally ringed space. One way of doing this is showing the following:
    • (\mathcal{O}_{X,p})_{red} is a local ring.
    • \mathcal{O}_{X_{red},p} \cong (\mathcal{O}_{X,p})_{red}, i.e.
 \left( \varinjlim_{V \ni p} \mathcal{O}_X(V) \right)_{red} = \varinjlim_{V \ni p} \mathcal{O}_{X_{red}}(V)
  • 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  \mbox{Spec}A \subset X that \mbox{Spec } A_{red} \cong (\mbox{Spec }A)_{red} (as locally ringed spaces!) to conclude that X is a scheme.
    • this consists of a map on topological spaces (the identity) and a sheaf map \mathcal{O}_{(\mbox{Spec }A)_{red}} \to \mathcal{O}_{ \mbox{Spec } A_{red} }.
    • define the sheaf map on distinguished affines and show its and isomorphism i.e.  (A_f)_{red} \cong (A_{red})_{\overline{f}}.
  • Define a map of schemes X_{red} \to X
    • Map on topological spaces is the identity.
    • For  U \subset X use the canonical projection  \mathcal{O}_X(U) \to \mathcal{O}_X(U)_{red} and the presheaf to sheaf map \mathcal{O}_X(U)_{red} \to \mathcal{O}_{X_{red}}(U) to get the desired sheaf map.
    • Check the map on local rings is \mathcal{O}_{X,p} \to \mathcal{O}_{X,p}/\sqrt 0 and hence a local homomorphism.

Part c)

  • The desired factorization on topological spaces is clear.
  • Show the desired factorization on the level of presheaves:
    • Use that \mathcal{O}_X(U) is reduced and the universal property of kernels to show there is a factorization \mathcal{O}_Y(U) \to \mathcal{O}_Y(U)_{red} \to \mathcal{O}_X(U).
  • 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 X \to Y_{red} \to Y as locally ringed spaces.
    • The above factorization gives a factorization \mathcal{O}_{Y,f(p)} \to \mathcal{O}_{X,p} into \mathcal{O}_{Y,f(p)} \xrightarrow{\alpha} \mathcal{O}_{Y_{red},p} \xrightarrow{\beta} \mathcal{O}_{X,p}
    • Use that \alpha is local (and surjective!) and that \beta \circ \alpha is local to conclude that \beta is also local.
  • Uniqueness follows essentially from the uniqueness of of the map in Prop-Def II.1.2.

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