219,340pages on
this wiki
Add New Page
Discuss this page0 Share

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


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.

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random wikia