# 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.