II.2.3

This problem has Hartshorne Height 1.

HAPPY
Part a) Part b) $$ \left( \varinjlim_{V \ni p} \mathcal{O}_X(V) \right)_{red} = \varinjlim_{V \ni p} \mathcal{O}_{X_{red}}(V)$$
 * Straightforward application of the definition stalk and thinking of elements of the stalk as equivalences classes $$\langle s, U \rangle$$ etc.
 * 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.
 * 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.