II.2.4

This problem has Hartshorne Height 1.

HAPPY

 * Define an inverse map: $$\mbox{Hom}_{Rings}(A, \Gamma(X,\mathcal{O}_X)) \to \mbox{Hom}_{Sch}(X, \mbox{Spec } A)$$
 * As topological spaces: use $$f \colon A \to B = \Gamma(X,\mathcal{O}_X) $$ and the restriction map $$ B \to \mathcal{O}_{X,p}$$ to determine what prime of $$A$$ the point $$p \in X$$ should map to; call the map m.
 * Should find $$m^{-1}(D_a) = X_{f(a)}$$, prove a lemma (or do II.2.16a) to show the latter subset is open; this shows continuity.


 * Define the map of sheaves on distinguished affines:
 * Use $$A \to B \to \mathcal{O}_X(X_{f(a)})$$ and the universal property of $$A \to A_a $$ to get a map $$\mathcal{O}_{\mbox{Spec }A}(D_a) =A_a \to \mathcal{O}_X(X_{f(a)}) = \mathcal{O}_X(m^{-1}(D_a))$$
 * Check these maps behave well with restrictions, i.e. if $$D_b \subset D_a$$ then $$A_a \to A_b \to \mathcal{O}_X(X_{f(b)})$$ is the same as $$ A_a \to \mathcal{O}_X(X_{f(a)}) \to \mathcal{O}_X(X_{f(b)})$$
 * Check that $$A_{m(p)} \to \mathcal{O}_{X,p}$$ is a local homomorphism.
 * Conclude there is a well defined map of sheaves $$\mathcal{O}_{\mbox{Spec }A} \to m_*\mathcal{O}_X$$


 * Check that the two maps defined between $$\mbox{Hom}_{Rings}(A, \Gamma(X,\mathcal{O}_X)) \leftrightarrow \mbox{Hom}_{Sch}(X, \mbox{Spec } A)$$ compose to give the identity on both sets.
 * For $$ \mbox{Hom}_{Rings}(A, \Gamma(X,\mathcal{O}_X)) \to \mbox{Hom}_{Sch}(X, \mbox{Spec } A) \to \mbox{Hom}_{Rings}(A, \Gamma(X,\mathcal{O}_X)) $$ one way to prove it is to say you know what the map of sheaves does on distinguished affines and note that $$ \mbox{Spec } A = D_1$$.
 * the other direction: $$ \mbox{Hom}_{Sch}(X, \mbox{Spec } A) \to \mbox{Hom}_{Rings}(A, \Gamma(X,\mathcal{O}_X)) \to \mbox{Hom}_{Sch}(X, \mbox{Spec } A)$$ requires more work:
 * The notation: $$(f,f^\#) \colon X \to \mbox{Spec } A$$, $$ \Gamma(f^\#) = \theta \colon A \to B$$, $$ r_x \colon B \to \mathcal{O}_{X,x}$$.
 * Show $$\theta^{-1}(r_x^{-1}(m_x)) = f(x)$$ (this requires some thought!) and probably a diagram involving $$A, A_{f(x)}, A_{\theta^{-1}(r_x^{-1}(m_x))},\mathcal{O}_{X,x}$$ and some arguments of uniqueness of certain maps.
 * Argue that the map of sheaves agree on distinguished affines (basically the restriction maps from the (f,f^\#) have to agree with the unique maps that come from localization ), hence the maps of schemes is the same.