# II.2.4

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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.