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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.

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