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II.2.1

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This problem has Hartshorne Height 1.

Commutative Algebra

  • the primes of  A_f are in bijection with the primes of A not containing f
    • this problem shows its not just a bijection but an isomorphism in terms of schemes.
    • the correspondence goes  Q \subset A_f \mapsto Q \cap A and  P \subset A \mapsto PA_f \subset A_f.
  • (A_f)_b \cong A_{fb} (can prove by universal property argument)

HAPPY

  • use the correspondence above to define a continuous map m\colon D_f \to \mbox{Spec} A_f.
  • define a map of sheaves \mathcal{O}_{\mbox{Spec} A_f} \to m_*\mathcal{O}_{D_f}
    • this can be done by defining it on distinguished affines and using the last result of commutative algebra.

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