II.2.1

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.