# 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.