= Solution
Restriction and division by powers of $f$ define a natural <ring homomorphism>
$$
\Phi:A_f\longrightarrow\Gamma(X_f,\mathcal O_{X_f}),
\qquad a/f^n\longmapsto f^{-n}a|_{X_f}.
$$
If $\Phi(a/f^n)=0$, then $a|_{X_f}=0$, and part (b) gives $f^ma=0$ for some $m$; this is exactly the criterion that $a/f^n=0$ in the <localization of a ring> $A_f$. Hence $\Phi$ is injective. Given $b$ in the target, part (c) gives $f^Nb=a|_{X_f}$ for some $a\in A$, so $b=\Phi(a/f^N)$. Hence $\Phi$ is surjective and
$$
\boxed{\Gamma(X_f,\mathcal O_{X_f})\cong A_f.}
$$
Back to article page
