I am interested in Prop II.2.5b in Hartshorne stating that if $D_+ (f) = \{p \in \textrm{Proj } B \mid f \notin p \}$ then there is a canonical homeomorphism $D_+ (f) \cong \textrm{Spec } B_{(f)}$ the latter notation meaning the elements of degree 0 in $B_f$.
Following Hartshorne's proof, there is a natural morphism $D_+ (f) \to \textrm{Spec } B_{(f)}$ by restricting the canonical map $D(f) = \textrm{Spec } B_f \to \textrm{Spec } B_{(f)}$ to $D_+ (f)$. Now Hartshorne claims "the properties of localization show that [this] is bijective."
I would like to be more explicit about surjection so I consulted Liu's book which said the following:
Explicitly, $\theta$ which is the restriction map described above is $p \mapsto (pB_f) \cap B_{(f)}$. I would like to show that $\theta(p) =(pB_f) \cap B_{(f)}= q$. At this point, it seems to be a matter of following the contractions and extensions of ideals with an added twist of having two maps. However, I am having some difficulties following the maps properly. Can someone lend me a hand?
I think the problem is complicated mainly by the grading.
$$p = \{x \in B \mid x/1 \in \sqrt{qB_f}\} = \{x \in B \mid \exists m \in \mathbf{Z}^+ x^m / 1 \in qB_f\}$$
so $pB_f$ looks like all the things of the form $xb/f^N$ with $x^m b^m / f^{mN} \in qB_f$. Intersecting with $B_{(f)}$ results in taking
$$ \deg (f) = r, \deg (x) = s, \deg (b) = Nr - s$$
Thus $(xb/f^N)^m \in qB_f \cap B_{(f)} = q$ and $q$ was prime so $xb/f^N \in q$.
Conversely, any element of $q$ is already of degree 0 and satisfies trivially the condition of being in the radical, namely taking $m = 1$.