$S$ is a graded ring (over non-negative integers), $f \in S\_{+}$ is a homogeneous element of positive degree, $D(f)$ the elements of Proj $S$ not containing $f$. I don't see the bijection between $D(f)$ and Spec $S\_{(f)}$. Here $S\_{(f)}$ is the zero-degree part of $S\_{f}$ obtained from $S$ by inverting f. I see the bijection from $D(f)$ to the homogeneous primes in $S\_{f}$, but is there 1-1 correspondence between primes in $S\_{(f)}$ and homogeneous primes in $S\_{f}$?



Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)}
The homeomorphism $D\_+(f) \to \text{Spec } S\_{(f)}$ is given by $\mathfrak{p} \mapsto \mathfrak{p} S\_f \cap S\_{(f)}$ with inverse map $\mathfrak{q} \mapsto \oplus\_n \{x \in S\_n : x^{|f|} / f^n \in \mathfrak{q}\}$. This can be checked by simple calcuations.


