Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)}

Question

$S$ 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 $S_{f}$?

Answer 1

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.