For a graded ring $S$, $\mathrm{Proj}S$ is constructed in Hartshorne's Algebraic Geometry. Then Proposition 2.5 on the same page says that $$(D_+(f), \mathcal O|_{D_+(f)}) \cong \mathrm{Spec}S_{(f)}$$ for any homogeneous $f \in S_+$. The bijection between the underlying topological spaces is given by $$\varphi: \mathfrak a \mapsto (\mathfrak a S_f) \cap S_{(f)}.$$
I think I need some concrete examples to understand this, so I take $S = \mathbb Z[X_0,X_1]$ where the degree of $X_0$ and $X_1$ is $1$. Let $f=X_0$, $\mathfrak a$ be the ideal in $S$ generated by $X_1$, then it is homogeneous, prime and does not contain $f$. Through $\varphi$, it is mapped to $(\mathfrak a S_f) \cap S_{(f)}$. But what is $\mathfrak a S_f$? I think since $\frac{1}{X_1} \in S_f$ and $X_1 \in \mathfrak a$, $\mathfrak a S_f$ is $S_f$ itself. But $S_{(f)}$ is a subring of $S_f$, $S_f \cap S_{(f)} = S_{(f)}$, which is not a prime ideal of $S_{(f)}$. So where is wrong?
Thank you very much.