Forgive me for the length of this post, but I feel this question is deserving of some detail.
Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = \bigoplus_{i=0}^{\infty} S_{i}, $ where $ S_{i} \cdot S_{j} \subset S_{i+j}. $
$f $ is a homogeneous element of $ S $ of degree 1, that is, $ f \in S_{1},$ and $ V(f) $ denotes the set of elements of $ \text{Proj}\;S $ which do not contain $ f $.
I am reading Eisenbud & Harris' "Geometry of Schemes". In Exercise III-6(a), the authors make the following statement:
The intersection $ (I \cdot S[f^{-1}]) \cap S[f^{-1}]_{0}) $ is generated by elements obtained by choosing a set of homogeneous generators of $ I $ and multiplying them by the appropriate negative powers of $ f.$
I think I understand this statement intuitively. But they go on to infer the correspondence from here, and proceed give the following hint concerning the manner of constructing the maps:
The correspondence is given by taking a prime $ \mathfrak{p} $ of $ S[f^{-1}] $ to $ \mathfrak{p} = \mathfrak{p} \cap S[f^{-1}]_{0} $ and taking the prime $ \mathfrak{q} $ of $ S[f^{-1}] _{0} $ to $ \mathfrak{q}S[f^{-1}].$
My difficulty boils down to not knowing how to use this hint to establish the result explicitly/mechanically.