In book Algebraic geometry R. Hartshorne
I'm confused in the proof of theorem 3.4 first chapter.
1) In the first paragraph of the proof the theorem 3.4: what is the significance of "this isomorphis sends $I(Y_i)$ to $I(Y)S_{(x_i)}$"?
My question is: $I(Y)S_{(x_i)}=I(Y)S_{x_i}\cap S_{(x_i)}$??