If we have a morphism between two affine Schemes $f: X rightarrow Y$ with $X = $ Spec $A$, and $Y = $ Spec $B$, is it true that $f^{-1}(D(g)) = D(f'(g))$? (where $f'$ is the associated map on the structure sheaves) If so, is there a simple proof? Otherwise, is there any other way to characterize the preimages of distinguished open sets?
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