I came across with the following problem:
If $f : X \rightarrow Y$ is a morphism of finite type over an affine scheme of a Noetherian ring $Y = \operatorname{Spec} A$ such that the base change $f' : X \times_Y \operatorname{Spec} \mathcal{O}_{Y,y} \rightarrow \operatorname{Spec} \mathcal{O}_{Y,y}$ is an isomorphism, then there exists an open neighborhood $V$ of y with $f^{-1}(V) → V$ an isomorphism?
This problem occurs when I followed the proof of Qing Liu's book, Proposition 4.4.2. If $X = \operatorname{Spec} B$ is also affine then I have a solution, but I don't know how to use this affine case for the general case.
Thanks.