I am not sure to understand the proof of the theorem 5.2.8 of Springer's LAG.
More precisely I am wondering why he can replace $X$ and $Y$ by open affine neighborhoods.
I have tried the following but I can't say whether this is correct:
We have to prove that $\phi$ is an isomorphism of ringed spaces. So we take $V\subset Y$ and we want to show that $\phi ^*: O_Y(V)\to O_X(\phi^{-1}(V))$ is a ring isomorphism. To do that we cover $V$ with affine opens $V = \cup _i V_i$. Now let $x\in U_i=\phi^{-1}(V_i)$. As $\phi$ is locally finite in $x$ by 5.2.6 there exists an affine open $U_{x_i}\subset U_i$ isomorphic (let $\nu$ this isomorphism) to an affine open subset $V_{x_i}^{'}$ of an affine variety $V_i^{'}$ which is finite over $V_i$ via a morphism $\mu$ such that we have $\phi|_{U_{i,x}}=\mu\circ\nu$. Then by normality and birationality, $\mu$ is an isomorphism.
As we have a sheaf, it follows that $\phi ^*: O_Y(V)\to O_X(\phi^{-1}(V))$ is an isomorphism and we are done.
Is this correct?
Thanks for yout help.