What does it mean that a proper, birational morphism (see here) of locally Noetherian schemes $\phi \colon X \to Y$ is an isomorphism over the regular points of $Y$? I have two possible definitions in mind that seem quite natural, and I am not sure which one is correct and whether they are equivalent (the second is of course stronger, a priori):
- For each $y \in Y$ regular, there is a unique preimage $x$ and $\phi_x^\#$ is an isomorphism of local rings
- For each $y \in Y$ regular, there is an open (affine) neighborhood $y \in U \subseteq \mathrm{Reg}(Y)$ such that $\phi \colon \phi^{-1}(U) \to U$ is an isomorphism
Edit: I found this terminology in the definition of a desingularization in the strong sense in Liu's "Algebraic Geometry and Arithmetic Curves", Definition 8.3.39. Note that a birational morphism satisfies my first condition for the generic points of $X$ and $Y$, so I believe that this is what is meant, but I wonder if more is true.