I have a qoestion about example 3.18 given in Liu's "Algebraic Geometry and Arithmetic Curves" on page 232:
Lui states that if $f:X \to Y$ is a morphism of finite type of regular locally Noetherian schmes then the condition on beeing l.c.i. can be simplified.
The problem is that I don't understand why if $f$ is of finite type cocally $X$ can be interpreted as closed subscheme of $\mathbb{A}^n _Y$?
The problem is that $f$ is not assumed to be affine!
By definition "of finite type" means that locally we have $U⊂X, V⊂Y$ with $f(U)⊂V$ the ring map $O_Y(V)→O_X(U)$ is of finite type. (my reference: https://stacks.math.columbia.edu/tag/01T0)
But since $f$ is not affine I cannot expect to find affine $U$ and $V$ with $f^{-1}(V)=U$ so to interpret $X$ locally as closed subscheme of $\mathbb{A}^n _Y$ makes to me no sense for me without the assumption that $f$ is affine.
Where is the error in my reasonings?