/sci/ - Science & Math

Cooking is so relaxing. This soup is going to last me for the whole weekend. Anyway...
>>8991401
>>8990505
This turned out to be more cumbersome than I anticipated. Here goes:

Since [math] X [/math] and [math] Y [/math] are integral, they have unique generic points, denoted [math] {\eta}_X [/math] and [math] {\eta}_Y [/math] respectively, and [math] {\eta}_X [/math] is the only point [math] f [/math] maps to [math] {\eta}_Y [/math]. Moreover, [math] {\eta}_X \in U \subset X [/math] where [math] U [/math] is a dense open separated sub-scheme, so [math] {\eta}_Y \nin f(X \setminus U) [/math].
[math] X \setminus U \rightarrow Y [/math] is quasi-compact so there exists [math] V \subset Y [/math] (dense) open, such that [math] f^{-1}(V) \subset U [/math], whereby [math] f(f^{-1}(V)) \rightarrow V [/math] is separated.
On the other hand, [math] f [/math] is of finite type so there exist [math] V' \subset V [/math] and [math] U' \subset f^{-1}(V) [/math] affine open, with [math] f(U') \subset V' [/math], [math] {\eta}_Y \in V' [/math] and [math] f^{-1}(\{ {\eta}_Y \}) = \{ {\eta}_X \} \subset U' [/math] such that [math] {f \mid}_{U'} \rightarrow V' [/math] is finite.
Thus [math] f^{-1}(V') = U' \amalg W [/math] for some open subscheme [math] W [/math].
But [math] {\eta}_X \in U' [/math] so [math] W = \emptyset [/math], i.e. [math] f^{-1}(V') = U' [/math].