More on universal homeomorphisms

Question

I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).

1. Is a universal homeomorphism of connected regular (excellent finite dimensional) schemes an isomorphism if these schemes are not positive characterstic ones?

2. Suppose that a finite morphism $f:X\to Y$ of connected regular (excellent finite dimensional) schemes is generically purely inseparable. Does $f$ restrict to a universal homeomorphism of some open (non-empty) subschemes of $X$ and $Y$?

Answer 1

1. Yes. Let $f\colon X \to Y$ be a universal homeomorphism of locally noetherian schemes. Assume that $X$ and $Y$ are integral and $Y$ is normal, and the function field $k(Y)$ has characteristic 0. Then $k(X) = k(Y)$, and $f$ is an isomorphism by Zariski's main theorem.

2. Under these conditions $f$ is a universal homemorphism. In fact $X$ is the normalization of $Y$ in $k(X)$, again by Zariski's main theorem. But $k(X)$ is purely inseparable over $Y$, so it is obtained by a successive extraction of $p^{\rm th}$ roots of 1. If $F \colon Y \to Y$ is the absolute Frobenius, some power of $F$ will factor through $X$, and then it is easy to see that $X$ is universally isomorphic to $Y$, since $F$ is a universal isomorphism.