descending smoothness along unramified morphisms

Question

Let $f:X\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms of schemes locally of finite type. Suppose that $f$ is unramified and surjective, $g$ is flat and $g\circ f$ is étale. Does $g$ is also étale?

Answer 1

Let $A\subset B$ be a finite etale extension of Dedekind domains (ie, smooth curves). Let $b\in B$ be such that $\text{Frac}(A)(b) = \text{Frac}(B)$, but $A[b]\subsetneq B$. Then the normalization of $A[b]$ is $B$, and hence consider $X = Spec\; B$, $Y = Spec\; A[b]$, and $Z = Spec\; A$, where $f : X\rightarrow Y$ is normalization (hence is surjective and unramified), and $Y\rightarrow Z$ is the obvious map.

Firstly, $B/A$ is clearly etale. Since $A[b]\ne B$, $A[b]$ is not normal, hence not regular (normal curves are regular!). If $A[b]/A$ were etale, then since $A$ is regular, $A[b]$ must also be regular, hence $A[b]/A$ is not etale. Since $A[b]\subset B$, $A[b]$ is an integral domain, and hence a torsion-free $A$-module. Since $A$ is Dedekind, torsion-free $\Rightarrow$ flat, so $A[b]/A$ is flat.

For an example, you could take $A$ to be a ring of integers of a number field with nontrivial class group, and $B$ the maximal unramified extension of $A$. Let $b'$ generate the extension of fraction fields of $B$ over $A$, and let $b := ab'$, where $a\in A$ is a non-unit. Then $A,B,b$ satisfy the conditions given above.

More specifically, for a field $k$ of characteristic not 2, you could take $A = k[x^2]$, $B = k[x]$, and $b = x^2\cdot x = x^3$, then $A[b] = k[x^2,x^3]\subsetneq k[x] = B$.

A related question can be found here: Power bases for unramified extensions of an affine genus 0 curve.