I have a question about a proof from Arithmetic Geometry (edited by Cornell & Silverman) on page 51:
The proof starts with "According to [6], we may and do assume $S = Spec \ k$, ...
[6] refers to Grothendieck, A. and Dieudonne, J. Elements de geometrie algebrique, IV, No 4.
Unfortunately the author hasn't give the concrete lemma or proposition which justifies this reduction step from $S$ to field. Has anybody an idea which result from EGA 4 on etale schemes the author here have in mind?
If $X\to S$ is an etale morphism of schemes, then the fiber $X_s$ is a disjoint union of spectra of finite separable field extensions of $k(s)$. There is a (partial) converse to this:
Lemma (Stacks 02GM or EGA IV4 Corollaire 17.6.2): If $X\to S$ is flat, locally of finite presentation, and for every $s\in S$ the fiber $X_s$ is a disjoint union of spectra of finite separable field extensions of $k(s)$, then $X\to S$ is etale.
This is the relevant result.