Is "very ampleness" etale/fpqc local on the base (under reasonable conditions?) Let $f : X\rightarrow S$ be a quasicompact morphism of schemes, and $\mathcal{L}$ a line bundle on $X$.
Let $g : S'\rightarrow S$ be a surjective map, and $f' : X_{S'}\rightarrow S'$ the base change.
Under what conditions on $f,g$ does $\mathcal{L}|_{X_{S'}}$ being very ample imply that $\mathcal{L}$ is very ample?
Certainly this holds if $g$ is a Zariski covering. Does it hold if $g$ is an etale covering? fpqc covering? (possibly with some conditions on $f$?)
Part of the problem is that I'm having difficulty finding cohomological characterizations of very ampleness.
 A: A good cohomological characterisation of very ampleness is criterion (4) of [Tag 01VR]: $f$ is quasi-separated, the counit $\psi \colon f^*f_* \mathscr L \to \mathscr L$ is surjective, and the induced map $r_{\mathscr L, \psi} \colon X \to \mathbf P(f_*\mathscr L)$ is an immersion.
Firstly, note that formation of $f_* \mathscr L$ (and therefore of $\mathbf P(f_*\mathscr L)$) commutes with flat base change [Tag 02KH]. Also note that $r_{\mathscr L,\psi}$ is automatically quasi-compact, since $f$ is quasi-compact and $\mathbf P(f_*\mathscr L) \to S$ is quasi-separated [Tag 03GI].
This allows us to check that all three criteria are fpqc-local on the target:

*

*$f$ is quasi-separated: this is fpqc-local by [Tag 02KR].

*the counit $\psi \colon f^*f_*\mathscr L \to \mathscr L$ is surjective: this is fpqc-local since formation of $f_* \mathscr L$ commutes with flat base change and since surjectivity of a morphism of quasi-coherent sheaves is fpqc-local (this follows from faithful flatness).

*the induced map $r_{\mathscr L,\psi} \colon X \to \mathbf P(f_*\mathscr L)$ is a (quasi-compact) immersion: this is fpqc-local by [Tag 02L8].

