Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for all $n \ge n_0(M)$ the sheaf $\mathcal{M} \otimes \mathcal{L}^{\otimes n}$ is generated by global sections, there exist a $r(n)$ such that the map
$$ \mathcal{O}_X^{r(n)} \to \mathcal{M} \otimes \mathcal{L}^{\otimes n}$$ induced by some appropr $r(n)$ global section of sheaf on rhs.
Questions:
Is it possible to verify ampleness on a faithfully flat cover $g: X' \to X$?Ie, to deduce that if $g^* \mathcal{L}$ ample on $X'$, then $\mathcal{L}$ is acutally ample? If yes, could somebody sketch the proof (idea)? If not, does it work with stronger additional assumptions on cover faithfully flat $g$?
#Edit: As Will Sawin's example shows it is reasonable to add finiteness assumption to $g$ otherwise a usual covering by disjoint affine opens of $X$ demonstrates that such approach cannot be expected to actually work for all faithfully flat coverings, as any invertible sheaf on affine scheme is known to be ample;
Other interesting case which not rely on finiteness assumption comming to my mind is what about covers $g$ (not neccessary finite) obtained from faithfully flat cover on the base $S$, eg for $S=\text{k}$ as base changing toitsalg closure. In other words, is in that case ampleness checkable by descent on the base?Does the analogous story work with very ample?
What about the relative case, ie we start with structure morphism $f: X \to S$ and would like to know if relative $f$-ampleness of an invertible $\mathcal{L}$ can be checked on faithfully flat $S$-cover $g: X' \to X$. So, if the argument in absolute case indeed work, can it be adapted to relative situation?
NB: Concrete applications I had in mind: reasoning on methods to check ampleness of canonical sheaf of stable curves needded to get an embedding in projective space for whole family (in absolute case, ie over $k$, and relative, ie over arbitrary base $S$)
