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 $S$)