Ampleness of a given line bundle descends under field extension

For every quasi-coherent sheaf $F$ on $X$, $\mathrm{\Gamma }(X,F){\otimes }_{k}K\cong \mathrm{\Gamma }(X_K,F_K)$. Indeed choose a finite affine cover; its intersections are affine by separatedness. The sheaf gluing equalizer for the modules of sections is exact, and tensoring by $K$ preserves that equalizer and finite products, just as in [F3]. On each affine chart the sections of the pulled-back quasi-coherent sheaf are the original module tensored with $K$, so the equalizer is exactly the global-section module of $F_K$.

Fix a coherent $F$. Its base extension is coherent, since its finite presentations base extend on affine charts. By [F1], for all sufficiently large $n$, the evaluation map for $F_K\otimes L_K^n$ is onto. By step 1.1 this is the base extension of the evaluation map $\mathrm{\Gamma }(X,F\otimes L^n){\otimes }_k{\mathcal{O}}_X\to F\otimes L^n$. On every affine chart its cokernel becomes zero after tensoring by $K$ and hence is zero by [F2]. Thus the original sheaf is globally generated for all such $n$. Since $F$ was arbitrary, [F1] gives ampleness of $L$. AC is inherited from [F1]; no descent of a newly chosen line bundle is assumed.