  We extend results on asymptotic invariants of line bundles on complex
projective varieties to projective varieties over arbitrary fields. To do so
over imperfect fields, we prove a scheme-theoretic version of the gamma
construction of Hochster and Huneke to reduce to the setting where the ground
field is $F$-finite. Our main result uses the gamma construction to extend the
ampleness criterion of de Fernex, K\"uronya, and Lazarsfeld using asymptotic
cohomological functions to projective varieties over arbitrary fields, which
was previously known only for complex projective varieties. We also extend
Nakayama's description of the restricted base locus to klt or strongly
$F$-regular varieties over arbitrary fields.
