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

The gamma construction and asymptotic invariants of line bundles over arbitrary fields