The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied. The differential geometers often call these 'jets of order n on X', but we reserve the term 'jet' for a somewhat different but related construction discussed in § 5. * Buium calls his construction a differential 'jet space',

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... conflicts badly with our terminology in several ways.