Ax–schanuel Condition in Arbitrary Characteristic

Abstract

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

A note on extending actions of infinitesimal group schemes

“…Then G corresponds to H, where the completion is taken with respect to the augmentation ideal of the Hopf algebra H. Since the map H → H is a k-algebra map and any continuous homomorphism H → R is determined by its values on H, we get a one-to-one map G(R) → G(R). An easy diagram chase shows that this map is a homomorphism of groups (see also Section 2.6 in [12]).…”