Lie algebras and separable morphisms in pro-affine algebraic groups

Abstract

Let K K be an algebraically closed field of arbitrary characteristic, and let f : G → H f:G\rightarrow H be a surjective morphism of connected pro-affine algebraic groups over K K . We show that if f f is bijective and separable, then f f is an isomorphism of pro-affine algebraic groups. Moreover, f f is separable if and only if (its differential) f o f^o is surjective. Furthermore, if f f is separable, then L ( Ker ⁡ f ) = Ker ⁡ f o {\mathcal L}(\operatorname {Ker}f)=\operatorname {Ker} f^o .