Are there any criteria for a morphism (as nice as you wish) in characteristic p to be etale that relates to the differential map on the associated sheaves of differentials or tangent spaces? I know this is a little vague but I encourage you to post any criterion that looks even a little related!
As an example, I think there is a criterion that a finitely presented schemes is etale if and only if the relative frobenius is an isomorphism. Is there a version of this for maps to be etale?