Differential basis, 𝑝-basis, and smoothness in characteristic 𝑝>0

Abstract

It is shown that every differential basis of a Noetherian ring R R of prime characteristic p p over an arbitrary subring k k is a p p -basis of R R over k k . Moreover, if k k is a field, then R R is smooth over k k , provided R R has a differential basis over k k and the ring R ⊗ k k p − 1 R{ \otimes _k}{k^{{p^{ - 1}}}} is reduced.