$p$-bases and differential operators on varieties defined over a non-perfect field

Let $k$ be a possibly non-perfect field of characteristic $p > 0$. In this work we prove the local existence of absolute $p$-bases for regular algebras of finite type over $k$. Namely, consider a regular variety $Z$ over $k$. Kimura and Niitsuma proved that, for every $\xi \in Z$, the local ring $\mathcal{O}_{Z,\xi}$ has a $p$-basis over $\mathcal{O}_{Z,\xi}^p$. Here we show that, for every $\xi \in Z$, there exists an open affine neighborhood of $\xi$, say $\xi \in \text{Spec}(A) \subset Z$, so that $A$ admits a $p$-basis over $A^p$. This passage from the local ring to an affine neighborhood of $\xi$ has geometrical consequences, some of which will be discussed in the second part of the article. As we will see, given a $p$-basis $\mathcal{B}$ of the algebra $A$ over $A^p$, there is a family of differential operators on $A$ naturally associated to $\mathcal{B}$. These differential operators will enable us to give a Jacobian criterion for regularity for varieties defined over $k$, as well as a method to compute the order of an ideal $I \subset A$.