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

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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 ξ∈ Z, the local ring OZ,ξ has a p-basis over OZ,ξᵖ. Here we show that, for every ξ∈ Z, there exists an open affine neighborhood of ξ, say ξ∈ Spec(A) ⊂ Z, so that A admits a p-basis over Aᵖ. This passage from the local ring to an affine neighborhood of ξ has geometrical consequences, some of which will be discussed in the second part of the article. As we will see, given a p-basis B of the algebra A over Aᵖ, there is a family of differential operators on A naturally associated to 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 ⊂ A.