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\(p\)-bases and differential operators on varieties defined over a non-perfect field. (English) Zbl 1428.13043

Summary: 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 \(\xi \in \mathrm{Spec}(A) \subset Z\), 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 \(I \subset A\).

MSC:

13N05 Modules of differentials
14B05 Singularities in algebraic geometry
13H05 Regular local rings
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References:

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