A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras

TL;DR

This paper extends the classical Zariski-Nagata theorem to mixed characteristic settings by introducing $p$-derivations, showing they accurately describe symbolic powers of prime ideals where previous methods failed.

Contribution

It introduces $p$-derivations into commutative algebra to characterize symbolic powers in mixed characteristic, a novel application of this concept.

Findings

$p$-derivations define new differential powers in mixed characteristic.

These differential powers coincide with symbolic powers of prime ideals.

First application of $p$-derivations in commutative algebra.

Abstract

In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the elements that vanish up to order $n$ on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use $p$-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of $p$-derivations to commutative algebra.