A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras
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 -th symbolic power of a given prime ideal consists of the elements that vanish up to order on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use -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 -derivations to commutative algebra.
Keywords
Cite
@article{arxiv.1709.01049,
title = {A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras},
author = {Alessandro De Stefani and Eloísa Grifo and Jack Jeffries},
journal= {arXiv preprint arXiv:1709.01049},
year = {2020}
}
Comments
In the previous version, the proof of Lemma 2.1 had a gap, due to a wrong claim about the structure of certain essentially smooth algebras. We removed the wrong statement, and fixed the proof of Lemma 2.1. The rest of the paper remains unchanged