A Zariski--Nagata theorem for smooth ℤ-algebras-Reference-Cited by-同舟云学术

A Zariski--Nagata theorem for smooth ℤ-algebras

Published:2018-05-16 Issue:761 Volume:2020 Page:123-140
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

De Stefani Alessandro¹^ORCID,Grifo Eloísa²^ORCID,Jeffries Jack³

Affiliation:

1. Department of Mathematics , University of Nebraska , Lincoln , NE 68588-0130 , USA

2. Department of Mathematics , University of Virginia , Charlottesville , VA 22904-4135 , USA

3. Department of Mathematics , University of Michigan , Ann Arbor , MI 48109-1043 , USA

Abstract

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.

Funder

National Science Foundation

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2018-0012/pdf

Reference20 articles.

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5. A. Buium, Differential characters of abelian varieties over p-adic fields, Invent. Math. 122 (1995), no. 2, 309–340.

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