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.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS #1606353
Funding statement: The third author was partially supported by NSF Grant DMS #1606353.
Acknowledgements
We would like to thank Alexandru Buium, Bhargav Bhatt, and Craig Huneke for helpful conversations and suggestions. We thank the anonymous referees for many helpful comments and suggestions. We thank Holger Brenner and Luis Núñez-Betancourt for allowing us to include a proof of Lemma 3.4. We would like to thank Rankeya Datta for bringing the error in Lemma 2.1 of a previous version of the article to our attention.
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