2018
DOI: 10.1515/crelle-2018-0012
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A Zariski--Nagata theorem for smooth 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 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 … Show more

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Cited by 11 publications
(13 citation statements)
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“…Finally, the last example shows that the condition of separability in Theorem 4.6(ii) cannot be avoided (also, see [7,Example 3.8]).…”
Section: Some Examples and Computationsmentioning
confidence: 99%
“…Finally, the last example shows that the condition of separability in Theorem 4.6(ii) cannot be avoided (also, see [7,Example 3.8]).…”
Section: Some Examples and Computationsmentioning
confidence: 99%
“…Another motivation comes from [9], where the authors use higher order differential operators to measure various kind of singularities in all characteristics. These higher order operators also play a key role in recent developments in the study of symbolic powers of ideals (see [10] and [9, Section 10] for details). We hope that the calculation of the level of a pair of polynomials might help in the understanding of these differential operators.…”
Section: > > < > >mentioning
confidence: 99%
“…One less-understood notion of ideal powers is the differential power of an ideal. The nth differential power of an ideal I, with respect to a base ring A, is denoted I n A and defined using the ring of A-linear differential operators on R. Differential powers of ideals have recently seen a renewed interest, having been used to been used to generalize the celebrated Zariski-Nagata Theorem ([DDSG + 18], [DSGJ20]). For prime ideals in polynomial rings of characteristic zero, the differential power of an ideal coincides with the symbolic power of an ideal [DDSG + 18]; one can even use a variant of the differential power to extend this result to any algebra essentially of finite type over a perfect field [CR21,Theorem 4.6].…”
Section: Introductionmentioning
confidence: 99%