A Zariski--Nagata theorem for smooth ℤ-algebras
A Zariski--Nagata theorem for smooth ℤ-algebras
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A Zariski--光滑 ℤ-代数的永田定理
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发表时间：
2017
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通讯作者：
J. Jeffries
中科院分区：
文献类型：
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作者：
Alessandro De Stefani;Eloísa Grifo;J. Jeffries
在完全域上的多项式环中，素理想的符号幂可以通过微分算子来描述：Zerkiki和Nagata的一个经典结果说，给定素理想的n次符号幂由相应簇上直到n阶为零的元素组成。然而，这种描述在混合特性方面失败。本文利用Buium和Joyal提出的p-导子概念，定义了一类新的混合特征微分幂，并证明了这类新对象与素理想的符号幂是一致的.这似乎是第一个应用p-导子交换代数。
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.