A Zariski - Nagata theorem for smooth ℤ-algebras
TLDR
In this article, the authors 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.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.read more
Citations
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Journal ArticleDOI
Quantifying singularities with differential operators
TL;DR: The F-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties, such as non-singularity and strong F-regularity.
Posted Content
Quantifying singularities with differential operators.
TL;DR: A numerical invariant for rings of characteristic zero or p>0 that exhibits many of the useful properties of the F-signature and a number of results on symbolic powers and Bernstein-Sato polynomials are obtained.
Journal ArticleDOI
Noetherian operators, primary submodules and symbolic powers
TL;DR: In this paper, the authors give an algebraic and self-contained proof of the existence of Noetherian operators for primary submodules over general classes of noetherian commutative rings.
Dissertation
Uniform Symbolic Topologies in Non-Regular Rings
TL;DR: In particular, for rational double point surface singularities over C, it was shown in this article that USTP solidarity is impossible for rational regular rings, and for rational toric rings, it is shown that the class of excellent regular rings enjoys class solidarity relative to uniform symbolic topology property.
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Symbolic analytic spread: upper bounds and applications
Hailong Dao,Jonathan Montaño +1 more
TL;DR: In this paper, the authors defined the symbolic analytic spread of an ideal in terms of the rate of growth of the minimal number of generators of its symbolic powers, and provided upper bounds for the analytic spread under certain conditions, including the Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity.
References
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Éléments de géométrie algébrique
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Book
Commutative Ring Theory
Hideyuki Matsumura,Miles Reid +1 more
TL;DR: In this article, the authors introduce the notion of complete local rings and apply it to a wide range of applications, including: I-smoothness, I-flatness revisited, and valuation rings.
Journal ArticleDOI
Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
Journal ArticleDOI
Uniform bounds and symbolic powers on smooth varieties
TL;DR: In this paper, it was shown how one can use multiplier ideals to establish uniform bounds on the multiplicative behavior of certain families of ideal sheaves on a smooth algebraic variety.