arXiv subjects

Kyle Maddox

Publications and source records attributed to Kyle Maddox.

10 recordsLinked to original sources

arXiv2026

$F$-depth and $F$-nilpotent rings: generalizations and applications

Computation of the Frobenius closure of ideals in rings of prime characteristic is a difficult problem. For a given ideal, its Frobenius test exponent provides a valuable degree of uniformity in performing the calculation. Hence, it is desirable to find uniform upper bounds on the Frobenius test exponent of ideals. Even in nice rings of low dimension, Brenner showed considering the collection of all ideals is generally hopeless, but for Cohen-Macaulay rings, Katzman-Sharp there are uniform upper bounds on the Frobenius test exponent for the class of parameter ideals. Subsequent efforts in controlling the Frobenius test exponents have typically involved studying the Frobenius action on local cohomology and the degree to which this action is nilpotent. The goal of this survey article is to examine the history of the Frobenius test exponent problem and its relationships to singularity types for local rings defined in terms of the Frobenius action on local cohomology. We also explore several generalizations of these prior results to a setting where less nilpotence in the Frobenius action is assumed.

math.AC↗

arXiv2026

A Buchsbaum theory for Frobenius closure

We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$. Here, $\mathfrak{q}^F$ is the Frobenius closure of $\mathfrak{q}$ and $e(\mathfrak{q})$ denotes the Hilbert--Samuel multiplicity of $\mathfrak{q}$. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma-Quy in the setting of tight closure.

math.AC↗

arXiv2025

The weak normalization of an affine semigroup

In this article, we define and explore the weak normalization of an affine semigroup. In particular, for a fixed prime integer, we provide a geometric description of the weak normalization of an affine semigroup with respect to that prime, which corresponds to the weak normalization of the affine semigroup ring over a field of that prime characteristic, similar to the description of the seminormalization of an affine semigroup given by Reid-Roberts. We then use this description to understand the singularities of an affine semigroup ring defined over a field of prime characteristic and provide several examples. In particular, we demonstrate that all affine semigroup rings defined over fields of prime characteristic have a uniform upper bound on the Frobenius test exponent of all ideals, which provides a large and important class of examples with a positive answer to a question of Katzman-Sharp on uniformity of Frobenius test exponents. Finally, we provide an algorithm and implementation to compute the weak normalization of an affine semigroup, as well as the Frobenius test exponent and Frobenius closures of ideals in the affine semigroup ring.

math.AC↗

arXiv2024

New versions of Frobenius and integral closure of ideals

In this article, we define three new operations on ideals which generalize integral closure and Frobenius closure of ideals, whose definitions incorporate an auxiliary ideal and a real parameter. These additional ingredients are common in adjusting old definitions of ideal closures in order to generalize them to pairs, with an eye towards further applications in algebraic geometry. In the case of tight closure, similar generalizations exist due to N. Hara and K.I. Yoshida, as well as A. Vraciu, and in the case of Frobenius closure, to K. Schwede. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, as applications, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and an $F$-nilpotent property for pairs.

math.AC↗

arXiv2023

Counting geometric branches via the Frobenius map and $F$-nilpotent singularities

We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes $F$-nilpotent curves. Further, we show that a reduced, local $F$-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to $F$-nilpotent affine semigroup rings.

math.AC↗

arXiv2022

Rees algebras and generalized depth-like conditions in prime characteristic

In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing a depth-like invariant which captures as special cases Lyubeznik's F-depth and the generalized F-depth from Maddox-Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F-nilpotence deforms.

math.AC↗

arXiv2021

Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

math.AC↗

arXiv2021

Generalized $F$-depth and graded nilpotent singularities

We address explicit constructions of new variants of $F$-nilpotent singularities. In particular, we explore how (generalized) weakly $F$-nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. From these results, explicit examples are produced and we provide bounds on their Frobenius test exponents. To accomplish these tasks, we introduce the {\it generalized $F$-depth} in analogy to Lyubeznik's $F$-depth. These depth-like invariants track (generalized) weakly $F$-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.

math.AC↗

arXiv2019

$F$-nilpotent rings and permanence properties

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local rings are open with respect to the Zariski topology whenever $R$ is $F$-finite or essentially of finite type over an excellent local ring.

math.AC↗

arXiv2019

A sufficient condition for finiteness of Frobenius test exponents

The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < \dim(R)$, then $\operatorname{Fte}(R)<\infty$.

math.AC↗