Austyn Simpson

activity

20192026

most cited

F

-nilpotent rings and permanence properties

9 citations · 11 across the 12 of their papers we have counts for

collaborators

14 papers

math.AC2026

$F$ -injectivity does not deform

Karl Schwede, Austyn Simpson

We show that there exists an $F$ -finite four-dimensional local domain $(R, m)$ of characteristic two which is not $F$ -injective but which admits a nonzerodivisor $f\in \m…

math.AC2026

A Buchsbaum theory for Frobenius closure

Kriti Goel, Kyle Maddox, Lance Edward Miller +2

We give a partial characterization for when the difference $e (q) - ℓ_{R} (R / q^{F})$ is independent of the choice of parameter ideal $q \subseteq R$ in a…

math.AG2025

Bertini's theorem for $F$ -rational $F$ -pure singularities

Alessandro De Stefani, Thomas Polstra, Austyn Simpson

Let $k$ be an algebraically closed field of characteristic $p > 0$ , and let $X \subseteq P_{k}^{n}$ be a quasi-projective variety that is $F$ -rational and $F$ -pure. We prove that…

math.AC2025

Hilbert-Kunz multiplicity and $F$ -signature can disagree

Seungsu Lee, Suchitra Pande, Austyn Simpson

We compute the $F$ -signature function of the ample cone of any nontrivial ruled surface over $P_{k}^{1}$ where $k$ is an algebraically closed field of prime characteristic. As…

math.AC2025

$F$ -injectivity does not imply $F$ -fullness in normal domains

Alessandro De Stefani, Thomas Polstra, Austyn Simpson

We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are $F$ -injective but not $F$ -full. Along the way, we find ex…

math.AC2025

On deformation of perfectoid purity in Gorenstein domains

Benjamin Baily, Karina Dovgodko, Austyn Simpson +1

If $(R, m)$ is a complete local ring of mixed characteristic $(0, p)$ and $R / pR$ is an $F$ -pure Gorenstein domain, we find a sufficient condition for $R$ to be perfectoid…

math.AC2023

Uniform arithmetic in local rings via ultraproducts

Clay Adams, Francesca Cantor, Anese Gashi +4

We reinterpret various properties of Noetherian local rings via the existence of some $n$ -ary numerical function satisfying certain uniform bounds. We provide such characterization…

math.AC2023

ACC for $F$ -signature: a likely counterexample

Clay Adams, Theodore J. Sandstrom, Austyn Simpson

Let $k = F_{2}$ and let $0 \neq = α \in k$ . We present a conjecture supported by computer experimentation involving the Brenner-Monsky quartic $g_α=α…

math.AC2023

Flat morphisms with regular fibers do not preserve $F$ -rationality

Eamon Quinlan-Gallego, Austyn Simpson, Anurag K. Singh

For each positive prime integer $p$ we construct a standard graded $F$ -rational ring $R$ , over a field $K$ of characteristic $p$ , such that $R \otimes_{K} K$ is not $F$ -rati…

math.AG2023

On $F$ -pure inversion of adjunction

Thomas Polstra, Austyn Simpson, Kevin Tucker

We analyze adjunction and inversion of adjunction for the $F$ -purity of divisor pairs in characteristic $p > 0$ . In this vein, we give a complete answer for principal divisors unde…

math.AC2022

On Localization of Tight Closure in Line- $S_{4}$ Quartics

Levi Borevitz, Naima Nader, Theodore J. Sandstrom +3

Building on work of Brenner and Monsky from 2010 and on a Hilbert-Kunz calculation of Monsky from 1998, we exhibit a novel example of a hypersurface over $F_{2}$ …

math.AC2020★ 2 cited

$F$ -purity deforms in $Q$ -Gorenstein rings

Thomas Polstra, Austyn Simpson

We show that $F$ -purity deforms in local $Q$ -Gorenstein rings of prime characteristic $p > 0$ . Furthermore, we show that $F$ -purity is $m$ -adically stable in loca…

math.AC2020★ 9 cited

$F$ -nilpotent rings and permanence properties

Jennifer Kenkel, Kyle Maddox, Thomas Polstra +1

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…

math.AG2019

Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

Rankeya Datta, Austyn Simpson

Let $k$ be an algebraically closed field of characteristic $p > 0$ . We show that if $X \subseteq P_{k}^{n}$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less…

F-injectivity does not deform

A Buchsbaum theory for Frobenius closure

Bertini's theorem for F-rational F-pure singularities

Hilbert-Kunz multiplicity and F-signature can disagree

F-injectivity does not imply F-fullness in normal domains

On deformation of perfectoid purity in Gorenstein domains

Uniform arithmetic in local rings via ultraproducts

ACC for F-signature: a likely counterexample

Flat morphisms with regular fibers do not preserve F-rationality

On F-pure inversion of adjunction

On Localization of Tight Closure in Line-S4​ Quartics

F-purity deforms in Q-Gorenstein rings

F-nilpotent rings and permanence properties

Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

$F$ -injectivity does not deform

Bertini's theorem for $F$ -rational $F$ -pure singularities

Hilbert-Kunz multiplicity and $F$ -signature can disagree

$F$ -injectivity does not imply $F$ -fullness in normal domains

ACC for $F$ -signature: a likely counterexample

Flat morphisms with regular fibers do not preserve $F$ -rationality

On $F$ -pure inversion of adjunction

On Localization of Tight Closure in Line- $S_{4}$ Quartics

$F$ -purity deforms in $Q$ -Gorenstein rings

$F$ -nilpotent rings and permanence properties