arXiv subjects

Thomas Polstra

Publications and source records attributed to Thomas Polstra.

At least 19 recordsLinked to original sources

arXiv2025

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

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq\mathbb{P}^n_k$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H \subseteq \mathbb{P}^n_k$ is a general hyperplane, then $X \cap H$ is also $F$-rational and $F$-pure. Of related but independent interest, we present a relationship between the characteristic and index of a $\mathbb{Q}$-Gorenstein variety with isolated non-$F$-regular locus which is $F$-pure but not $F$-regular.

math.AG↗

arXiv2025

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

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 examples of two-dimensional local geometrically normal domains which are $F$-injective but not $F$-anti-nilpotent. A crucial theme of our constructions is the behavior of $F$-injectivity along a purely inseparable finite base change.

math.AC↗

arXiv2024

Strong $F$-regularity and the Uniform Symbolic Topology Property

We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain $R$. Let $R$ be a normal domain of prime characteristic $p>0$ that is $F$-finite or essentially of finite type over an excellent local ring. Assume there exists a finite extension $R\to S$ so that the non-strongly $F$-regular locus of $\mathrm{Spec}(S)$ consists only of isolated points, then there exists a constant $C$ such that for all ideals $I \subseteq R$ and $n \in \mathbb{N}$, the symbolic power $I^{(Cn)}$ is contained in the ordinary power $I^n$. In other words, $R$ enjoys the Uniform Symbolic Topology Property. Moreover, if $R$ is $F$-finite and strongly $F$-regular, then $R$ enjoys a property that is proven to be stronger: there exists a constant $e_0 \in \mathbb{N}$ such that for any ideal $I \subseteq R$ and all $e \in \mathbb{N}$, if $x \in R \setminus I^{[p^e]}$, then there exists an $R$-linear map $\varphi: F^{e+e_0}_*R \to R$ such that $\varphi(F^{e+e_0}_*x) \notin I$.

math.AC↗

arXiv2024

Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property

We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain essentially of finite type over a field, there exists a constant $C$ so that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mbox{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.

math.AC↗

arXiv2023

On $F$-pure inversion of adjunction

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 under $\mathbb{Q}$-Gorenstein assumptions but without divisibility restrictions on the index. We also give a detailed analysis relating the $F$-purity of the pairs $(R,\Delta + D)$ and that of $(R_D, \text{Diff}_D(\Delta))$ motivated by Kawakita's log canonical inversion of adjunction via reduction to prime characteristic.

math.AG↗

arXiv2023

On the Equality of Test Ideals

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly $F$-regular ring whose anti-canonical algebra is Noetherian on the punctured spectrum.

math.AC↗

arXiv2023

Primary Decompositions of Regular Sequences

Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence of elements in $R$. Then there exists a finite set of primes $\Lambda$ and natural number $C$ so that for all $n_1,\ldots,n_t$ there exists a primary decomposition $(x_1^{n_1},\ldots,x_t^{n_t})=Q_1\cap \cdots \cap Q_\ell$ so that $\sqrt{Q_i}\in \Lambda$ and $\sqrt{Q_i}^{C(n_1+\cdots + n_t)}\subseteq Q_i$ for all $1\leq i\leq \ell$.

math.AC↗

arXiv2020

$F$-purity deforms in $\mathbb{Q}$-Gorenstein rings

We show that $F$-purity deforms in local $\mathbb{Q}$-Gorenstein rings of prime characteristic $p>0$. Furthermore, we show that $F$-purity is $\mathfrak{m}$-adically stable in local Cohen-Macaulay $\mathbb{Q}$-Gorenstein rings.

math.AC↗

arXiv2020

Compatible ideals in Gorenstein rings

Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$ such that the ideal $I$ is the sum of images of all $R$-linear maps $S\to R$.

math.AC↗

arXiv2020

A theorem about maximal Cohen-Macaulay modules

It is shown in a local strongly $F$-regular ring there exits natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen-Macaulay module then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

math.AC↗

arXiv2019

Continuity of Hilbert-Kunz multiplicity and F-signature

We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\m$-adic topology.

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

Equimultiplicity Theory of Strongly $F$-regular rings

We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article provide a unified approach to the study of these $F$-invariants under localization and as measurements of singularities.

math.AC↗

arXiv2019

Local cohomology bounds and test ideals

We find sufficient conditions which imply equality of the finitistic test ideal and test ideal in rings of prime characteristic. Utilizing recent progress from the prime characteristic minimal model program we equate the notions of $F$-regular and strongly $F$-regular for 4-dimensional rings essentially of finite type over a field of prime characteristic $p>5$.

math.AC↗

arXiv2019

Covers of rational double points in mixed characteristic

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.

math.AG↗

arXiv2019

Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has $F$-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.

math.AC↗

arXiv2018

Global Frobenius Betti numbers and F-splitting ratio

We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair $(R,\mathscr{D})$, where $\mathscr{D}$ is a Cartier algebra, is equivalent to the positivity of the global F-signature ${\rm s}(R,\mathscr{D})$ of the pair. This extends a result previously proved by these authors, by removing an extra assumption on the Cartier algebra.

math.AC↗

arXiv2018

$F$-signature under birational morphisms

We study $F$-signature under proper birational morphisms $π: Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq π^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.

math.AG↗