Searcharxiv⌕ Search

arXiv subjects

Lance Edward Miller

Publications and source records attributed to Lance Edward Miller.

At least 19 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

Arithmetic partial differential operators on ramified extensions of $\ZZ_p$

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_π$ with uniformizer $π$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $π$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.

math.NT↗
arXiv2026

Derived jet and arc spaces

We study jet schemes and arc spaces in the context of derived algebraic geometry. Explicitly, we consider the jet and arc functors in the category of schemes and study their animations to the category of derived schemes -- what we call the derived jet and arc spaces. We show that the derived constructions agree with the classical versions when the base scheme is smooth, or more generally for local complete intersection log canonical singularities, giving a derived interpretation to a theorem of Mustaţă. For more singular spaces we get new singularity invariants in the form of higher homotopy groups. We also study cotangent complexes for derived jet and arc spaces, generalizing previous formulas for sheaves of differentials of classical jet and arc spaces. Several applications are obtained. Specifically, we revisit recent results on the local structure of arc spaces from the lens of cotangent complexes, giving more unified proofs and removing unnecessary hypotheses. In particular, we extend a version of Reguera's curve selection lemma for arc spaces to the case of non-perfect base fields.

math.AG↗
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

Higher dimensional geometry of $p$-jets

In this work, we prove a quantitative version of the prime-to-$p$ Manin--Mumford conjecture for varieties with ample cotangent bundle. More precisely, let $A$ be an abelian variety defined over a number field $F$, and let $X$ be a smooth projective subvariety of $A$ with ample cotangent bundle. We prove that for every prime $p\gg 0$, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is finite and explicitly bounded by a summation involving cycle classes in the Chow ring of the reduction of $X$ modulo $p$. This result is a higher dimensional analogue of Buium's quantitative Manin--Mumford for curves. Our proof follows a similar outline to Buium's in that it heavily relies on his theory of arithmetic jet spaces. In this context, we prove that the special fiber of the arithmetic jet space associated to a model of $X$ is affine as a scheme over $\mathbb{F}_p^{\text{alg}}$. As an application of our results, we use a result of Debarre to prove that when $X$ is $\mathbb{Q}^{\text{alg}}$-isomorphic to a complete intersection of $c > \text{dim}(A)/2$ many general hypersurfaces of $A_{\mathbb{Q}^{\text{alg}}}$ of sufficiently large degree, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is bounded by a polynomial that depends only on $p$, the dimension of the ambient abelian variety, and intersection numbers of certain products of the hypersurfaces.

math.AG↗
arXiv2023

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↗
arXiv2022

Differentially fixed ideals in toric varieties

This article concerns monomial ideals fixed by differential operators of affine semi-group rings over $\mathbb{C}$. We give a complete characterization of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.

math.AC↗
arXiv2022

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↗
arXiv2022

Arithmetic differential geometry in the arithmetic PDE setting, I: connections

This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois group of a $p$-adic field. Existence and uniqueness of geodesics and of Levi-Civita and Chern connections are proved in this context. In a sequel to this paper a theory of arithmetic Riemannian curvature and characteristic classes will be developed.

math.NT↗
arXiv2021

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. We also prove structure results for the spaces of "PDE differential modular forms defined on the ordinary locus." We also produce a system of differential equations satisfied by our PDE modular forms based on Serre and Euler operators.

math.NT↗
arXiv2019

Perfectoid spaces arising from arithmetic jet spaces

Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $δ$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $δ$-morphisms appearing in the theory such as the $δ$-characters of elliptic curves and the $δ$-period map on modular curves.

math.NT↗
arXiv2019

Singularities of Rees-like Algebras

Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conjecture. Their construction involved two new ideas: Rees-like algebras and step-by-step homogenization. Yet, all of these varieties are singular and the nature of the geometry of these projective varieties was left open. The purpose of this paper is to study the singularities inherent in this process. We compute the codimension of the singular locus of an arbitrary Rees-like algebra over a polynomial ring. We then show that the relative size of the singular locus can increase under step-by-step homogenization. To address this defect, we construct a new process, we call prime standardization, which plays a similar role as step-by-step homogenization but also preserves the codimension of the singular locus. This is derived from ideas of Ananyan and Hochster and we use this to study the regularity of certain smooth hyperplane sections of Rees-like algebras, showing that they all satisfy the Eisenbud-Goto Conjecture, as expected. On a more qualitative note, while Rees-like algebras are almost never Cohen-Macaulay and never normal, we characterize when they are seminormal, weakly normal, and, in positive characteristic, F-split. Finally, we construct a finite free resolution of the canonical module of a Rees-like Algebra over the presenting polynomial ring showing that it is always Cohen-Macaulay and has a surprising self-dual structure.

math.AC↗
arXiv2018

Witt differentials in the h-topology

Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.

math.AC↗
arXiv2017

The s-multiplicity function of 2x2-determinantal rings

This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $λ( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.

math.AC↗
arXiv2017

On Lower Bounds for $s$-multiplicities

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.

math.AC↗