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Mordechai Katzman

Publications and source records attributed to Mordechai Katzman.

At least 19 recordsLinked to original sources
arXiv2026

Segre-Determinantal Loci and the Image Variety for Three Flatland Cameras

Motivated by applications of algebraic geometry to reconstruction problems in computer vision, we initiate a study of the equations of degeneracy loci associated with linearly dependent points on Segre varieties. When these points are constrained to lie on a common hyperplane, we prove that the vanishing ideals of these loci are prime, Cohen-Macaulay, and generated by the natural maximal minors, and that these minors form a universal Gr\"{o}bner basis.

math.AG↗
arXiv2025

A note on freeness

In this brief note we show that for a field extension $K/F$, $S=K[\![\mathbf{x}]\!]$ is a free $R=F[\![\mathbf{x}]\!]$-module precisely when $K/F$ is finite. We then raise the question \emph{what is the projective dimension of $S$?}

math.AC↗
arXiv2024

Stanley-Reisner Ideals with Pure Resolutions

This paper investgates Stanley-Reisner ideals with pure resolutions. We first describe two infinite families of such ideals associated to highly symmetric complexes. We then prove a partial analogue to the first Boij-S\"oderberg Conjecture for Stanley-Reisner ideals, by detailing an algorithm for constructing Stanley-Reisner ideals with pure Betti diagrams of any given shape, save for an initial shift.

math.AC↗
arXiv2022

Lyubeznik numbers, $F$-modules and modules of generalized fractions

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring $R$ of prime characteristic. The methods used employ Lyubeznik's $F$-modules over $R$, particularly his $F$-finite $F$-modules, and also the modules of generalized fractions of Sharp and Zakeri. It is shown that many modules of generalized fractions over $R$ have natural structures as $F$-modules; these lead to $F$-module structures on certain local cohomology modules over $R$, which are exploited, in conjunction with $F$-module structures on injective $R$-modules that result from work of Huneke and Sharp, to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2.

math.AC↗
arXiv2019

The TestIdeals package for Macaulay2

This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given $p^{-e}$-linear map.

math.AC↗
arXiv2018

FFRT Properties of hypersurfaces and their F-signature

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite $F$-representation Type (FFRT) and we compute the $F$-signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations.

math.AC↗
arXiv2018

Multiplicity bounds in prime characteristic

We extend a result by Huneke and Watanabe bounding the multiplicity of $F$-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen-Macaulay ring of prime characteritic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen-Macaulay rings in characteristic zero which have dense $F$-injective type.

math.AC↗
arXiv2018

Strong $F$-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.

math.AC↗
arXiv2017

Global parameter test ideals

This paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings. We do this for Cohen-Macaulay localizations (resp., completions) of non-local rings, for generalized Cohen-Macaulay rings, and for non-local rings with isolated non Cohen-Macaulay points, each being an isolated non $F$-rational point. The tools used to prove this results are constructive in nature and as a consequence our results yield algorithms for the computation of these global parameter test ideals. Finally, we illustrate the power of our methods by analyzing the HSL numbers of local cohomology modules with support at any prime ideal.

math.AC↗
arXiv2017

$D$-module and $F$-module length of local cohomology modules

Let $R$ be a polynomial or power series ring over a field $k$. We study the length of local cohomology modules $H^j_I(R)$ in the category of $D$-modules and $F$-modules. We show that the $D$-module length of $H^j_I(R)$ is bounded by a polynomial in the degree of the generators of $I$. In characteristic $p>0$ we obtain upper and lower bounds on the $F$-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of $R/I$. The obtained upper bound is sharp if $R/I$ is an isolated singularity, and the lower bound is sharp when $R/I$ is Gorenstein and $F$-pure. We also give an example of a local cohomology module that has different $D$-module and $F$-module lengths.

math.AC↗
arXiv2017

The support of local cohomology modules

We describe the support of $F$-finite $F$-modules over polynomial rings $R$ of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analyzed. To the best of our knowledge, this is the first algorithm to avoid extensive use of Gröbner bases and hence of substantial practical value. We also use the idea behind this algorithm to prove that the support of $H^j_I(S)$ is Zariski closed for each ideal $I$ of $S$ where $R$ is noetherian commutative ring of prime characteristic with finitely many isolated singular points and $S=R/gR$ ($g\in R$).

math.AC↗
arXiv2014

An extension of a theorem of Hartshorne

We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal primes of a local ring.

math.AC↗
arXiv2013

Annihilators of Artinian modules compatible with a Frobenius map

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field. The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the $F$-finite case.

math.AC↗
arXiv2013

Rings of Frobenius operators

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

math.AC↗
arXiv2012

Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings

In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exists for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $2\times 2$ minors in a $2\times 3$ matrix of indeterminates form discrete sets.

math.AC↗
arXiv2012

An algorithm for computing compatibly Frobenius split subvarieties

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $ϕ: F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $ϕ$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $ϕ$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.

math.AC↗
arXiv2011

Some properties and applications of $F$-finite $F$-modules

The purpose of this paper is to describe several applications of finiteness properties of $F$-finite $F$-modules recently discovered by M. Hochster to the study of Frobenius maps on injective hulls, Frobenius near-splittings and to the nature of morphisms of $F$-finite $F$-modules. Among the results in the paper we show that morphisms of $F$-finite $F$-modules have a particularly simple form, and we show that certain Frobenius near-splittings have finitely many compatible submodules, thus generalizing a result of M. Blickle and G. Böckle to the non-$F$-finite case.

math.AC↗