arXiv subjects

Justin Lyle

Publications and source records attributed to Justin Lyle.

At least 19 recordsLinked to original sources

arXiv2026

Trace ideals of syzygies

We study the behavior of trace ideals under taking syzygies. In particular, when $R$ is numerical semigroup ring and $I$ is a homogeneous ideal in $R$, we obtain an upper estimate for $\operatorname{tr}_R(\Omega^1_R(I))$, and we show this estimate is sharp when $I$ is the conductor ideal $\mathfrak{c}_R$ of $R$. Using this result, we characterize the numerical semigroup rings for which $\operatorname{tr}_R(M) \subseteq \operatorname{tr}_R(\Omega^1_R(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which $\operatorname{tr}_R(\Omega^1_R(\mathfrak{c}_R))=\mathfrak{c}_R$.

math.AC↗

arXiv2026

A counterexample to the localization problem for AB rings

We construct a complete local Gorenstein ring $R$ of dimension $1$ with a prime ideal $\mathfrak{p}\in \operatorname{Spec}(R)$ such that $R$ is an AB ring, but the localization $R_{\mathfrak{p}}$ is not an AB ring. This settles the localization problem for AB rings posed by Huneke and Jorgensen in the negative.

math.AC↗

arXiv2026

Derived complete intersections and polynomial growth of Betti numbers over dg-algebras

A theorem of Gulliksen states that a local ring is a complete intersection if and only if the Betti numbers of its finitely generated modules grow polynomially. We prove a derived version of Gulliksen's Theorem. More precisely, we prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth. As a key ingredient in the proof, we establish the existence and uniqueness of minimal models and acyclic closures of morphisms of dg-algebras in a broader setting than was previously known. We also extend to dg-algebras a theorem of Halperin on the vanishing of deviations of local rings, recovering Gulliksen's Theorem as an immediate consequence.

math.AC↗

arXiv2026

The derived depth formula for modules of finite quasi-projective dimension

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous depth formula, a variation of the Derived Depth Formula for width, an extended version of Ischebeck's Formula, and a Dependency formula in the vein of Jorgensen. Several special cases of our main results are new even under stronger assumptions on the vanishing of various complete intersection dimensions.

math.AC↗

arXiv2025

Centers of Endomorphism Rings and Reflexivity

Let $R$ be a local ring and let $M$ be a finitely generated $R$-module. Appealing to the natural left module structure of $M$ over its endomorphism ring and corresponding center $Z(\operatorname{End}_R(M))$, we study when various homological properties of $M$ are sufficient to force $M$ to have a nonzero free summand. Consequences of our work include a partial converse to a well-known result of Lindo describing $Z(\operatorname{End}_R(M))$ when $M$ is faithful and reflexive, as well as some applications to the famous Huneke-Wiegand conjecture.

math.AC↗

arXiv2025

G-levels of perfect complexes

We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or Gorenstein injective $R$-modules, of $R$-complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated $R$-modules.

math.AC↗

arXiv2025

Ordinary and symbolic powers of matroids via polarization

In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goal: (i) recover and extend several well--known results in the literature, especially regarding Stanley--Reisner ideals of matroids, and (ii) provide short, elementary proofs for these results. Among them, we provide simple proofs of two celebrated results of Minh and Trung, Varbaro, and Terai and Trung elegantly characterizing the Cohen-Macaulay property, or even Serre's condition $(S_2)$, of symbolic and ordinary powers of squarefree monomial ideals in terms of their combinatorial (matroidal) structure. Our work relies on the interplay of several combinatorial and algebraic concepts, including dualities, polarizations, Serre's conditions, matroids, Hochster-Huneke graphs, vertex decomposability, and careful choices of monomial orders.

math.AC↗

arXiv2025

On the depth of tensor products over Cohen-Macaulay rings

Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions.

math.AC↗

arXiv2024

Annihilators of (co)homology and their influence on the trace Ideal

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $\omega_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of $\mathfrak{m} \operatorname{Ext}_R^i(\omega_R,R)$ should force the trace ideal of $\omega_R$ to contain $\mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting.

math.AC↗

arXiv2023

Generalized trace submodules and centers of endomorphism rings

Let $R$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. We introduce a general form of the classically studied trace map that unifies several notions from the literature. We develop a theory around these objects and use it to provide a broad extension of a result of Lindo calculating the center of $\operatorname{End}_R(M)$. As a consequence, we show under mild hypotheses that in dimension $1$, the canonical module of $Z(\operatorname{End}_R(M))$ may be calculated as the trace submodule of $M$ with respect to the canonical module of $R$.

math.AC↗

arXiv2023

On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity

This paper provides a method to get a noetherian equicharacteristic local UFD with an isolated singularity from a given noetherian complete equicharacteristic local ring, preserving certain properties. This is applied to invesitgate the (non)vanishing of Ext modules. It is proved that there exist a Gorenstein local UFD $A$ having an isolated singularity such that $\operatorname{Ext}_A^{\gg0}(M,N)=0$ does not imply $\operatorname{Ext}_A^{\gg0}(N,M)=0$, a Gorenstein local UFD $B$ having an isolated singularity such that $\operatorname{Tor}_{>0}^B(M,N)=0$ does not imply $\operatorname{depth}(M\otimes_B N)=\operatorname{depth} M+\operatorname{depth} N-\operatorname{depth} B$, and a Cohen-Macaulay local UFD $C$ having an isolated singularity such that $\operatorname{Ext}_C^{>0}(M,C)=0$ does not imply the total reflexivity of $M$.

math.AC↗

arXiv2023

Endomorphism algebras over commutative rings and torsion in self tensor products

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where $\operatorname{End}_R(M)$ has an $R^*$-algebra structure, and prove that if $M$ is indecomposable, then $M \otimes_{\operatorname{End}_R(M)} M$ must always have torsion in this case under mild hypotheses.

math.AC↗

arXiv2023

On a generalization of Ulrich modules and its applications

We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications observed in the literature. Additionally, we reveal their significance as obstructions to Cohen-Macaulay properties of tensor products. Leveraging this insight, we show the utility of these modules in testing the finiteness of homological dimensions across various scenarios.

math.AC↗

arXiv2020

Exterior powers and Tor-persistence

A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.

math.AC↗

arXiv2019

Minimal Cohen-Macaulay Simplicial Complexes

We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball, the dunce hat, and recently discovered non-partitionable Cohen-Macaulay complexes. We further provide various ways to construct such complexes.

math.CO↗

arXiv2019

Extremal growth of Betti numbers and trivial vanishing of (co)homology

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke.

math.AC↗

arXiv2019

Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum

We say that a Cohen-Macaulay local ring has finite $\operatorname{\mathsf{CM}}_+$-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite $\operatorname{\mathsf{CM}}_+$-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite $\operatorname{\mathsf{CM}}_+$-representation type are exactly the local hypersurfaces of countable $\mathsf{CM}$-representation type, that is, the hypersurfaces of type $(\mathrm{A}_\infty)$ and $(\mathrm{D}_\infty)$. We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite $\operatorname{\mathsf{CM}}_+$-representation type.

math.AC↗

arXiv2018

Rank Selection and Depth Conditions for Balanced Simplicial Complexes

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that rank selected subcomplexes of balanced simplicial complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial compex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.

math.AC↗