arXiv subjects

Justin Lyle

Publications and source records attributed to Justin Lyle.

At least 19 recordsLinked to original sources

arXiv2026

The residue field as an infinite syzygy

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian ring which is either local or positively graded over the field $k$. We provide a method based on liftings for constructing free coresolutions of $k$ over $R$. We apply this method to twisted trivial extensions of the node $k[u,v]/(uv)$ to build a family of positively graded $k$-algebras of dimension $1$ over which $k$ is an infinite graded syzygy, resolving an open question of Iyengar in the negative. We also prove a formula for alternating sums of lengths of Ext modules when $R$ is positively graded, and use it to show that a graded free coresolution of $k$ cannot dualize to a resolution unless $R$ is Artinian Gorenstein.

math.AC↗

arXiv2026

On End-Semidualizing Modules

We introduce and study the notion of End-semidualizing modules which are generalizations of semidualizing modules in the classical sense, and also extend the notion of $S-R$ semidualizing bimodules studied by Holm-White and Wakamatsu. By constrast with these notions, End-semidualizing modules exist abundantly. We characterize these modules by the inclusion of projective and flat modules in their Auslander class, or injective modules in their Bass class, and give criteria for when these Foxby classes satisfy the two-out-of-three property. Over commutative Noetherian rings, we prove base-change and duality results, and characterize local depth inequalities by the inclusion of all modules of finite flat or injective dimension in the corresponding classes.

math.AC↗

arXiv2026

Frobenius multiplicities and F-signatures of Grassmannians of two-planes

Raedschelders, Špenko and Van den Bergh proved that the Plücker coordinate ring $C_n$ of the Grassmannian $\operatorname{Gr}(2,n)$ has finite Frobenius representation type when the characteristic $p$ satisfies $p\ge\max\{n-2,3\}$, and they determined the indecomposable summands of its Frobenius pushforwards up to nonzero multiplicity. We give nonrecursive finite formulas for every entry of the Frobenius multiplicity matrix and prove that each entry is a polynomial in $p$ of degree at most $2n-3$. Consequently the $F$-signature of $C_n$, and the asymptotic proportion of every indecomposable summand, is a single rational function of $p$ throughout the characteristic range. Its limit as $p\to\infty$ is the leading coefficient of the first free multiplicity, which we evaluate in closed form through Bernoulli numbers. For $\operatorname{Gr}(2,4)$ we find $s_p(C_4)=(13p^2+8)/(5(3p^2+2))$, and we calculate the analogous rational functions for $\operatorname{Gr}(2,5)$ and $\operatorname{Gr}(2,6)$ with computer assistance. We also construct the full characteristic polynomial from the Riemann-Roch weights and two residual matrices whose entries have degree at most $n-1$ in $p$. In the sequel, we use this work to show rank-two determinantal rings have FFRT, and to calculate their $F$-signatures.

math.AC↗

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 $\operatorname{tr}_R(Ω^1_R(I))$, 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 $\operatorname{tr}_R(M) \subseteq \operatorname{tr}_R(Ω^1_R(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which $\operatorname{tr}_R(Ω^1_R(\mathfrak{c}_R))=\mathfrak{c}_R$.

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

On the totally acyclic condition and the Auslander-Reiten Conjecture

We prove new cases of the Auslander-Reiten conjecture under the assumption that the base ring satisfies the totally acyclic condition $(\mathbf{tac})$. A consequence of our work is that Auslander-Reiten conjecture holds in the graded setting whenever $R$ is Gorenstein and reduced. We also give an example of a local ring $R$ which does not satisfy $(\mathbf{tac})$ but for which the condition $\operatorname{Ext}^{i>0}_R(M,R)=0$ implies $M$ is totally reflexive for any finitely generated $R$-module $M$.

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↗

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

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

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

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↗

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 $ω_R$. 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 $\mathfrak{m} \operatorname{Ext}_R^i(ω_R,R)$ should force the trace ideal of $ω_R$ 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↗

arXiv2024

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↗

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

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 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↗