Papers

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen–Macaulay $R$-modules, introduced by Iyama–Wemyss and Iyama–Kalck–Wemyss–Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen–Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama–Kalck–Wemyss–Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.

Let $R$ be a commutative noetherian local ring. We extend the work of the author and Takahashi by investigating the structure of irreducible monomorphisms originating from free modules in the category of finitely generated $R$-modules. In the case where $R$ is complete, we further study the structure of irreducible monomorphisms and epimorphisms to and from injective modules in the category of artinian $R$-modules.

Given a point set $[n]$ and a collection $B$ of $k$-subsets of $[n]$, we create a monomial ideal whose Betti numbers completely determine whether the pair $([n],B)$ is a Steiner system $S(t,k,n)$. In particular, our construction characterizes Steiner triple systems and Steiner quadruple systems solely on the basis of Betti numbers of monomial ideals. Moreover, these monomial ideals allow us to reformulate the renowned prime power conjecture as follows: If a monomial ideal $M$ of $S = K[x_1, \dots, x_{q^2+q+1}]$ generated by squarefree monomials of degree $q+1$ has Betti numbers $\mathrm{b}_s(S/M) = {{q^2+q+1} \choose s} \text{ for all } s=0,\ldots,q$ then $q$ is a prime power.

We study asymptotic multiplicity sequences of graded families of arbitrary ideals in Noetherian local rings. We characterize convergence of the zeroth component and prove that the first positive component not forced to vanish always converges under natural equidimensionality and catenarity assumptions. In contrast, higher components may exhibit arbitrary asymptotic behavior: even in $k[[x,y]]$, integrally closed filtrations with fixed radical, height, and analytic spread can realize any prescribed lower and upper limits. For Noetherian graded filtrations, however, all asymptotic components exist, are finite and rational, and satisfy Rees-type and valuative criteria for integral dependence. In the reduced pure-dimensional complex analytic setting, a common principalizing modification provides a second convergence mechanism without finite-generation assumptions. We also develop a diagonal mixed theory: for graded $\mathfrak m$-primary families, the top diagonal components are sums of Cutkosky's mixed multiplicities, while on a common analytic model all diagonal mixed components converge. The latter result extends the one-family common-model theorem.

Let $R$ be a commutative ring and $\Phi$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $\Phi$-reduced and $\Phi$-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with $\Phi$ are determined by $\mathcal{L}$. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When $\Phi$ is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module-theoretic framework for studying generalized torsion and completion with respect to families of ideals.

In this paper we define a novel product on polyominoes called the tensor product of polyominoes. Using a special case of this product which we name a dilation, we construct a new class of polyominoes called the dilated closed paths. This class is a ”thick” generalization of Cisto and Navarra's class of closed paths (arXiv:2006.13935). We prove the Zig-Zag Walk Conjecture for this new class thereby fully characterizing primality for its associated polyomino ideals.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We classify linear systems of parameters of $S/I$ over any field $K$ using linear algebra and show explicit linear systems of parameters when $K$ has at least $n$ elements. If $I(G)$ is the edge ideal of a perfect graph $G$, a cycle or the complement of a cycle, we show that $S/I(G)$ has a 0-1 linear system of parameters, and for graphs with independence number equal to $2$, we characterize when $S/I(G)$ has a 0-1 linear system of parameters.

This article focuses on defining the Euler class group of stably free module and analyzes the condition for existence of unimodular element in corresponding projective module.This is an improvisation of work by Banerjee and Das, On a question of Nori:Obstructions,improvements and appliaction. If A is a smooth affine algebra over certain affaine algebras.

In this article, we extend the classical relationship between ADE singularities and their maximal Cohen-Macaulay modules to quotient singularities of types A and D. We identify projected hypersurfaces which allow us to construct explicit matrix factorizations, and in return, families of maximal Cohen-Macaulay modules. These hypersurfaces also yield the dual resolution graphs of the corresponding quotient singularities via Oka's method. Finally, we show that all special Cohen-Macaulay modules over the original quotient singularity can be recovered from those over the projected hypersurfaces.

In 2006, Katzman showed that the first six Betti numbers of the Stanley–Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let $\tau(d)$ be the least number of vertices of a flag complex whose $d$-th reduced integral homology has torsion. We prove that $\tau(d)\geq d+10$ for every $d\ge0$. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph $G$, the first seven Betti numbers of the edge ideal $I(G)$ are field independent.

Let $q=p^k$ be prime power, let $F= \mathbb{F}_q$ and let $V$ be the vector space of 2 by 2 matrices over $F$ with trace zero. Let $G = \mathrm{SL}_2(F)$. Then $G$ acts on $V$ via conjugation. Let $\Omega = S(V^*) \otimes \Lambda(V^*)$ be the algebra of differential forms on $V$. A minimal generating set for $\Omega^G$ when $q=3$ was computed by the author and Meyer. In this article we compute a minimal generating set for $\Omega^G$ for all $q$.

Let $I_h(a,b)$ be the ideal generated by the $h\times h$ permanents of a generic $a\times b$ matrix over a field of characteristic different from two. We prove that $\dim K[X]/I_h(a,b)=\max(a,b)(h-1)$ whenever $1\le h\le\min(a,b)$ and $h<\max(a,b)$. Consequently, the critical locus of the permanent of order $n$ has codimension $2n$ for every $n\ge 2$. We also prove that the maximal permanents of a generic $k\times(k+1)$ matrix generate a geometrically reduced complete intersection. The dimension argument uses a pivot expansion and cubic relations in the border variables. Two successive comparisons with highest-degree equations reduce the relevant fibers to a coordinate-support calculation. Reducedness follows from a separate regular-sequence argument and a dimension estimate excluding components over the exceptional locus.

We study generating sets for the kernels of the Frobenius polarization operators associated with elements of finite groups. By applying the discrete Fourier transform on each coset, we describe these kernels in terms of additive semigroups defined by zero-sum relations among roots of unity. We show that the monomials corresponding to the minimal nonzero elements of these semigroups form generating sets of minimum cardinality. We further describe the minimal nonzero elements in the general case in terms of those in the cyclic case and obtain a formula for the minimum number of generators. For the operator associated with a generator of a cyclic group, we also show, as a consequence of known results on integer zero-sum relations and circulant determinant representations, that its kernel and the subalgebra generated by these circulant determinants have the same field of fractions.

Let $(\mathcal{X},\mathcal{Y})$ be an admissible balanced pair in a locally presentable abelian category $\mathcal{A}$, and let $\mathcal{E}$ be its induced exact structure. We study the class $G(\mathcal{X})$ of cycles of two-sided complexes with terms in $\mathcal{X}$ that are both right and left $\mathcal{X}$-acyclic. If $\mathcal{X}$ is closed under direct summands and finite direct sums and is $\kappa$-Kaplansky for every sufficiently large regular cardinal $\kappa$, we prove that $(G(\mathcal{X}),G(\mathcal{X})^{\perp *})$ is a hereditary cotorsion pair relative to $\mathcal E$. For relative purity determined by finitely presented modules, we give two criteria for functorial completeness and show that relative periodic modules generate a functorially complete hereditary cotorsion pair. We also compute the relative Gorenstein objects, their right orthogonal, and explicit approximations for a family of exact structures over Dedekind domains.

We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound $\operatorname{deg}(TV) \leq \operatorname{deg}(V)^2$ for every smooth irreducible affine variety $V \subseteq \mathbb{A}^n$ and $TV\subseteq \mathbb{A}^{2n}$ its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.

We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel–Şahin–Singh–Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family \[ \Gamma_m(n)=\langle m,m+1,\ldots,\widehat{m+n},\ldots,2m-1\rangle, \qquad 2\leq n<m, \] where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity $m$ and $3\leq n<m$, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.

In the 1980s, Stanley and J. Watanabe proved that over a field of characteristic zero every monomial complete intersection has the strong Lefschetz property. Determining which monomial complete intersections in positive characteristic have the weak Lefschetz property has been the subject of ongoing research. In this paper, we give a complete description of the locus of exponent vectors corresponding to monomial complete intersections over a field of positive characteristic that fail the weak Lefschetz property. We describe this locus explicitly in terms of translates of positive-orthant lattice balls in the taxicab metric.

In this paper, using the equivalence of two extensions, we obtain an improved upper bound for the perturbation index, improving the bounds of Quy and N. V. Trung.

We prove that, over a commutative Noetherian ring, every finitely generated module of finite complete intersection dimension has finite quasi-projective dimension. Our proof adapts Bergh's technique of reducing complexity, originally used to establish virtual smallness for complexes of finite complete intersection dimension over local rings, by successively enlarging the perfect locus of the cones and controlling their homology. This gives an affirmative answer to part of a question of Jorgensen, Takahashi, and the third author. We also establish a dual injective version of our result for rings admitting a dualizing complex. Finally, we show that, over such rings, the three injective analogues of complete intersection dimension appearing in the literature coincide for bounded complexes with finitely generated homology, while two of them coincide even without a dualizing complex. We also give an example showing that the finite-generation hypothesis is necessary, thereby providing a full answer to a question of Sather-Wagstaff. We conclude by studying the localization behavior of these dimensions, obtaining a partial answer to a question of Sather-Wagstaff and Totushek.

In this paper we study whether there is a global to local principle for the homology of order complexes of finite lattices. We explore whether the non-vanishing of the homology of a lattice in degree $a+b-2$ forces the existence of a pair of lattice-complements whose open intervals below have non-vanishing homology in degrees $a$ and $b$ respectively. We consider the same question also in the language of multigraded free resolutions. We provide positive evidence for the existence of such pairs and give counterexamples to strengthenings of the question.

Let $f=(f_1,\dots,f_r)$ be a complete factorization of a central hyperplane arrangement $D$ in $X=\mathbb{C}^n$. For a monoid ideal $K\subseteq \mathbb{N}^r$ we study the Bernstein-Sato ideal $B^K_f$ of $f$ along $K$, that is, the $\mathbb{C}[s]$-annihilator of $\mathcal{D}_X[s]f^s/\sum_{m\in K}\mathcal{D}_X[s]f^{s+m}$. When $D$ is free we compute two families of these ideals with the help of AI. For the unit shift $K=\langle e_i\rangle$ we prove that $B^{-e_i}_f$ is generated by an explicit product of linear forms indexed by the dense edges of $D$ contained in $D_i$. This determines all the Bernstein-Sato ideals $B^{a,b}_f=\operatorname{Ann}_{\mathbb{C}[s]}\mathcal{D}_X[s]f^{s-a}/\mathcal{D}_X[s]f^{s-b}$, $a\geq b$, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of $\mathcal{D}_X[s]f^s/\mathcal{D}_X[s]f^{s+e_i}$ along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of $D$; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal $K=\langle e_1,\dots,e_r\rangle$ we show that $B^K_f$ is generated by one Euler relation for each irreducible factor of the essential quotient of $D$. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in $\mathbb{C}^2$: for $f=(x,y,x+y,x+2y)$ and $K=\langle 3e_1,3e_2\rangle$ we compute $B^K_f$ exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

In this paper we present a new result that exhibits in a non-recursive fashion a primary decomposition of any monomial ideal. Our result is derived from the classical quotient-sum recursive procedure for determining a primary decomposition of any proper ideal in a commutative Noetherian ring. We also show how our result can be used as an algorithm for producing a primary decomposition directly from the exponent vectors of the minimal generators of the ideal.

We prove the Eisenbud–Huneke–Ulrich conjecture on powers of m-primary ideals with partially linear minimal free resolutions in characteristic zero.

An effective strategy for constructing compact representations of integral coefficients in scattering amplitudes is to make their singularity structure manifest. This is naturally achieved by treating them as elements of fraction fields of polynomial quotient rings, where poles correspond to algebraic vanishing loci. Simplification via partial fraction decomposition relies on exploiting the structure of the numerators and is governed, at least in a first approximation, by their behaviour on irreducible codimension-two varieties. These are identifiable through primary decompositions of the associated ideals. Despite the availability of general algorithms, primary decompositions remain computationally challenging and often require tailored strategies. In this work, we extend an approach based on ideal saturation, identify an issue related to the breaking of Lorentz covariance in the choice of ideal generators, and present a solution based on fitting ansätze in special kinematic limits. We also provide substantial improvements to an algorithm for the generation of numerical points in proximity of given varieties and to one for testing primality of unmixed ideals, both of which rely on a semi-numerical procedure for computing independent sets and variety dimensions. Finally, we present new codimension-two primary decompositions for five-point massless amplitudes at higher powers in the dimensional regulator, and for finite remainders with five-point one-mass kinematics, where a new spurious singular locus appears.

Let $M$ denote a module over a local Noetherian ring $(R,\mathfrak{m})$ with residue field $\Bbbk$ and its injective hull $E_R$. Let $D_R(\cdot) = \operatorname{Hom}_R(\cdot,E_R)$ denote the Matlis duality functor. Then $M$ is called Matlis reflexive whenever the injection $M \to D_R^2(M)$ is an isomorphism. The following results are proved: (1) For an ideal of $R$ a reflexive module is quasi-complete, generalizing part of Belshoff's results (see \cite{Br}). (2) We improve Zöschinger's numerical criterion for Matlis reflexivity (see \cite{Zh2}). (3) As shown by Enochs (see \cite{Ee}) for a prime ideal and the injective hull $E_R(R/\mathfrak{p})$ it follows that $D_R(E_R(R/\mathfrak{p}))$ is the completion of a free $R_{\mathfrak{p}}$-module of rank $\tau_{\mathfrak{p}}$, the Enochs number of $\mathfrak{p}$. We characterize when $\tau_{\mathfrak{p}} = 1$, extending author's result (see \cite{Sp16}).

We describe the ideals of an idealization $D\ltimes M$ by triples $(I,N,\varphi)$, where $I$ is an ideal of $D$, $N$ is a submodule of $M$ with $IM\subseteq N$, and $\varphi\in\operatorname{Hom}_D(I,M/N)$. For a domain $D$ and torsion-free $M$, the nonzero proper ideals of $D\ltimes M$ with nonzero annihilator are exactly $0\ltimes N$. As an application, for $R=\mathbb R[[t]]\ltimes\mathbb R[[t]]$, the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality $2^{\aleph_0}$. This answers negatively a question of Behboodi and Rakeei.

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a “next step” after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog–Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

We present an elementary proof of the bivariate nested Artin-Popescu approximation theorem: it ensures the existence of a nested algebraic power series solution of a given polynomial (functional) equation. Such equations often appear in enumerative combinatorics, e.g., when counting lattice walks in the first quadrant. The ad hoc proofs of the algebraicity of the generating functions, as e.g. proposed by Bousquet-Mélou and Jehanne , can thus be replaced by applying the theorem without need to resort to the extremely difficult general case solved by Popescu. Our method of proof follows closely the arguments of Denef and Lipshitz who treat a more general case (namely, the Artin approximation for Weierstrass systems). This is combined with the techniques of Hauser and Woblistin for the description of the overall geometry of the infinite dimensional variety formed by all power series solutions. Putting both approaches together now provides combinatorialists with an accessible proof for the bivariate nested Artin-Popescu approximation theorem.

Transcendence questions of periods and quasi-periods of Anderson modules, as well as the hyperderivatives thereof, are of major interest in number theory over function fields. In the case of Drinfeld modules, this question was answered by the second author. In this paper, we show that these results hold for general Anderson $t$-modules under certain assumptions on their Galois representations. The proofs use the explicit description of the $\mathfrak{p}$-adic Galois representation of the first author by means of a rigid analytic trivialization of the associated $t$-motive, as well as prolongations of $t$-modules and $t$-motives. The general result is applied to determine algebraic relations between periods and quasi-periods, and their hyperderivatives, of finitely many Drinfeld modules. Along the way, our results provide more evidence for the Mumford-Tate conjecture for Anderson modules.