Papers

Iima and Yoshino asked for an ideal $I$ in $S=k[x_1,x_2,\ldots]$, with $\operatorname{deg} x_i=i$, and a monomial order such that $S/I\cong k[x_i:i\equiv\pm1\pmod5], \operatorname{in}(I)=(x_i^2,x_ix_{i+1}:i\geq1).$ We construct such an ideal and monomial order over every field $k$ of characteristic different from $5$ containing an element $c$ with $c^2+c=1$. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to $I$ and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we introduce the notion of ranks of triangulated categories and study Verdier quotients of $\operatorname{\mathsf{D^b}}(R)$ from the viewpoint of this invariant. Our main result determines the rank of the Verdier quotient $\operatorname{\mathsf{D^b}}(R)/\operatorname{\mathsf{thick}}(R\oplus k)$ in terms of the ranks of the categories of maximal Cohen-Macaulay modules on the punctured spectrum.

A question posed in the literature asks whether the divisoriality of a prime can alternate indefinitely along an ascending chain of overrings of a Prüfer domain or a descending chain of same-quotient-field underrings, and what happens at the limit. We construct such chains with all finite-stage domains Bézout and show, in both directions, that the limit prime can be either divisorial or nondivisorial.

Matlis duality, first introduced by Eben Matlis in 1958, stands as one of the most elegant and powerful tools in commutative algebra, establishing a striking contravariant equivalence between Artinian and Noetherian modules over complete local rings. In this paper, we undertake a thorough and systematic investigation of Matlis reflexive modules — those modules for which the canonical evaluation homomorphism into the double Matlis dual is an isomorphism. Our exposition begins with a detailed historical narrative, tracing the intellectual trajectory from classical Pontryagin duality through Grothendieck's dualizing complexes to the modern formulation of Matlis duality. We then develop the core theory from first principles, demonstrating that the class of Matlis reflexive modules constitutes a Krull-Schmidt category — a result that generalizes the classical decomposition theorems for modules of finite length. Over complete Noetherian local rings, we prove that Matlis reflexive modules coincide precisely with the class of minimax modules (those possessing a Noetherian submodule whose quotient is Artinian). We establish full closure properties, including stability under submodules, quotients, extensions, and finite direct sums, with rigorous and self-contained proofs. The paper also surveys recent applications to generalized local cohomology, change-of-rings results, and connections to pure-injective modules and linear compactness, and concludes with a discussion of open problems and future research directions.

We prove that continuous $A$-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity.

In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh's conjecture. For the depth, we first show that, for each $k\geq2$, the inequality $\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G)^{(k)}/I(G)^{k}$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.

We show that weak polynomial completeness does not imply almost polynomial completeness by constructing an explicit extension of domains \(D\subseteq A\) such that \(\Int(D)\subseteq\Int(A)\) but \(\Int(D^2)\nsubseteq\Int(A^2)\).

We give an exact translation between fat-point interpolation in complex projective space and a dark-state problem for fixed-particle-number bosons. Under the unitary Fischer–Bargmann realization, homogeneous degree becomes particle number, the Veronese lift becomes a bosonic symmetric product state, and order-$m$ vanishing becomes orthogonality to all coherent jet excitations of orders below $m$. The initial degree of a symbolic power is therefore the first particle number admitting a dark state of a positive constraint Hamiltonian, and the Waldschmidt constant is the asymptotic particle cost per suppression order. After developing both sides of this dictionary from first principles, we prove a finite-scale no-dark-state amplification theorem. Frobenius amplifies the associated full-rank certificate in one good positive-characteristic fiber; a nonzero maximal minor returns it to characteristic zero. Its asymptotic consequence is Demailly's conjectured inequality. The paper is written so that the projective statement, the quantum reformulation, and the arithmetic part of the solution can be read independently and then identified exactly.

In this paper, we study thick subcategories of the (stable) module categories of finite-dimensional weakly symmetric algebras with radical cube zero. When the spectral radius of the Ext matrix is two, thick subcategories containing the algebra, equivalently thick subcategories of the stable category, are classified by Serre subcategories of a certain abelian category; when it is not two, only the trivial thick subcategories occur. As an application, this yields, to the best of our knowledge, the first examples of commutative Noetherian Gorenstein dominant local rings that are not complete intersections.

For $2 \times 2$ matrices $A_1,\dots,A_k$, write $[A_1,\dots,A_k]$ for the left-normed iterated commutator $[\dots[[A_1,A_2],A_3],\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \geq 2$ over any field of characteristic $\neq 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k / I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k = P_2 \cap \cdots \cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2 \times 2$ minors of an explicit $2 \times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.

Let $(R,\mathfrak m)$ be a Noetherian local ring and $J$ be an arbitrary ideal of $R$. Suppose that $f_1,\ldots,f_r$ is a $J$-filter regular sequence in $R$ and $I=(f_1,\ldots,f_r)$. In this paper, we establish an improved upper bound for the perturbation index of the associated graded ring $\mathrm{gr}_J(R/I)$, refining the bound obtained by Quy and N. V. Trung.

In this paper, we relate multiplicities of Bernstein–Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain “mod” b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f, and we provide an algorithm to compute them. Second, we show that they can be approximated by the multiplicities of roots of power b-functions (the b-functions with respect to powers of f). Further, we give a sharp upper bound for the Hodge level of elements given by a certain sum of such multiplicities. Next, we give an effective asymptotic solution to the Gelfand problem by determining an explicit threshold after which every integer shift of a root of b_f(s) is a pole of the Archimedean zeta function of f. We also show that the order of these poles is equal to the nilpotency index of the logarithmic monodromy operator, which we further express as the limit of the multiplicities of roots of power b-functions. We define several filtrations, relating them to the weight and Hodge filtrations, based upon which we leave some open questions that we address in the affirmative in the case when f has a homogeneous isolated singularity, or it is a hyperplane arrangement, or it is a semi-invariant on a spherical variety. We give several immediate applications to our results, including a positive answer to a question of Torelli assuming the hypersurface has log canonical singularities: 1/f lies in the intersection complex of the hypersurface of f if and only if -1 is a simple root of b_f(s).

The purpose of this work is twofold. In the first part, we consider layered mixed matrices introduced by Murota, relate them to existing notions in combinatorial commutative algebra, and investigate the irreducibility of their determinants. Furthermore, for a layered mixed matrix in combinatorial canonical form, we determine the sparsity structure of its inverse. That is, we characterize which entries of the inverse are nonzero. In the second part, we establish for the first time a formal connection between these algebraic results and the theory of buffering structures for reaction networks developed by Mochizuki and Okada. We identify the lattice of buffering structures with the lattice of order ideals of the block poset of the combinatorial canonical form of the associated layered mixed matrix. This allows us to characterize the reducibility of the symbolic Jacobian determinant as a polynomial in the reaction-rate derivatives, as well as the nonzero sensitivity responses of species concentrations to reaction-rate perturbations.

Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal $I(G)$ of a connected graph $G$, we show that $\mathrm{v}(I(G)^k)=\mathrm{v}(\overline{I(G)^k}) = 2k-1$ for all $k \geq 1+|E(G)|$. Moreover, when $G$ is disconnected, we prove that $\mathrm{v}(\overline{I(G)^k})\leq\mathrm{v}({I(G)^k})$ for all sufficiently large $k$.

We determine the ozone group of \(U_q^{+}(B_2)\). More precisely, if \(\ell=\operatorname{ord}(q^2)\) when \(q\) is a root of unity, we obtain $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} \mu_2 , & \text{if \(q\) is not a root of unity},\\[2mm] \mu_{\gcd(\ell,2)}, & \text{if \(q\) is a primitive \(m\)-th root of unity, \(m\geq5\).} \end{cases} $$ We also study several homological properties of \(U_q^{+}(B_2)\), showing that it is Artin–Schelter regular of global dimension \(4\), Auslander-regular, Cohen–Macaulay, strongly Noetherian, and skew Calabi–Yau. Using two different iterated Ore presentations, we show that its Nakayama automorphism is trivial for every $q \in \K^*$, so $U_q^{+}(B_2)$ is Calabi–Yau of dimension $4$; for $q$ not a root of unity this also follows from work of Goodearl and Yakimov. When $q$ is a primitive $m$-th root of unity with $m\geq5$, we deduce that the center is Gorenstein, we check directly that the ozone group acts with trivial homological determinant, and we describe the nonzero normal elements in terms of the ozone group.

Classes of squarefree monomial ideals were intensively studied being an important connection between two main areas in mathematics: commutative algebra and combinatorics. Many algebraic invariants of the squarefree monomial ideals are given in terms of properties of the combinatorial associated objects (graphs, simplicial complexes etc.).

A finite simple graph $G$ is called a cograph if it does not contain the path on four vertices $P_4$ as an induced subgraph. It is classically known that the family of cographs are well-quasi-ordered by the induced subgraph relation \cite{D}. In preceding work of Knudsen and the third author \cite[Theorem 7.2]{KR}, it was shown that this well-quasi-order statement admitted a categorification, which allowed those authors to prove universal finite generation statements about the homology groups of configuration spaces on cographs \cite[Theorem 1.5]{KR}. In this work, we expand \cite[Theorem 7.2]{KR} to be compatible with the family of polynomial rings on the vertex sets of cographs. By consequence, we are able to prove a number of universality results related with edge and toric ideals of these polynomial ring, partially generalizing and expanding upon work of Kahle \cite{kahle2019binomial}. We also conclude strong restrictions on the kinds of topologies that can arise from graph complexes and anchored configuration spaces associated to cographs, as well as combinatorial constraints on the possible combinatorics of hyperplane arrangements of cographs.

We prove a necessary condition conjecture ([27, Conjecture B]) for the maximal singularity of the Hilbert scheme of points in 3D for a large class of degrees, which in particular implies the 1978 Briançon-Iarrobino Conjecture for a tetrahedral degree, and vastly generalises the result in [19]. The novelty lies in introducing a new upper bound on the dimensions of the tangent spaces of the claimed non-maximally singular ideals, constructing canonical ideals that satisfy the claimed maximal singularity condition, providing a lower bound on the dimension of their tangent spaces, and finally comparing the two cases.

For a numerical semigroup $S$ with embedding dimension four, we study the relationship between its type $t(S)$ and the cardinality of its minimal presentations $\eta(S)$. Using an approach based on the geometry of the Apéry set, we prove that $4t(S) + 5 \geq \eta(S) \geq t(S)-11$. This resolves a problem of Moscariello and Sammartano asking whether $t(S)$ is bounded by a function of $\eta(S)$, and improves the previously known bound $9t(S) + 4 \geq \eta(S)$ due to Bresinsky.

Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom-Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein-Sato root can be detected. Keywords. Bernstein-Sato polynomial, monodromy conjecture.

The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik–Solomon algebra. It is known that the defining ideal of the Orlik–Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik–Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik–Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik–Terao algebras coming from non-supersolvable arrangements.

Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.

Let $R$ be a commutative semiring with unity $(1\neq0)$, and let $S$ be a proper multiplicatively closed subset of $R$. In this paper, we introduce the notion of quasi $S$-$n$-ideals in commutative semirings. We the study basic property of quasi $S$-$n$-ideals and establish their relationships with quasi $n$-ideals, $S$-$n$-ideals, $S$-prime ideals, and $S$-primary ideals. We also investigate their behavior under localization. Finally, we determine quasi $S$-$n$-ideals in the quotient polynomial semiring $R[x]/\langle x^m\rangle$ and study their natural extensions via the constructions of idealization and amalgamation.

Let $(G_n)_{n\ge n_0}$ be a family of graphs on $[n]$ whose edge ideals $I_n\subseteq R_n=k[x_1,\dots,x_n]$ form an $\mathrm{Inc}(\mathbb{N})$-invariant chain, $I_{n+r}=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I_n)$ for $n\ge n_0,\ r\ge0$. We determine which pairs $(n,r)$ make $R_{n+r}/I_{n+r}$ Cohen–Macaulay, for five classical families: line graphs $L_n$, complements of line graphs $L_n^c$, complete graphs $K_n$, cyclic graphs $C_n$, and complements of cyclic graphs $C_n^c$. For line graphs we give the generators of $I_{n+r}$, the height and Krull dimension of $R_{n+r}/I_{n+r}$, and a complete classification: for $n\ge5$, $R_{n+r}/I_{n+r}$ is Cohen–Macaulay if and only if $r=n-4$. For complements of line graphs, complete graphs, and complements of cyclic graphs, we show the chain is Cohen–Macaulay unconditionally, for every $r\ge0$; the first and third arise from the same underlying phenomenon, in which the $\mathrm{Inc}(\mathbb{N})$-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove $\mathrm{Inc}(\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$ once $r\ge n-3$, giving Cohen–Macaulayness in this range, and conjecture – with supporting computational and structural evidence – a complete classification: Cohen–Macaulayness holds if and only if $r\ge\lfloor(n-4)/2\rfloor$ and $r\ne n-4$.

Let $(F|K,w)$ be an immediate valued function field of transcendence degree one over a rank-one perfect valued field $(K,v)$ of characteristic $p>0$. It is henselian rational if $F^h=K(Y)^h$ for some $Y\in F^h$. Kuhlmann proved henselian rationality over tame fields; we investigate how far his method extends to perfect fields. Relative approximation degrees are a central ingredient in Kuhlmann's approach. We first complete their theory over henselian fields by proving the existence of the relative approximation degree and constant of every polynomial, including for pseudo-convergent sequences of algebraic type. Using the $j$-invariants of associated monomial valuations, we describe these invariants directly through Taylor expansions and extend the henselian degree bound of Kuhlmann and Vlahu. We next study the Artin–Schreier reduction underlying the henselian rationality argument. Over perfect fields, every polynomial is Artin–Schreier equivalent to one whose relative approximation degree lies in ${1,p}$. We construct an explicit rank-one example showing that $p$ cannot always be reduced to one modulo the Artin–Schreier image of $K[X]$. Nevertheless, reduction to degree one becomes possible in this example after passing to equivalence modulo the Artin–Schreier image of $K(X)^h$, and the resulting Artin–Schreier function field is henselian rational. Finally, assume that $K$ equals its absolute ramification field, and let $L=IC(F|K,w)$ be the relative algebraic closure of $K$ in $F^h$. We prove that $F^h$ is henselian rational over $L$, and that henselian rationality descends to $K$ whenever $L|K$ is finite. This finiteness condition holds whenever some separating transcendental element induces an extension of Type II, yielding henselian rationality in this case.

Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in $\mathrm{SL}_2(F[X^{\pm1},Y^{\pm1}])\setminus \mathrm{E}_2(F[X^{\pm1},Y^{\pm1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.

This work concerns ideals invariant under the action of the group of linear base changes on a generic matrix; we call these GL-invariant ideals. We determine the ordinary powers, their saturations with respect to determinantal ideals, and the symbolic powers of GL-invariant ideals. We give explicit formulas for asymptotic invariants known as (skew) Waldschmidt constants and asymptotic resurgence, which measure the growth of these families and compare the ordinary and symbolic topologies. We also prove that the generalized symbolic Rees algebras associated with these ideals are Noetherian.

For a numerical semigroup $S$ with Frobenius number $F$ and gaps $G$, a polynomial $P\left(z\right)=\sum_{g\in G}c_{g}z^{g}$ of degree $F$ is called a gap polynomial. We show that for a symmetric numerical semigroup, the power sums of the critical points of a gap polynomial vanish for every power $g\in G$. For a pseudo-symmetric numerical semigroup, the same result holds if the term $z^{F/2}$ is omitted from $P\left( z\right) $. We give several corollaries, including that the sum of the critical values of a gap polynomial vanishes.

We construct finite geproci sets in every projective dimension and in every positive characteristic by introducing $\mathbb{F}_N$-Artin-Schreier configurations. In $\mathbb{P}^3$, we characterize exactly when such configurations are geproci: an $\mathbb{F}_N$-Artin-Schreier configuration on $q$ lines spanning $\mathbb{P}^3$ is $(q,N)$-geproci if and only if $q\leq N$. We then develop a lifting construction which produces geproci sets in $\mathbb{P}^n$ for every $n\geq 3$.

We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and hypergraphs (arXiv:2512.17619), in which invariants of the face ring such as graded Betti numbers and f- and h-vectors are persisted across a filtration. That framework is built from the minimal free resolution and is thus Tor-theoretic; we work instead on the injective side, and the resulting modules record information localized at a single vertex, complementing the global picture given by maximal-support local cohomology. For a vertex prime $p_i = (x_j : j \neq i)$ we prove an exact $\mathbb{Z}^n$-graded decomposition of $H^q_{p_i}(k[\Delta])$ into the maximal-support local cohomology of the deletion and of the link of the vertex $i$, the first in $x_i$-degree zero and the second repeated in every positive $x_i$-degree; at the level of graded dimensions this recovers the vertex-prime case of Rahimi's bigraded formula. With Hochster's formula this yields a closed combinatorial description of every multigraded piece. Building on this structure we introduce per-vertex persistent local cohomology numbers, prove a persistent links-Hochster formula, obtain interval decompositions of the resulting reversed-arrow persistence modules and a bottleneck stability theorem, retain multiplication by the uninverted variable as a morphism of persistence modules that the two barcodes alone do not determine, and extend the theory to an arbitrary coordinate prime, where the multiplication maps of the uninverted variables assemble into a commuting Boolean diagram of persistence modules.