Papers

In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg–Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.

We show that the characteristic cycle multiplicities of $A$-hypergeometric systems are upper semicontinuous in the parameter for every rational projective weight, thereby proving a conjecture of Schulze and Walther [Duke Math. J. 142 (2008), 465–509]. The proof uses a specialization formula for filtered families satisfying a fixed product support condition, expressing the difference between the special and generic characteristic cycles as the characteristic cycle of the first Tor module. Combined with the Gevrey index theorem, this yields upper semicontinuity of generic Gevrey irregularity and of each of its graded dimensions. We also construct non-Cohen–Macaulay semigroup rings whose Gevrey dimensions are constant in the parameter despite the presence of a genuine irregular slope, showing that the Gevrey jump locus need not coincide with the full exceptional arrangement defined by Ext.

To a polycube $\mathcal P$ in $\mathbb R^n$, we associate an ideal $I_{\mathcal P}$ generated by the join–meet relations of vertices that are corners of a cuboid of $\mathcal P$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\geq 3$ the ideal $I_{\mathcal P}$ is prime if and only if $\mathcal P$ is a cuboid. For convex polycubes in $\mathbb R^3$ of uniform thickness whose vertex set is a sublattice of $\mathbb N^3$, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen–Macaulay ones in terms of a poset on the vertices of $\mathcal P$; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened $L$-convex polyomino is Koszul, and we characterize when it is Cohen–Macaulay.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of polynomial type and let $r_{I, N}^j(M)$ be its degree. In general we have $r_{I, N}^j(M) \leq r_{I,N}^2(\text{Syz}^d_A(k))$ for $j \geq d + 2$. If $A$ is also dominant in the sense of Takahashi and $M$ is non-free then we show $\limsup_{j \rightarrow \infty} r_{I, N}^j(M) = r_{I,N}^2(\text{Syz}^d_A(k))$. We prove an analogous result when $V$ is a Cohen-Macaulay module of finite injective dimension and the polynomial type function $n \rightarrow \ell(\text{Ext}_A^j(M, V/I^{n+1}V))$

Let $S$ be a Cohen-Macaulay normal domain containing a field $k$ with an action of a finite group $G$ with $|G|$ invertible in $k$. We prove that $G$-equivariant noncommutative crepant resolutions of $S$ induce noncommutative crepant resolutions of $S^G$ if the $G$-action is small. We also discuss similar results for equivariant maximal modification algebras. We introduce the notion of $G$-maximal modifying ($G$-MM) module, and we show that $G$-MM module over $S$ induces MM $S^G$-module in some special cases. Furthermore, we prove that Gorenstein MMAs of three-dimensional $\mathbb{Q}$-Gorenstein Cohen-Macaulay rings are derived equivalent. As an application of these results, we prove the existence of an MM module and derived equivalences of MMAs for three-dimensional terminal singularities of index two.

We establish some basic properties of t-flat fine log schemes, i.e., fine log schemes whose underlying schemes are flat over their log structure (in a suitable sense). Notably, we show that the depth of a t-flat fs log scheme is simply determined by its logarithmic stratification. We also show that there is a good theory of log cotangent complexes for t-flat fine log schemes, and use the log cotangent complex to give a homological criterion for t-flatness.

Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $\nu$ be a valuation of $K/k$. We show $\nu$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.

Let $\delta=xz-yt$. We give an explicit tame factorization of the Anick automorphism $(x+\delta t,y+\delta z,z,t)$ of $F[x,y,z,t]$. After a linear change of coordinates, the factorization consists of sixteen elementary automorphisms, all fixing $z$. The proof uses a commutator construction with variables already present in the algebra. A complete list of the elementary factors is included in Section 5. The explicit polynomial factorization identities are defined over $\ZZ[1/2]$; no conclusion in characteristic $2$ is asserted.

We develop two new general frameworks for constructing finitely generated residually finite groups that possess infinite girth, satisfy no non-trivial group law, and contain no non-abelian free subgroups. While such groups are rare–with known examples including the first Grigorchuk group, Thompson's group F, and certain iterated wreath products–our approach provides a more explicit construction of such groups realized as subgroups of direct products of symmetric groups. We investigate the relationship between the girth of a group and that of its finite quotients, leading to the introduction of a "marked" girth criterion, which we establish as a necessary and sufficient condition for infinite girth within the class of residually finite groups. Furthermore, we explore the connection between girth and profinite completions and provide a congruence realization result for infinite girth in linear groups. We conclude by posing several open questions that arise naturally from our results.

For a finite group \(G\), let \(N(G)\) be the simplicial complex of subsets that do not generate \(G\). We separate a general Frattini-topological reduction from the structure special to finite \(p\)-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of \(G/\Phi(G)\). For a finite \(p\)-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of \(\mathrm{PG}(r-1,p)\), where \(r=d(G)\) and \(q=|\Phi(G)|\). We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the \(\mathbb Z\)-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen–Macaulay classification.

Suppose $f_1, \dots, f_{n+1}$ are elements of a regular ring $R$, where $n := \dim R$. The first author proved that $f_1^n \cdots f_{n+1}^n \in (f_1^{n+1}, \dots, f_{n+1}^{n+1})R$; this uses the Brian\c con-Skoda theorem — no simpler proof is known, as far as we are aware. We show here that this result is optimal in many respects. On the other hand, when the $f_i$ are homogeneous general polynomials of a fixed degree in $R := \mathbb{F}[x_1,\dots,x_n]$, for $\mathbb{F}$ a field of characteristic zero, we prove that $f_1 \cdots f_{n+1} \in (f_1^2, \dots, f_{n+1}^2)R$.

We give a combinatorial criterion for detecting roots of Bernstein-Sato polynomials of ideals. As applications, we prove the strong monodromy conjecture for determinantal and Pfaffian ideals.

We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs $G$ where ${\rm Ind}(G)$ is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where ${\rm Ind}(G)$ is shellable but not vertex decomposable.

We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy $F$-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and $\delta$-vector, the Newton polytope of the generic monodromy $F$-polynomial is independent of the order.

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\ldots,x_n)^2\subsetneq k[[x_1,\ldots,x_n]]$ has an isolated critical point and $\operatorname{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R/I$ is a nonzero Noetherian $k$-algebra and $a=[f]\in A$ is nilpotent with $d_{A/k}a=0$, then $I:f\nsubseteq(I,f)$. For a local ring $(R,\mathfrak n)$ we obtain the stronger noncontainment $I:f\nsubseteq(I,f)+\mathfrak n(I:f)$. The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies $J_f:f\subseteq\overline{J_f}$, disproving the integral-closure strengthening proposed by Ma and Zuo.

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S = \k[x_1, \ldots, x_n]$. For each integer $t \ge 1$, let $\alpha_t(G)$ denote the maximum size of an ordered matching $M$ of $G$ with alternating path length $\ell(M) \le 2t - 1$. Hang, Tam, and Vu established that \[ \operatorname{depth}\!\left(S/J(G)^{(t)}\right) \le n - 1 - \alpha_t(G) \] for all $t \ge 1$, where $J(G)^{(t)}$ is the $t$-th symbolic power of $J(G)$. In this paper, we introduce a natural hereditary class of graphs $\mathcal{P}_\k$ for which this inequality holds with equality for every $t \ge 1$. We prove that $\mathcal{P}_\k$ contains all distance-hereditary graphs, and we show that within the class of chordal graphs, $\mathcal{P}_\k$ coincides with the class of distance-hereditary (or gem-free) graphs.

The discriminant of a square matrix is defined as a discriminant of its characteristic polynomial. In this paper we provide a description of linear maps preserving the matrix discriminant assuming that the ground field is large enough. Our description generalizes the known results to the case where the field may have a finite cardinality and its characteristic may divide the size of considered matrices. To obtain this description, we extend the approach used in our previous work on the subject and, in addition to that, introduce new invariants for the action of all bijective linear maps preserving a given polynomial, which are on independent interest and could be applied to other problems in the theory of linear preservers.

Using differential Galois theory—specifically, Magid's theory of the complete Picard-Vessiot closure—we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for small numbers of points, while in arbitrary dimension and arbitrary cardinality the best general result was the stable version, valid on one dense Zariski-open set for all sufficiently large $r$. We show that the recent proof of Demailly's conjecture for arbitrary finite point sets by Hà and Sivakumar removes the stability threshold. The key observation is that the strict generic Waldschmidt estimate of Bisui and Nguyẽn allows one to select a single finite symbolic level $m_0$. Equality of the corresponding interpolation number is a Zariski-open condition. Demailly's inequality then propagates this one finite condition to lower bounds for every symbolic power, and a standard regularity criterion converts those bounds into the Harbourne–Huneke containment for every $r$. Consequently, for every $N\ge2$ and every $s\ge1$, one dense Zariski-open family of $s$-point configurations in $\mathbb{P}^N$ satisfies the Harbourne–Huneke containment simultaneously for all $r\ge 1$. The argument also clarifies a quantifier issue in the earlier literature: before the full Demailly theorem, estimates on the generic fibre naturally yielded either all $r$ on a very general set, or one open set only for $r\gg 0$. The new theorem replaces infinitely many symbolic conditions by a single finite interpolation condition.

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.

We study the diagonal Specht ideals $I_\lambda\subseteq\mathcal{R}_{m,n}:=\mathbb{C}[\mathbf{x}_1,\dots,\mathbf{x}_m]$, $\mathbf{x}_r=(x_{r,1},\dots,x_{r,n})$, generated by the $\lambda$-isotypic component of $\mathcal{R}_{m,n}$ for the diagonal action of $S_n$. We characterize the set of zeros of $I_\lambda$ as $V_\lambda=\bigcup_{\mu\not\trianglelefteq\lambda}H_\mu$ and prove that $\lambda\mapsto V_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{V_\lambda:\lambda\in \mathcal{P}_n\},\supseteq)$ for all $m$, while $\lambda\mapsto I_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{I_\lambda:\lambda\in \mathcal{P}_n\},\subseteq)$ if and only if $m=1$ or $n\leq3$. Unlike the case $m=1$, where Specht ideals are always radical, radicality in the diagonal setting depends on $\lambda$ and $m$. We prove radicality for hook partitions of length at most three by providing explicit Gröbner bases, and we develop three criteria for non-radicality via content, multidegree, and total degree, showing in particular that $I_\lambda$ is not radical for any non-hook partition $\lambda$ when $m\geq\operatorname{len}(\lambda)$. Using the $\operatorname{GL}_m(\mathbb{C})$-action on $\mathcal{R}_{m,n}$, we show that $I_\lambda$ is radical for all $m$ if and only if it is radical for $m=n$. Finally, for $m\geq2$, we prove that $\mathcal{R}_{m,n}/I_\lambda$ is Cohen-Macaulay if and only if $\lambda=(n)$ or $\lambda=(n-1,1)$, and the same is true for $\mathcal{R}_{m,n}/\operatorname{rad}(I_\lambda)$.

We introduce complementary determinantal edge ideals. For a graph $G$ on $n$ vertices, the complementary determinantal edge ideal $J_{c}(G)$ is generated by the maximal minors of a generic $(n-2)\times n$ matrix obtained by deleting the two columns indexed by each edge of $G$. We completely characterize the graphs for which $J_{c}(G)$ is Cartwright–Sturmfels with respect to the grading by columns. We prove that this property is equivalent to the existence of a multilinear universal Gröbner basis and characterize the graphs satisfying these equivalent conditions. In particular, the ideals satisfying these equivalent conditions are radical.

Over a field of characteristic zero, we characterize the Hilbert functions that determine the degree reverse lexicographic generic initial ideal among Artinian quotients in four variables with the $2$-strong Lefschetz property. Our criterion is a finite numerical test that strictly extends the quasi-symmetry condition of Harima and Wachi. The proof parametrizes the possible generic initial ideals by flags of lower sets in the Borel orders on their common three-variable section. We also give nonrecursive formulas for the standard monomials and minimal generators of the almost reverse lexicographic ideal, and specialize them to generic complete intersections in at most four variables without restrictions on the degrees.

We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert–Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.

We study finite determinacy and modality in positive characteristic through three complementary constructions: prescribed-jet transversality, Frobenius deformation directions, and a finite flat critical cover. For matrix germs over an arbitrary field, prescribed-jet transversality to the rank stratification shows that finite left-right determinacy is equivalent to finite codimension of the tangent image. Analogous results hold for pure right equivalence, for one-sided actions over infinite fields, and in specified ranges over finite fields. These methods also yield a characterization of finite contact determinacy for positive-dimensional ideals and imply expected-height, Cohen-Macaulayness, reducedness, and normality results for determinantal ideals. For isolated hypersurface singularities over an algebraically closed field of characteristic $p>0$, we introduce a Frobenius algebra whose length and Loewy structure govern the Milnor number, determinacy, and essential corank. Using a finite flat critical cover, we prove that proper modality coincides with right modality. It follows that, in fixed characteristic, bounded right modality gives an ambient-dimension-independent bound on determinacy and, up to nonsingular quadratic suspension, a finite algebraic parametrization by modular families. The same Frobenius methods yield uniform finiteness results for $F$-jumping numbers and show that the $F$-pure threshold is determined at a single Frobenius level.

We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie algebra of every maximal nested unipotent subgroup has the special property of being what we call "integral"; we also show that this fails for X = affine space of dimension > 3. As an application of our results, we show that Nagata's automorphism is contained in a unique Borel subgroup.

We study the reflexivity property of ideals in numerical semigroups. In particular, we characterize irreducible reflexive ideals and determine the Apéry set of a reflexive ideal. Moreover, we study how reflexivity allows us to characterize some classes of numerical semigroups and the interaction of reflexivity with the property of being a trace ideal.

We investigate derivation modules of hyperplane arrangements associated with the monomial groups $G(r,p,\ell)$. Motivated by Mühlherr's separator-based construction for graphic arrangements, we extend this approach by combining separators with integral expressions. For a subarrangement $\mathcal{A}_G$ defined by a graph $G$, we show that, when $p<r$, the derivation module $\mathcal{D}(\mathcal{A}_G)$ is generated by the standard derivations $\eta_0,\dots,\eta_{\kappa(G)}$ together with a finite set of separator-based derivations. This result generalizes the complete graph case and provides new evidence toward the broader applicability of separator methods in the study of hyperplane arrangements.

It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules $J$ over a commutative noetherian ring $A$, we show that K-injectivity of $J$ can be detected locally: if $(A,\mathfrak{m})$ is a noetherian local ring, then $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(\widehat{A},J)$ is K-injective over $\widehat{A}$, the $\mathfrak{m}$-adic completion of $A$; and for an arbitrary commutative noetherian ring $A$, $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(A_{\mathfrak{m}},J)$ is K-injective over $A_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ in the singular locus of $A$. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.

Given a finite Cartesian set $S=X \times Y$ and a decreasing set of monomials $\mathcal M$, we call extremal polynomials for $\mathcal M$ over $S$ those that have the maximum number of zeros in $S$ and whose support belongs to $\mathcal M$. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If $\max(\mathcal M)$, taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If $|\max(\mathcal M)|=2$, either all extremal polynomials are canonical, or the problem reduces to the case where $\max(\mathcal M)=\{x^{d_x},y^{d_y}\}$. In the latter case, we prove that the existence of noncanonical extremal polynomials depends on finding families of subsets whose elementary symmetric functions agree. This condition is more restrictive than the classical Prouhet–Tarry–Escott problem, which asks for two sets whose elementary symmetric functions agree. We determine the number of triples $(X, Y, h)$, where $h$ is a quadratic noncanonical extremal polynomial. We apply extremal polynomials to coding theory via minimum-weight codewords.