Papers

Christophersen's problem predicts that the connected component of the automorphism group of a finite-dimensional local algebra $A$ of dimension $\ell$ over an algebraically closed field of characteristic zero has dimension at least $\ell-1$, with equality if and only if $A$ is isomorphic to $\mathbf{k}[t]/(t^{\ell})$. We settle this problem in the class of monomial algebras. Using the combinatorial structure of the irredundant irreducible decomposition of a monomial ideal, we prove the predicted inequality and characterize the equality case. We further obtain stronger lower bounds depending on whether the defining monomial ideal is reducible or irreducible, and we determine all monomial algebras attaining each of these bounds.

We define and study the $F$-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety $X$, we show that the $F$-signature function defines a continuous function on the big cone of $X$, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the $F$-signature of any big $\mathbb{R}$-divisor on $X$ is equivalent to the global $F$-regularity of $X$. As applications, we prove a birational transformation rule for the $F$-signature in terms of the $F$-signature of certain graded ideal sequences. We also prove that in a reduction modulo $p$ situation, a characteristic independent lower bound for the $F$-signature of $X$ with respect to any ample divisor $L$ spreads out to a similar bound for arbitrary big divisors on $X$.

We describe the Macaulay2 package MixedHodgeModules, which computes Hodge and weight filtrations on localizations S_f of a polynomial ring S at a non-constant reduced polynomial f, as well as on twisted localizations S_f f^{-\alpha}, where \alpha\in QQ is rational. The package includes functionality for computing Hodge ideals, weighted Hodge ideals, higher multiplier ideals, generation levels of Hodge filtrations, weight levels of elements, p-functions, and lengths of weight filtrations. It also provides routines for computing graded de Rham complexes, Du Bois complexes, and intersection Du Bois complexes of hypersurfaces.

We consider the following three conditions for polynomials in several variables. Condition (1) holds for a polynomial if, for every affine integral domain, the following is true: whenever the result of substituting algebraically independent elements into the polynomial lies in the kernel of a locally nilpotent derivation, the elements themselves must lie in that kernel. A polynomial satisfies condition (2) if it does not belong to the kernel of any nonzero LND of the polynomial algebra. It satisfies condition (3) if the quotient algebra modulo the ideal generated by the polynomial is rigid, that is, it admits no nontrivial LNDs. For irreducible binomials, we prove that all three conditions are equivalent. For reducible binomials, we establish all existing implications between these conditions.

Let $R$ be a ring and $M$ a left $R$-module. We prove that $M$ is an $\omega_1$-indexed inverse limit of injective modules with split epimorphic connecting maps, and $M$ is also an $\omega$-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of $\omega$-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring $R$, they are precisely the Enochs cotorsion modules if and only if $\operatorname{gldim}R\leq1$ and $\operatorname{Spec}R$ is countable.

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $\Gamma_E(R)$ and the associate class graph $\Gamma_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $\Gamma(R)$, $\Gamma_A(R)$ and $\Gamma_E(R)$. We prove that the count for $\Gamma_A(R)$ is bounded below by the count for $\Gamma_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $\Gamma_A(R)\cong\Gamma_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $\Gamma(R)\cong\Gamma_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.

We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.

The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the Kähler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$, where ${\rm char}(K)=0$ and $P=K[x_1,\dots,x_n]$, we prove a vanishing theorem for $H_{\rm dR}^m(R)$ based on the shape of a Macaulay basis of $dI\wedge \Omega^m_{P/K}$. For an Artinian local algebra $A=P/\langle f_1,\dots,f_n\rangle$, where $\{f_1,\dots,f_n\}$ is a super regular sequence, we show that $H_{\rm dR}^{\bullet}(A)$ is non-trivial in general, but trivial when the natural system of generators of the relation module of $\Omega^m_{A/K}$ is a standard basis. Moreover, we provide a detailed study of the dimension of $H_{\rm dR}^0(A)=\ker(d_A)$ for Artinian local rings $A=P/I$. The case of arbitrary affine zero-dimensional schemes is reduced to this case using a Galois splitting, and as a result we obtain the de Rham cohomology of a fat point scheme. Many explicitly computed examples and counterexamples support the results and indicate how subtle the de Rham cohomology of a zero-dimensional scheme is in general.

We prove an $\Omega(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first super-linear lower bound for this fundamental algebraic problem for any explicit polynomial. However, the authors of this note were unable to follow the details and verify the argument in arXiv:2606.13628, in spite of considerable effort on their part. The proof we provide here is short, (almost) self-contained and seemingly simpler.

We study embeddings of Cohen–Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summand. Motivated by almost Gorenstein rings, we prove an Ext-vanishing criterion for finite projective dimension, which implies G-regularity and the generalized Auslander–Reiten condition. We characterize this summand condition for one-dimensional fiber products and numerical semigroup rings. For Stanley–Reisner rings of graphs, we characterize the corresponding condition for graded embeddings in terms of the graph and show that it is equivalent to a strict multiplicity inequality.

The principal matroid determinant $E_L$ of a linear space $L \subseteq \mathbb{P}^n$ has been introduced in recent work by Matsubara-Heo and Telen. In this paper, we give the complete factorization of this polynomial into its irreducible components, proving a conjecture of the aforementioned authors. Our methods rely on bounding local multiplicities and an étale-local description of the strata of reciprocal linear spaces developed by Elias, Proudfoot and Wakefield. We also discuss analogous questions for other coordinate-wise powers of linear spaces.

We establish strong and explicit forms of a one-sided Nullstellensatz for polynomial rings D[x1, . . . , xn] over arbitrary division rings D with central indeterminates. To this end, we employ the theory of integral dependence over left ideals and the notion of G-left ideals, which plays a role analogous to that of G-ideals in the classical theory. We prove that the Jacobson radical of any left ideal of D[x1, . . . , xn] coincides with the intersection of all G-left ideals containing it. As a consequence, we obtain an explicit description of Jacobson radicals of left ideals of polynomial rings over division rings in terms of integral dependence.

For a finite simple graph $G = (V, E)$, we consider the quotient algebra $\mathcal M(G)$ of the polynomial ring in the variables $u_e$ (for $e \in E$) by the ideal generated by all products $u_e u_f$ for non-disjoint edges $e$ and $f$ (including all squares $u_e^2$). This quotient is called the matching algebra of $G$, since it has a basis indexed by the matchings of $G$. In this quotient, we define a subalgebra $\mathcal D(G)$ generated by the signed incidence sums $\theta_v=\sum_{e=(v,w)}u_e-\sum_{e=(w,v)}u_e$ for all $v \in V$ (where all edges of $G$ are oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space, $\mathcal D(G)$ is dual to the span of all polynomials $p_M = \prod_{(i,j) \in M} (x_i - x_j)$, where $M$ ranges over all matchings of $G$. For the complete graph $K_n$, the latter span is a direct sum of two-row Specht modules (one in each degree); thus its Hilbert series is that of the Catalan triangle, and, in characteristic zero, the Dunkl matching algebra can be presented by linear and quadratic relations. For arbitrary graphs, we formulate the saturation problem of deciding when the selected matching Specht generators $p_M$ in a given degree $k$ span the full two-row Specht module $S^{(n-k,k)}$. We show that saturation in degree $k$ forces $k$-connectivity, that saturation in degree $2$ is equivalent to $2$-connectivity, and that $k$-linked graphs are saturated in degree $k$. We also prove saturation in every possible degree whenever the complement of $G$ is a matching. We show that the Dunkl matching algebra equals the full matching algebra exactly for forests, and give an explicit Hilbert series formula for unicyclic graphs.

We generalize a construction of Herzog and Takayama pertaining to differential graded algebra structures on mapping cones. Via this generalization, we provide the minimal free resolution of the edge ideal of a new graph built from suspension over a vertex cover of an old graph, and we discuss when this process preserves differential graded algebra structure on the respective resolutions.

For every nonnegative integer $n$, we construct a commutative total ring of quotients of weak global dimension exactly $n$. For $n\geq1$ the examples are reduced and non-coherent. The construction applies to any nonzero local ring $(A,\m)$: one adjoins countably many residue-field coordinates subject to eventual agreement with the residue of the $A$-coordinate. The resulting ring $\T(A)$ satisfies $Q(\T(A))=\T(A)$ and $\wgd\T(A)=\wgd A$. Its prime spectrum, maximal localizations, nilradical, Jacobson radical, and idempotents are described explicitly. For every pair of modules, positive-degree Tor over $\T(A)$ is naturally identified with Tor over $A$ after quotienting by a projective ideal generated by orthogonal idempotents. We prove that $\T(A)$ is coherent if and only if $A$ is a field. Specializing to regular local polynomial rings yields the prescribed dimensions, with explicit Koszul witnesses for the lower bounds. We also examine the minimal spectrum, the distinction between global and local Prüfer conditions, and a finitely presented cyclic module of flat dimension one and projective dimension two.

Let $(R,\mathfrak m,k)$ be a Noetherian local ring, let $M$ be a nonzero finitely generated $R$-module, and let $J_1,\ldots,J_s$ be arbitrary proper ideals of positive height on $M$. We introduce the notion of directional mixed multiplicities, obtained from the multigraded quotients $J^{\mathbf u}M/J^{\mathbf u+\mathbf e_j}M$ by retaining one distinguished direction $j$. These invariants refine the mixed multiplicity sequences of arbitrary ideals while remaining compatible with the two classical extremes: for one ideal they recover the positive components of the Achilles–Manaresi multiplicity sequence, and for $\mathfrak m$-primary ideals their top-degree terms recover the usual mixed multiplicities individually. We establish additivity, an associativity formula, superficial reduction, and an exact decomposition of the diagonal coefficients into directional components. Over an infinite residue field, every positive directional coefficient is realized as an eventual length by a sufficiently general directional $(FC)$-system. We then identify an additional synchronization condition under which the directional coefficient becomes the Hilbert–Samuel multiplicity of a parameter ideal, yielding a Rees-type theorem and recovering the classical theorem in the $\mathfrak m$-primary case. In addition, we show a multihomogeneous product formula for $J_1^{n_1}\cdots J_s^{n_s}$, which specializes to the classical Bhattacharya–Rees–Teissier formula.

We study the dimension-indexed marked-root polynomial maps introduced by Harish. For every $n\geq 4$, their lower coefficient block extends to a polynomial coordinate system, giving three-dimensional Keller maps of geometric degree $n(n-2)$ for every choice of the fixed coefficients. We determine the generic monodromy of the original maps and all successive coordinate descents: it is the cyclic wreath product for odd $n$, and an explicit subgroup of index two for even $n$. Its order is $(n-2)^n n!/\gcd(2,n-2)$. We construct the smooth finite normalization and identify its complement of the affine source as $n-2$ disjoint affine divisors. One is ramified; the others record unramified loss of inverse sheets. The normalization has Picard group $\mathbb{Z}^{n-2}$ and the homotopy type of a wedge of $n-2$ two-spheres. Its rational deck group is cyclic, while the polynomial deck group of the source map is trivial. By base change, the normalization describes the preimage of any subvariety of the target; when the target polynomial factors, a component with only constant units is an open subset of a Kummer cover of the simple roots of one factor. Finally, the monodromy separates these maps, for $n\geq 4$, from single weighted lifts under stable polynomial left-right equivalence.

One of the most important problems in computational algebra is the computation of real solutions of a polynomial system. Lasserre, Laurent, and Rostalski introduced a numerical method to compute the real radical of an ideal defining a finite set of real points. Their method is based on moment matrices and reduces the problem to solving a semidefinite program. In this work, we build on this method and on the work of Laurent and Mourrain to exploit the sparsity of the input polynomials. For this purpose, we extend the theory of moment matrices to the setting of affine toric varieties. We prove that, for sparse polynomial systems, this approach allows us to recover real points on affine toric varieties. Moreover, we show that a projection of the spectrahedron associated with the semidefinite program is, in fact, a polytope, and we propose a variation of the method to compute only one real solution of the system.

In this paper, we mainly prove Hovey's conjecture: for any commutative coherent ring, there is a lattice isomorphism between the lattice of wide subcategories of the category of finitely presented modules and that of thick subcategories of the perfect derived category. We construct, for every finitely presented module, a bounded finite free complex whose zeroth homology is the given module and whose homology lies in its abelian closure. This realization gives an inverse to Hovey's correspondence without a closure operation. In addition, we study finite extension filtrations, their behavior under change of rings, and the limits of reconstructing wide subcategories of arbitrary modules from finitely presented modules.

Let $R_{n,p}$ be the determinantal hypersurface ring defined by the determinant of a generic $n \times n$ matrix in characteristic $p$ where $n \geq 2$. In this paper, we obtain explicit bounds for the $F$-signature of $R_{n,p}$. For the upper bound, we prove that this $F$-signature is decreasing in $n$, so the $F$-signature of $R_{2,p}$ serves as an upper bound, which is $2/3$. For the lower bound, we find a toric degeneration $B_n$ of $R_{n,p}$, so the $F$-signature of $B_n$ serves as a lower bound. This $F$-signature can be computed using either Gröbner basis or Han-Monsky machinery, and its value is $\frac{(n!)^2}{(2n-1)!}$.

Let $G_n$ be the bipartite complement of the even cycle $C_{2n}$, and let $J_{G_n}$ be its binomial edge ideal. For $n\geq5$, we study the interaction between the combinatorial structure of $G_n$ and the algebraic invariants of $S_{G_n}/J_{G_n}$. Our main combinatorial ingredient is a classification of the subsets of $V(G_n)$ having the cut point property, obtained through an analysis of disconnected induced subgraphs of $G_n$. We use this classification to describe the minimal primes and to study dimension and degree-theoretic invariants of $J_{G_n}$ and $S_{G_n}/J_{G_n}$. We also study local Vasconcelos numbers, graded Betti numbers and the Hilbert series, and obtain information on projective dimension and depth. Finally, we analyze induced paths in $G_n$ and derive corresponding bounds for the Castelnuovo–Mumford regularity.

Fix a square-free monomial $m = x_{i_1} \cdots x_{i_s}$ in $S = \K[x_1, \dots, x_n]$, where $\K$ is a field. The ideal generated by all square-free monomials that are Borel moves of $m$ is referred to as the square-free principal Borel ideal, denoted by $\mathrm{sfBorel}(m)$. While the Waldschmidt constant of $\mathrm{sfBorel}(m)$ has been partially studied in the literature, we consider the remaining cases and provide bounds for the Waldschmidt constant of $\mathrm{sfBorel}(m)$. In some cases, we explicitly compute its exact value. Furthermore, we study this invariant for square-free Borel and square-free lex-segment ideals, establishing both bounds and exact values for special cases.

Let $F=(f_1,\dots,f_r)$ be a family of nonzero polynomials on $\mathbb C^n$. More than ten years ago, Budur conjectured that $B_F$, the Bernstein–Sato ideal of $F$, is generated by products of linear polynomials with rational coefficients. With the aid of AI, we construct two counterexamples to Budur's conjecture. The examples indicate that main results in [Wu26] provide the optimal linearity properties that the zero locus of Bernstein-Sato ideals satisfies in general.

We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_\alpha$. We first relate their structure to the conjugates of $\alpha$, characterize when $S_\alpha$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_\alpha^\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $\alpha$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[\alpha]$ is a finite factorization domain for every algebraic number $\alpha$, establish criteria ensuring that $S_\alpha$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[\beta]$ for arbitrary algebraic $\beta$, and prove that, for algebraic integers $\alpha$, unique factorization ascends from $\mathbb{Z}[\alpha]$ to $\mathbb{Z}[\alpha/n]$ for every $n\in\mathbb{N}$.

Let $I\subseteq S$ be a monomial ideal with minimal generator degrees between $e$ and $d$, where $2\leq e\leq d$. We prove that if $t_a(S/I)<ea$, then $$ t_{a+b}(S/I)\leq t_a(S/I)+ \left\lceil\frac{(2d-1)b}{2}\right\rceil $$ for $a\geq2$, $b\geq0$, and $a+b\leq\operatorname{pd}_S(S/I)$. Consequently, $$ t_{a+b}(S/I)\leq ae+db-1-\left\lfloor\frac b2\right\rfloor. $$ In particular, if $t_2(S/I)<2e$, then $$ t_i(S/I)\leq \left\lceil\frac{(2d-1)i}{2}\right\rceil-2(d-e) $$ for $2\leq i\leq\operatorname{pd}_S(S/I)$. We also obtain a regularity bound and give a quadratic family in which our estimate for the last shift is smaller than every bound obtained from ordinary subadditivity. The constant $2d-1$ is sharp for $d=2$.

Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.

The $b$-invariant of a normal graded ring was defined recently by Okuma, Watanabe, and Yoshida. We study this for invariant rings of finite groups, proving that it is nonpositive in the case of abelian groups, but not necessarily otherwise; more generally, we give a description of the anticanonical module for invariant rings of finite groups. Turning to infinite groups, we record the anticanonical module and the $b$-invariant for families of determinantal rings. We also construct standard-graded Cohen-Macaulay rings of characteristic zero, with isolated singularities and negative $b$-invariant, that are not of strongly $F$-regular type, thereby disproving a conjecture of Okuma-Watanabe-Yoshida.

Let $C_n^m$ be the $m$-th power of the cycle on $n$ vertices, and let $\Gnm$ be its complement, where $m\ge1$ and $n\ge3m+1$. We prove that every induced subcomplex of the clique complex of $C_n^m$ is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of $R/I(\Gnm)$. We also prove that $R/I(\Gnm)$ is a Buchsbaum ring of depth $2$ and that the independence complex of $\Gnm$ is a triangulated $m$-manifold with boundary for $m\ge2$, namely a Möbius band or an annulus when $m=2$. For the complement $\overline H$ of every graph $H$ in the above class, we show that $\reg(I(\overline H)^k)=2k+\indm(\overline H)-1$ for all $k\ge2$, where $\indm$ denotes the induced matching number. In particular, some power of $I(\Gnm)$ has a linear resolution if and only if $n\ge4m+1$. In this range, we further prove that the colon ideals $(I^{k+1}:M)$, where $I=I(\Gnm)$ and $M$ is a minimal monomial generator of $I^k$, have regularity $2$.

Let $(A,I)$ be a henselian pair such that $A$ is a domain of characteristic zero and $I\ne0$. We prove that any flat affine henselian scheme of finite presentation over $(A,I)$ has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in $V(I)$ has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.

We prove that the support of every double Grothendieck polynomial is an $M^\natural$-convex set. Our main new tool is a rigidity result for the $K$-classes of multiprojective varieties with rational singularities.