Papers

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

We study Cohen–Macaulayness, in the sense of Hamilton–Marley, of non-Noetherian rings arising as big Cohen–Macaulay algebras. Motivated by Bhatt's notion of cohomological Cohen–Macaulayness, we call a locally finite-dimensional ring CCM if its structure sheaf satisfies this condition. Our main comparison theorem shows that every locally finite-dimensional CCM ring is locally HMCM. Using this theorem, we prove that if $A$ is Noetherian and $R$ is an integral $A$-algebra that is locally balanced big Cohen–Macaulay over $A$, then $R$ is CCM and hence locally HMCM. In particular, if $A$ is an excellent Noetherian domain, $p$ is a prime, $n\geq1$, and $A/pA\neq0$, then $A^+/p^nA^+$ is CCM and locally HMCM. We also show, using finite-dimensional valuation domains, that CCM is strictly stronger than locally HMCM. Finally, for a Noetherian ring $A$ of characteristic $p>0$ and its perfection $A_{\mathrm{perf}}$, we prove that the following conditions are equivalent: $A$ is locally weakly $F$-nilpotent; $A_{\mathrm{perf}}$ is a locally balanced big Cohen–Macaulay $A$-algebra; and $A_{\mathrm{perf}}$ is CCM. Under these equivalent conditions, $A_{\mathrm{perf}}$ is locally HMCM.

Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.

Let $J_G$ denote the binomial edge ideal of a finite graph $G$ in the polynomial ring $S$. We determine all triples $(n,t,d)$ with $n\geq 3$ for which there exists a finite connected graph $G$ on $n$ vertices with ${\mathrm{depth}}(S/J_{G})=t$ and $\dim(S/J_G)=d$.

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We isolate the local ultrafilter hypothesis $\textsf{LUH}$: the existence of a strongly compact cardinal implies $\textsf{LUH}$, while $\textsf{LUH}$ implies the existence of a measurable cardinal. Assuming $\textsf{LUH}$, we construct a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ which is not Gorenstein flat; hence $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$.

We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen–Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen–Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen–Macaulayness are equivalent.

In this article, we study Hankel determinantal rings, a special class of determinantal rings defined by minors of Hankel matrices of indeterminates. We discuss the question posed in \cite[Question~4.8]{conca2018hankel} by showing that the test ideal of a Hankel determinantal ring coincides with the ring itself.

We initiate a systematic study of Inc-invariant chains of graphs, the combinatorial counterparts of Inc-invariant chains of edge ideals arising in the theory of equivariant Noetherianity. Such a chain consists of graphs on growing vertex sets whose edge sets are compatible with the action of the monoid of strictly increasing maps on the positive integers. We show that several associated combinatorial invariants exhibit rigid asymptotic behavior. The independence number eventually stabilizes, and every fixed entry of the $f$-vector and the $h$-vector of the independence complex is eventually linear. For clique complexes, every fixed entry of the $f$-vector is eventually polynomial, whereas the entries of the $h$-vector are eventually quasi-polynomial. Moreover, the clique and chromatic numbers are eventually quasi-linear, and their difference is eventually at most one. We also prove that the matching number eventually attains the maximal value $\lfloor n/2\rfloor$. Finally, admissible and minimal paths eventually have lengths at most $3$ and $5$, respectively, and their maximal lengths stabilize. These results reveal strong asymptotic regularity in graph families governed by increasing symmetry.

We prove an asymptotic structural property of generic initial ideals. This single phenomenon yields results on Hilbert functions, persistence, hyperplane restriction, graded Betti numbers, combinatorial shadow minimization, and Lefschetz properties.

This paper is a continuation of arXiv:2606.17088, where I studied precartesian coextensions of commutative monoids by systems of abelian groups. In this paper, we generalise it to study coextensions by systems of commutative monoids. Among other things, we showcase that this version is able to simultaneously generalise Leech's version of coextensions and Redei's version, also called Schreier coextensions. We show that what going from systems of abelian groups to systems of commutative monoids costs us is quasi-inverses in $\mathsf{Pcoex}(M, \mathcal{L})$. Specifically, instead of a symmetric categorical group, they now only become symmetric monoidal groupoids. Moreover, though not explicitly stated in the body of the paper, another core difference is that for a monoid scheme $X$, regarded as a monoid functor, a precartesian coextension no longer need to be a monoid scheme. Thus, $\mathsf{Pcoex}(X, \mathcal{L})$ is, as stated, ineffective at studying monoid scheme coextensions. The rest of the theory goes through, but requires developing a new cohomological approach.

A multiplicity-free subvariety of a product of projective lines is a subvariety with the property that the expansion of its Chow class in the usual basis has all coefficients equal to 0 or 1. We show that, over a field of positive characteristic, a multiplicity-free subvariety has F-rational singularities. This strengthens Brion's result that multiplicity-free subvarieties are normal and Cohen–Macaulay, and that they have rational singularities over a field of characteristic 0.

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

Let R be a principal ideal domain and let $HR$ denote the Hurwitz series ring over R. We first give a recursive splitting formula for a Hurwitz series whose constant coefficient is a product of two coprime nonunits. This construction leads to irreducibility tests for prime-power constant coefficients, decompositions indexed by the distinct prime divisors of the constant coefficient, and upper and lower bounds for the length of every irreducible factorization. Over a discrete valuation domain, the valuation of any selected coefficient yields a sharper length bound. We then introduce the Hurwitz–Newton polygon and prove a product rule on ranges in which the relevant binomial coefficients are units. Primitive one-edge polygons consequently provide Dumas-type irreducibility criteria. Finally, localization is used to combine independent criteria at several prime elements. Examples over Z, localized polynomial rings, and the Gaussian integers illustrate the results.

We give two related but different methods for constructing rings $R$ of Krull dimension 1 and $K_{-d}(R) \neq 0$. The first works for $d = 2$, and the second works for any $d \geq 2$.

We establish Chudnovsky's Conjecture and, more generally, Demailly's Conjecture for any finite set of points in $\mathbb{P}^n_k$, where $k$ is an algebraically closed field of characteristic 0. The key idea of passing to positive characteristic, as well as the broad plan of the proof, was suggested by ChatGPT-5.6 Sol. All mathematical arguments and proofs in this paper were developed, written, and verified by the authors.

For a simplicial complex $\Delta$ and integers $n_i\ge 2$, set $A_{\Delta,\mathbf n}=\mathbb{k}[x_1,\ldots,x_m]/(I_\Delta+(x_1^{n_1},\ldots,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant ghost vertices. The formula extends the free-face formula of Gasanova–Herzog–Hibi–Moradi for square-zero flag face algebras and recovers, in the simplex-boundary case, a special case of their formula for monomial almost complete intersections. As a first consequence, we classify the nearly Gorenstein algebras in this family: on each connected component $C$ of $\Delta^{(1)}$, the induced complex is either the simplex $2^C$, with arbitrary exponents, or the boundary $\partial 2^C$, with every exponent equal to two. We also compute the Teter number on this nearly Gorenstein locus. For flag complexes the trace is generated by the free-face monomials for arbitrary exponents, and we characterize the equalities $\operatorname{tr}_A(\omega_A)=\mathfrak m_A^q$. In the square-zero one-dimensional case we isolate the additional contribution coming from triangle components.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and residue field $k$. Let $M$ be a maximal Cohen-Macaulay $A$-module. Let $e(M)$ be the multiplicity of $M$ and let $\mu(M)$ denote the number of its minimal generators. (1) Assume $A$ is not a complete intersection. If $\text{curv}(M) < \text{curv}(k)$ then we prove that under mild conditions, $e(M) \geq \mu(M)(1 + \text{curv}(k))$. (2) Assume $A$ is a complete intersection. If $\text{cx}(M) < \text{cx}(k)$ then we prove that $e(M) \geq 2\mu(M)$. In both cases we give examples which shows our results are sharp.

In this paper, we investigate algebraic properties of two classes of monomial ideals associated with graphs, namely path ideals and closed neighborhood ideals. Our primary focus is on the strong persistence property and several homological invariants, including depth and the $v$-number. We first prove that the path ideal of length $n-1$ associated with a path graph on $n\geq4$ vertices attached to an arbitrary graph satisfies the strong persistence property. We then study closed neighborhood ideals and establish the strong persistence property for graphs containing a leaf vertex. Furthermore, we identify a broader class of graphs satisfying this property, namely those whose vertex set admits a decomposition $V(G)=C\sqcup\{v\}\sqcup X,$ where $C$ is a nonempty clique, every vertex of $C$ is adjacent to each vertex of $X\cup\{v\}$, and the vertex $v$ has no neighbors in $X$. In addition, we investigate algebraic invariants of ideals associated with Helm graphs. Specifically, we compute the depth of the edge ideal and determine the Krull dimension, the depth, and the $v$-number of the closed neighborhood ideal of the Helm graph. These results contribute to a deeper understanding of the interplay between graph-theoretic structures and the algebraic properties of their associated monomial ideals.

We show that there exists an $F$-finite four-dimensional local domain $(R,\mathfrak{m})$ of characteristic two which is not $F$-injective but which admits a nonzerodivisor $f\in \mathfrak{m}$ such that $R/fR$ is $F$-injective.

A subsemiring $S$ of the nonnegative cone of the real line containing $1$ is called a positive semiring. The Bi-UF Positive Conjecture, posed by Baeth, Chapman, and Gotti in 2021, states that the prototypical semiring $\mathbb{N}_0$ is the only positive semiring whose additive monoid $(S,+)$ and nonzero multiplicative monoid $(S^\bullet,\cdot)$, where $S^\bullet=S\setminus\{0\}$, are both factorial. In this paper, we prove their conjecture.

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ with $|V(G)|$ variables. We prove that the local $v$-number of the symbolic powers $J(G)^{(t)}$ is linear for all $t \ge 1$ when $G$ is bipartite, and quasi-linear with period two for all $t \ge 2$ when $G$ is non-bipartite. Furthermore, we provide explicit formulas for these invariants in terms of the combinatorial data of $G$.

Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0 < k < \min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood–Richardson tableaux.

We obtain estimates for the roots of the Bernstein-Sato polynomial in terms of a log resolution by an orbifold.

Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$. We construct finitely many correlated Gaussian moments that determine $P$ up to an orthogonal change of variables. The number of copies and the rational covariance matrices are specified in terms of $d,m$. On coefficient balls, the induced map on orthogonal orbits has a Hölder continuous inverse with an explicit exponent. For $m\geq3$, the normalised local stationary-phase expansion at the critical branch through the origin equals the Gaussian moment series. Its coefficients recover the same finite moment vector, without assumptions on the other critical points. If the leading homogeneous part has an isolated complex critical point, we compute the twisted de Rham ranks and construct formal lattices on each fixed coupling support.

The Quot scheme $\operatorname{Quot}^{q}(\mathcal{O}_{\mathbb{P}^n}^{r})$ parametrizes the quotients of the trivial vector bundle of rank $r$ on $n$-dimensional projective space that have Hilbert polynomial $q$ and are flat over a base scheme. We identify numerical conditions on the polynomial $q$ that completely determine when this Quot scheme is smooth and irreducible. Our approach also uncovers further geometric features of the projective scheme $\operatorname{Quot}^{q} (\mathcal{O}_{\mathbb{P}^n}^{r})$ including the smoothness of the lexicographic point.

We characterize essential ideals in polynomial rings $R[(X_\lambda)_{\lambda\in\Lambda}]$ over a commutative ring $R$ with $1$. We present several characterizations, letting $R$ vary among notable classes of rings. The idea is to detect essentiality by checking the intersections with principal ideals generated by polynomials in a test class. In the general commutative case, we apply McCoy's zero-divisor criterion to construct an efficient test class of polynomials in terms of annihilators of content ideals. We then refine the test class and strengthen the result in several ways, assuming further properties on the coefficients ring $R$. Assuming that $R$ is Noetherian, we reduce the test class using the associated primes of $R$. When $R$ satisfies Serre's condition $(S_1)$, it suffices to use minimal primes. We also study the cases of $R$ Artinian, $R=\mathbb{Z}/n\mathbb{Z}$ and $R$ equal to the ideal-adic completion of an excellent ring, among others.

This article studies dominant local rings, a notion introduced by Takahashi. We prove that, for a flat local homomorphism R–>S between commutative noetherian local rings, dominance descends from S to R, thereby answering Takahashi's descent question affirmatively. The proof makes use of Krause's realization of the idempotent completion of the singularity category as the subcategory of compact objects in the homotopy category of acyclic complexes of injective modules.

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. Let $\Delta$ be a subset of $\operatorname{Spec} R$, and let $\operatorname{Ass}^{-1}\Delta$ stand for the full subcategory of $\operatorname{mod} R$ consisting of finitely generated $R$-modules whose associated prime ideals belong to $\Delta$. In this paper, we consider classifying torsion pairs in $\operatorname{Ass}^{-1}\Delta$ and some other full subcategories of $\operatorname{mod} R$.

By using an advanced model of AI, it is proved that there exists a commutative ring having a purely-prime ideal which is not purely-maximal. This gives a counterexample to \cite[Conjecture 5.8]{Tarizadeh}.

In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly $\aleph_1$-presented module $M$ satisfying Ext$_R^{\ge 1}(M,D^{(\aleph_1)})=0$ belongs to the left orthogonal class of $\overline{D}$, where $\overline{D}$ is the definable of $D$. This yields the consequence that every $\aleph_1$-generated strongly Gorenstein projective module is Gorenstein flat. Furthermore, we investigate when a flat Gorenstein projective module is projective. Finally, for any two-sided $\aleph_1$-coherent ring $R$, we prove the identity $\operatorname{silp}R+\operatorname{silp}R^{\mathrm{op}}=\operatorname{spli}R+\operatorname{spli}R^{\mathrm{op}}$.