1,345 papers
An infinite series of Gorenstein local algebras failing the affine homogeneity property
We provide an infinite series of commutative finite-dimensional Gorenstein local algebras $A_n$ for $n \ge 2$. We give an elementary proof that the maximal ideal of every algebra $A_n$ possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of $A_n$. The latter implies that the algebras $A_n$ fail the affine homogeneity property. We also discuss some consequences concerning additive actions on projective hypersurfaces, related to the generalized Hassett-Tschinkel correspondence for these algebras.
2604.04793Apr 2026
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2604.04252The Bourbaki degree of the syzygy module of 2 $\times$ 4 matrices
We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $Θ$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the syzygy module \(\operatorname{Syz}(Θ)\), generalizes previous constructions for plane curves and Jacobian matrices. Our main result is an explicit formula expressing the Bourbaki degree in terms of the degrees of the rows, the initial degree of a syzygy, and the first two Hilbert coefficients of the cokernel module \(\mathcal{Q} = \operatorname{coker}(Θ)\). We apply this framework to two important cases. First, matrices with constant first row, which are determined by a three-equigenerated ideal \(J = (f_1, f_2, f_3)\), where we show the Bourbaki degree measures how far \(J\) is from being a perfect ideal, and we completely characterize its smaller and larger values. Second, for a linear matrix, we use the Kronecker--Weierstrass classification to determine all possible Bourbaki degrees and homological types. This classification reveals the existence of a linear matrix with Bourbaki degree equal to 2, a value that does not occur for Jacobian matrices. Finally, in the geometric context of \(\mathbb{P}^3\), we provide a sufficient condition for \(\operatorname{Syz}(Θ)\) to define a codimension one distribution and obtain bounds on the Bourbaki degree when the initial degree is small.
2604.04252Apr 2026
View2604.02561On topologies on the space of valuations and the valuative tree
In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,Λ)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,Λ)$ and describe the supremum of an increasing family of valuations in a special subtree. We also view the valuative tree as a subset of the product $(Λ_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology.
2604.02561Apr 2026
View2604.00638Explicit minimal generating sets of a family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring
We display a new family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring over a field of characteristic zero. These primes are obtained as the kernel of a quasi-monomial algebra homomorphism. Up to constant coefficients, determined by some specific linear systems with binomial entries, we describe their minimal generating polynomial sets. The advantage of our family with respect to some previous work is, on the one hand, the explicit description of the generating sets and, on the other hand, the simplicity of the exponents of the aforementioned quasi-monomial homomorphism. We also provide a code in Python which states and solves the linear systems that lead to a complete description of the minimal generating sets with a "Gröbner-free" approach.
2604.00638Apr 2026
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Cohen-Macaulay and Gorenstein Properties of Bi-Amalgamated Algebras with Applications to Algebroid Curves
Let $A \bowtie^{f,g} (J,J')$ be the bi--amalgamation of a commutative ring $A$ with $(B,C)$ along the ideals $(J,J')$ with respect to the ring homomorphisms $(f,g)$. In this article, we study the basic homological properties of the bi--amalgamated algebra construction. We first calculate the dimension and depth of the bi--amalgamated algebra under fairly general circumstances and derive necessary and sufficient conditions for Cohen--Macaulayness in terms of maximal and big Cohen--Macaulay modules of $A$. Furthermore, we characterize the Gorenstein property of the bi--amalgamated algebra through the canonical modules of $f(A)+J$ and $g(A)+J'$. We apply our results to the theory of curve singularities by constructing Gorenstein algebroid curves through bi--amalgamated and amalgamated algebras. We also give a brief remark concerning the universally catenary property of $A\bowtie^{f,g}(J,J')$.
2604.00266Mar 2026
View2603.29978The van der Waerden Simplicial Complex and its Lefschetz Properties
The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(n,k)$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A(n,3)$ that have the Weak Lefschetz Property. We conjecture that $A(n,k)$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A(n,k)$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.
2603.29978Mar 2026
View2603.29714Toward the theory on local cohomologies at the ideals given by simplicial posets
For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x \mid x \in P \setminus \{\widehat{0} \}]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded, and $I_P$ is not a monomial ideal. To develop the theory on the local cohomology $H_{I_p}^i(S)$ and its injective resolution, this paper establishes the foundation. Specifically, we give an explicit description of the graded injective envelope ${}^*\! E_S(S/\mathfrak{p}_x)$, where $\mathfrak{p}_x$ is the prime ideal associated with $x \in P$. We also analyze morphisms between them.
2603.29714Mar 2026
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The $\v$-number of generalized binomial edge ideals of some graphs
Let \(G\) be a finite connected simple graph, and let \(\calJ_{K_m,G}\) denote its generalized binomial edge ideal. By investigating the colon ideals of \(\calJ_{K_m,G}\), we derive a formula for the local \(\v\)-number of \(\calJ_{K_m,G}\) with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has \(\v\)-numbers 1 or 2. When \(G\) is a connected closed graph, we compute the local \(\v\)-number of \(\calJ_{K_2,G}\) by generalizing the work of Dey et al. Additionally, under the condition that \(G\) is Cohen--Macaulay, we derive formulas for the \(\v\)-number of \(\calJ_{K_m,G}\) and \(\calJ_{K_2,G}^k\), and show that the \(\v\)-number of \(\calJ_{K_2,G}^k\) is a linear function of \(k\).
2603.29516Mar 2026
View2603.29104Regular rings over valuation rings
Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring $A$ essentially finitely presented over a valuation ring $V$. We discuss several properties of this notion of regularity for such $A$, obtaining results parallel to results for regularity of Noetherian local rings. We include classical and modern topics: openness of loci, perfectoid big Cohen--Macaulay algebras, and cotangent complexes. We also give an application to Noetherian rings, showing a version of Kodaira's vanishing theorem in large enough residue characteristics.
2603.29104Mar 2026
View2603.28293Symplectic completion over smooth affine algebras
In this article, we prove the following results:\\ \noindent \text{(1).} Let $R$ be a smooth affine algebra of dimension $3$ over an algebraically closed field $K$ with $3!\in K$, then we show that $\Um_4(R)=e_1\Sp_4(R)$ and $\Um_4(R [X])=e_1\Sp_4(R[X])$. \noindent \text{(2).} We also show that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_3(R)=e_1\SL_3(R)$. As a consequence it is shown that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_4(R)=e_1\Sp_4(R)$. \noindent \text{(3).} We show that if $R$ is a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R[X])=e_1\Sp_4(R[X])$. \noindent \text{(4).} We also show that if $R=\oplus_{i\geq 0}R_i$ is a graded ring over a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R)=e_1\Sp_4(R)$.
2603.28293Mar 2026
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Private neighbors, perfect codes and their relation with the $\vt$-number of closed neighborhood ideals
In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\vt$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\vt$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\vt$-number is a lower bound for the regularity of bipartite and well-covered graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\vt$-number of some special family of graphs.
2603.28247Mar 2026
View2603.27166Supports of intersections of thick subcategories and proxy smallness
Let R be a commutative noetherian ring. Let D^b(R) be the bounded derived category of finitely generated R-modules. Let X and Y be thick subcategories of D^b(R). In this paper, we consider the question asking when the equality Supp(X\cap Y)=Supp X\cap Supp Y holds, and give several answers. As applications, we obtain a characterization of the proxy small subcategories, and classifications of certain thick subcategories.
2603.27166Mar 2026
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Relative Gröbner bases of modules and applications in persistence theory
Finitely generated modules over the polynomial ring in $n$ indeterminates are isomorphic to quotients of finite rank free modules. We introduce a theory of relative Gröbner bases for those quotients of free modules and, equivalently, for pairs of submodules; we prove corresponding Buchberger- and Schreyer-type theorems. As applications of this theory, we consider three problems in persistence theory, which can be solved by relative Gröbner bases. First, we show that the relative Schreyer's theorem can be used to compute free presentations of complexes of finitely generated torsion-free modules. In contrast to previous approaches, this allows computation of free presentations for multicritical persistent homology directly at the chain module level without additional topological constructions. Second, any finitely generated Artinian module embeds in an Artinian injective hull, giving rise to a flat-injective presentation. We represent the embedding of the module in this injective hull by a quotient of a free module and apply the relative Schreyer's theorem to construct an algorithm for the computation of a free presentation from a flat-injective presentation. Third, we investigate how free presentations, and more generally free resolutions, obtained by the two preceding applications can be minimized by standard reduction techniques.
2603.27149Mar 2026
View2603.26574Modules of logarithmic derivations in weighted projective spaces and applications to free divisors
We introduce a weighted version of the module of logarithmic derivations of a divisor in weighted projective space, and provide a generalization of Saito's criterion for freeness in terms of weighted multiple eigenschemes (wME-schemes). Freeness of the nonstandard Z-graded module allows one to consider big families of free divisors in affine and standard projective space, i.e. when the module of logarithmic derivations of the divisor is free over the respective coordinate rings. We present a method to identify and construct these new families of free divisors in affine and projective space in any dimension, and give numerous explicit examples.
2603.26574Mar 2026
View2603.26289On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action
T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.
2603.26289Mar 2026
View2603.25604On Bass numbers of graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals in mixed characteristic
Let $A$ be a Dedekind domain of characteristic zero such that for each height one prime ideal $\mathfrak{p}$ in $A$, the local ring $A_{\mathfrak{p}}$ has mixed characteristic with finite residue field. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e., $\operatorname{deg} A=\underline{0}\in \mathbb{N}^n$ and $\operatorname{deg}(X_j)=e_j\in \mathbb{N}^n$. Let $I$ be a $\mathfrak{C}$-monomial ideal of $R$ and let $M:= H^i_I(R)=\bigoplus_{\underline{u}\in \mathbb{Z}^n}M_{\underline{u}}$. Recently, the second author and S. Roy [2025, J. Algebra 681, 1-21] proved that for a fixed $\underline{u}\in\mathbb{Z}^n$, the Bass numbers $μ_i(\mathfrak{p},M_{\underline{u}})$ are finite for each prime ideal $\mathfrak{p}$ in $A$ and for every $i\geq 0$. Let for a subset of $U$ of $\mathcal{S}=\{1, \ldots, n\}$, define a block to be the set $\displaystyle\mathcal{B}(U)=\{\underline{u} \in \mathbb{Z}^n \mid u_i \geq 0 \mbox{ if } i \in U \mbox{ and } u_i \leq -1 \mbox{ if } i \notin U \}$. Note that $\bigcup_{U\subseteq \mathcal{S}}\mathcal{B}(U)=\mathbb{Z}^n$. In this article, the main result we establish is that for a fixed prime ideal $\mathfrak{p}$ in $A$ and $i\geq 0$, the set of Bass numbers $\{μ_i(\mathfrak{p},M_{\underline{u}})\mid \underline{u}\in \mathbb{Z}^n\}$ is constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1, \ldots, n\}$. Our idea is to prove this by carrying out a comprehensive study of the structure theorem for the graded components of $M$ when $A$ is a complete DVR of mixed characteristic with finite residue field.
2603.25604Mar 2026
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On Stanley-Reisner Rings with Minimal Betti Numbers
We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For example, the Betti numbers of their Stanley-Reisner rings are given by the binomial coefficients, and their full subcomplexes are homotopy equivalent either to a point or to a sphere. These properties make it possible for us to either classify them or construct them inductively from instances with fewer vertices.
2603.25540Mar 2026
View2603.25013Quasi-factorially closed subalgebras of Laurent polynomial rings
Let $R$ be a domain and $B=R[x_1^{\pm1},\ldots,x_n^{\pm1}]$ the Laurent polynomial ring over $R$. In this paper we study pre-factorially closed (pfc) and quasi-factorially closed (qfc) $R$-subalgebras of $B$, which generalize the notion of factorially closed subalgebras. We first establish a localization criterion for the qfc property. Using this criterion, we investigate monoid algebras $A=R[M]$ associated with submonoids $M\subset \mathbb{Z}^n$. We prove that $R[M]$ is qfc in $B$ if and only if the group generated by $M$ is a direct summand of $\mathbb{Z}^n$. This provides a complete characterization of the qfc property in terms of the lattice structure of the associated group. As a consequence, when $n=1$ and $M\subset\mathbb{N}$, the algebra $R[M]$ is qfc in $B$ precisely when $M$ is a numerical semigroup. For a general $R$-subalgebra $A\subset B$, we introduce an invariant $\mathrm{Gap}(A)$. We show that if $\mathrm{Gap}(A)$ is finite, then $A$ is qfc in $B$. Moreover, we clarify how the pfc and qfc conditions are related to other notions that naturally appear for subalgebras, such as retracts, being algebraically closed in $B$, and normality.
2603.25013Mar 2026
View2603.24404Describing Multivariate Polynomial Subalgebras Using Equations
Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[ A = A_{0} \subset A_{1} \subset \ldots \subset A_m = \mathbb{K}[x_{1}, \ldots, x_n], \] where each $A_i$ can be written as the kernel of some linear functional $L_{i + 1} : A_{i + 1} \to \mathbb{K}$, and each $L_i$ is either a derivation or of the form $L_i : f \to c(f(\mathbfα) - f(\mathbfβ))$ for some $\mathbfα, \mathbfβ \in \mathbb{K}^{n}$ and $c \in \mathbb{K}$. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such $L_i$ which is a derivation may be written as a linear combination of partial derivatives evaluated at points of $\mathbb{K}^{n}$.
2603.24404Mar 2026
View2603.23709On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$
In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.
2603.23709Mar 2026
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