交换代数
While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…
In this paper, we study the canonical trace of Schubert cycles and determinantal rings. As an application, we give an explicit description of the non-Gorenstein locus and show that its structure is compatible with the known representations…
Let $X=(x_{ij})_{m\times n}$ be a matrix of indeterminates and let $S=\mathbb{k}[x_{ij} \mid 1\leq i\leq m,\ 1\leq j\leq n]$ be a polynomial ring over an infinite field $\mathbb{k}$. Let $I$ be an ideal generated by a subset of the set of…
The homological shift algebra and the projective dimension function of complementary edge ideals are investigated. Let $G$ be a connected graph, and let $I$ be its complementary edge ideal. For bipartite graphs $G$, we show that the…
We provide necessary and sufficient conditions for simplicial complexes whose determinantal facet ideals admit reduced Grobner bases under diagonal term orders. Building on and extending foundational results for binomial edge ideals and…
An arrangement of hypersurfaces in projective space is strict normal crossing (SNC) if and only if its Euler discriminant is nonzero. We study the critical loci of arbitrary Laurent monomials in the equations of the smooth hypersurfaces.…
Zonotopal algebras (external, central, and internal) of an undirected graph G introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional commutative graded algebras whose Hilbert series contain a wealth of combinatorial…
Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the…
We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher…
We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type…
Let R be a commutative ring with identity. The concept of second submodule of an R-module (as a dual notion of prime submodules) was introduced and studied by S.Yassemi in 2001. This notion has obtained a great attention by many authors and…
MacWilliams identities relate the weight enumerators of a code with those of its dual and are classically formulated with respect to the Hamming weight. For other metrics, however, these identities often fail when considering the weight…
In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first…
In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a…
It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the…
Let K be a field of characteristic 0. Let P_K(5,103,169) be the defining ideal of the space monomial curve {(t^5,t^{103},t^{169})}. In this paper we shall prove that the symbolic Rees algebra R_s(P_K(5,103,169)) is not Noetherian, that is,…
We extend to multiplicative lattices a theorem of Anderson and Roitman characterizing the cancellation ideals of a commutative ring.
In this paper, we prove that the open neighborhood ideal of a TD-unmixed tree is geometrically vertex decomposable. This result implies that the associated Stanley-Reisner complex is vertex decomposable. We further demonstrate that…
As the binomial edge ideal of a graph is always generated by homogeneous quadratic polynomials corresponding to the edges of the graph, the question of when a binomial edge ideal defines a Koszul algebra has been studied by many authors…
We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong…