Related papers: Nonincreasing depth functions of monomial ideals

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with non-negative entries and non-zero columns using combinatorial optimization and integer…

Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\depth S/I_G^k$ stabilizes is denoted by $\dstab(I_G)$. We show that $\dstab(I_G)<\ell(I_G)$ where $\ell(I_G)$ denotes the…

In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its…

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal $\mathcal{I}$ on $\omega$ we denote $\mathcal{D}_{\mathcal{I}}=\{f\in\omega^\omega: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in \omega$}\}$ and…

In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It follows easily that the Stanley Conjecture holds for the factor if and only if it holds…

Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $\ depth \ A \geq 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) =…

Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq…

Assume that $K$ is a field and $I_{1}\subsetneq ...\subsetneq I_{t}$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_{1},,...,x_{m}]$, for some $m\geq 1$. Suppose that $I_{j}$ is generated by polynomials of…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials…

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked…

For a maximal ideal $\mathfrak m$ of some anemic Hecke algebra $\mathbb{T}^S_\xi$ of a similitude group of signature $(1,d-1)$, one can associate a Galois $\overline{\mathbb F}_l$-representation $\overline \rho_{\mathfrak m}$ as well as a…

Let $A=\{{\bf a}_1,...,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $I_A \subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of…

Let $R$ be a commutative noetherian ring of dimension $d$ and $M$ be a commutative$,$ cancellative$,$ torsion-free monoid of rank $r$. Then $S$-$dim(R[M]) \leq max\{1, dim(R[M])-1 \} = max\{1, d+r-1 \}$. Further$,$ we define a class of…

We study the weak and strong Lefschetz properties for $R/\mathrm{in}(I_t)$, where $I_t$ is the ideal of a polynomial ring $R$ generated by the $t$-minors of an $m\times n$ matrix of indeterminates, and $\mathrm{in}(I_t)$ denotes the initial…

The degree excess function $\epsilon(I;n)$ is the difference between the maximal generating degree $d(I^n)$ of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear…

Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~…

In the paper we show that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid $\mathbf{ID}_{\infty}$ of all partial cofinite isometries of…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $\sdepth(M)$, and conjectured that $\depth(M) \le \sdepth(M)$ for all finitely generated $S$-modules $M$.…