Related papers: Betti tables of $p$-Borel-fixed ideals

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the…

Let $\mathcal R$ be a principal ideal domain and $\mathcal K = {\rm quot}(\mathcal R)$. Assume that $P_1,\ldots P_n\in \mathcal K[X]$ are polynomials which take $\mathcal R$ to $\mathcal R$, and $P$ is their product. If the $P_i$ satisfy…

We consider the ideal of inner $2$-minors $I_{\mathcal{P}}$ of a finite set of cells $\mathcal{P}$, which we call the cell ideal of $\mathcal{P}$. A nice interpretation for the height of an unmixed ideal $I_{\mathcal{P}}$, in terms of the…

Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that…

Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is…

We extend a result of Caviglia and Sbarra to a polynomial ring with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal…

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal…

Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable…

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$,…

This paper concerns the homological properties of a module $M$ over a commutative noetherian ring $R$ relative to a presentation $R\cong P/I$, where $P$ is local ring. It is proved that the Betti sequence of $M$ with respect to $P/(f)$ for…

The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general…

In this paper, we investigate which classes of monomial ideals have a quasi-additive property of homological shift ideals. More precisely, for a monomial ideal $I$ we are interested to find out whether $HS_{i+j}(I)\subseteq HS_i(HS_j(I))$.…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

We consider a combinatorial property isolated in the field of ideal convergence, a P-property for two ideals on natural numbers. We show that among selected ideals induced by disjoint families, not all pairs satisfy P-property for two…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…

Let $P$ be a finite partially ordered set with unique minimal element $\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\in P$ for which the open interval $(\hat{0}, q)$ is acyclic. Using basic simplicial topology,…

Given two finite sequences of positive integers $\alpha$ and $\beta$, we associate a square free monomial ideal $I_{\alpha,\beta}$ in a ring of polynomials $S$, and we recursively compute the algebraic invariants of $S/I_{\alpha,\beta}$.…

In this note, we study Serre's property $(S_i)$, and its relation to Alexander duality for monomial ideals in a polynomial ring over a field. We describe ideals that define the non-Cohen-Macaulay- and the non-$(S_i)$-loci of finitely…

In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our…

Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gr\"obner bases relative to the graded reverse…