Antonino Ficarra

• Postdoc
• University of Messina

In the present paper, motivated by a conjecture of Jahan and Zheng, we prove that componentwise polymatroidal ideals have linear quotients. This solves positively a conjecture of Bandari and Herzog. We introduce componentwise discrete polymatroids, as the combinatorial counterpart of componentwise polymatroidal ideals, and show that they are shella...

We classify all graphs $G$ satisfying the property that all matching powers $I(G)^{[k]}$ of the edge ideal $I(G)$ are bi-Cohen-Macaulay for $1\le k\le\nu(G)$, where $\nu(G)$ is the maximum size of a matching of $G$.

Let $G$ be a permutation graph. We show that $G$ is Cohen-Macaulay if and only if $G$ is unmixed and vertex decomposable. When this is the case, we obtain a combinatorial description for the $a$-invariant of $G$. Moreover, we characterize the Gorenstein permutation graphs.

In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal $I$, we conjecture that every symbolic power $I^{(k)}$ is componentwise linear and $$ \text{reg}\,I^{(k)}=\text{reg}\,I^k $$ for all $k \ge 1$. We prove that $\text{reg}\,I^{(k)}\ge\t...

In this paper, we study the Dao numbers 𝔡1(I), 𝔡2(I) and 𝔡3(I) of an ideal I of a Noetherian local ring (R, 𝔪,K) or a standard graded Noetherian K-algebra. They are defined as the smallest ℓ ≥ 0 such that I𝔪k is 𝔪-full, full, weakly 𝔪-full, respectively, for all k ≥ ℓ. We provide general bounds for the Dao numbers in terms of the Castelnuovo–Mumfor...

We present the theory of cotangent functors following the approach of Palamodov, and a conjecture of Herzog relating the vanishing of certain cotangent functors to the property of being a complete intersection.

We introduce the concept of matching powers of monomial ideals. Let I be a monomial ideal of S = K[x1,…,xn], where K is a field. The kth matching power of I is the monomial ideal I[k] generated by the products u1⋯uk where u1,…,uk is a sequence of support-disjoint monomials contained in I. This concept naturally generalizes the notion of squarefree...

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an...

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th homological shift algebras $\text{HS}_i(\mathcal{R}(I))=S\oplus\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. These algebras have the structure of a finitely generated bigraded module over the Rees algebra $\...

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring R with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo–Mumford regularity of appropriate...

Let $\Gamma$ be a $d$-flag sortable simplicial complex. We consider the toric ring $R_{\Gamma}=K[{\bf x}_Ft:F\in \Gamma]$ and the Rees algebra of the facet ideals $I(\Gamma^{[i]})$ of pure skeletons of $\Gamma$. We show that these algebras are Koszul, normal Cohen-Macaulay domains. Moreover, we study the Gorenstein property, the canonical module, a...

In this paper, we study the componentwise linearity of symbolic powers of edge ideals. We propose the conjecture that all symbolic powers of the edge ideal of a cochordal graph are componentwise linear. This conjecture is verified for some families of cochordal graphs, including complements of block graphs and complements of proper interval graphs....

Let [Formula: see text] be a finite simple graph on [Formula: see text] non-isolated vertices, and let [Formula: see text] be its binomial edge ideal. We determine almost all pairs [Formula: see text], where [Formula: see text] ranges over all finite simple graphs on [Formula: see text] non-isolated vertices, for any [Formula: see text].

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analogue of the famous Tutte theorem regarding graphs having a perfect matching. For chor...

We give a new, elementary proof of the celebrated Herzog-Hibi-Zheng theorem on powers of quadratic monomial ideals.

Let S be a standard graded polynomial ring over a field K in a finite set of variables, and let 𝔪 be the graded maximal ideal of S. It is known that for a finitely generated graded S-module M and all integers k ≫ 0, the module 𝔪kM is componentwise linear. For large k we describe the pattern of the Betti table of 𝔪kM when depthM > 0. Moreover, we sh...

We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application, we determine the canonical trace \(tr (\omega _R)\) of a Cohen–Macaulay ring R of codimension two, which is generically Gorenstein. It is shown that if the defining ideal I of R is generated by n elements,...

In the present paper, we investigate a conjecture of J\"urgen Herzog. Let $S$ be a local regular ring with residue field $K$ or a positively graded $K$-algebra, $I\subset S$ be a perfect ideal of grade two, and let $R=S/I$ with canonical module $\omega_R$. Herzog conjectured that the canonical trace $\text{tr}(\omega_R)$ is obtained by specializati...

A very well-covered graph is a well-covered graph without isolated vertices such that the size of its minimal vertex covers is half of the number of vertices. If G is a Cohen–Macaulay very well-covered graph, we deeply investigate some algebraic properties of the cover ideal of G via the Rees algebra associated to the ideal, and especially when G i...

Let $K$ be a field, $I\subset R=K[x_1,\dots,x_n]$ and $J\subset T=K[y_1,\dots,y_m]$ be graded ideals. Set $S=R\otimes_KT$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions $\text{v}(I^k)$ and $\text{v}(J^k)$ is investigated. When $I$ and $J$ are monomial ideals, we describe $\text{v}(L^...

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo-Mumford regularity of appropria...

Let [Formula: see text] be the standard graded polynomial ring, with [Formula: see text] a field, and let [Formula: see text], [Formula: see text], be a [Formula: see text]-tuple whose entries are non-negative integers. To a t-spread ideal [Formula: see text] in [Formula: see text], we associate a unique [Formula: see text]-vector and we prove that...

Let \(I\subset S\) be a graded ideal of a standard graded polynomial ring S with coefficients in a field K, and let \({\text {v}}(I)\) be the \({\text {v}}\)-number of I. In previous work, we showed that for any graded ideal \(I\subset S\), then \({\text {v}}(I^k)=\alpha (I)k+b\), for all \(k\gg 0\), where \(\alpha (I)\) is the initial degree of I...

In this note, we classify all the weighted oriented forests whose edge ideals have the property that one of their matching powers has linear resolution.

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein,...

The v-function of a graded filtration I = {I [k] } k≥0 is introduced. Under the assumption that I is Noetherian, we prove that the v-function v(I [k]) is an eventually quasi-linear function. This result applies to several situations, including ordinary powers, and integral closures of ordinary powers, among others. As another application, we invest...

The Cohen-Macaulay vertex splittable ideals are characterized. As a consequence, we recover several Cohen-Macaulay classifications of families of monomial ideals known in the literature by new simpler combinatorial proofs.

In the present paper, we study the Dao numbers d 1 (I), d 2 (I) and d 3 (I) of an ideal I of a Noetherian local ring (R, m, K) or a standard graded Noetherian K-algebra. They are defined as the smallest ℓ ≥ 0 such that Im k is m-full, full, weakly m-full, respectively, for all k ≥ ℓ. We provide general bounds for the Dao numbers in terms of the Cas...

For an ideal I in a Noetherian ring R, the Fitting ideals Fitt j (I) are studied. We discuss the question of when Fitt j (I) = I or Fitt j (I) = √ I for some j. A classical case is the Hilbert-Burch theorem when j = 1 and I is a perfect ideal of grade 2 in a local ring.

In the present paper, motivated by a conjecture of Jahan and Zheng, we prove that componentwise polymatroidal ideals have linear quotients. This solves positively a conjecture of Bandari and Herzog.

We introduce the Macaulay2 package MatchingPowers. It allows to compute and manipulate the matching powers of a monomial ideal. The basic theory of matching powers is explained and the main features of the package are presented.

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the d...

We introduce the concept of matching powers of monomial ideals. Let I be a monomial ideal of S = K[x 1 ,. .. , x n ], with K a field. The kth matching power of I is the monomial ideal I [k] generated by the products u 1 · · · u k where u 1 ,. .. , u k is a monomial regular sequence contained in I. This concept naturally generalizes that of squarefr...

Let I ⊂ S be a graded ideal of a standard graded polynomial ring S with coefficients in a field K, and let v(I) be the v-number of I. In previous work, we showed that for any graded ideal I ⊂ S generated in a single degree, then v(I k) = α(I)k + b, for all k ≫ 0, where α(I) is the initial degree of I and b is a suitable integer. In the present pape...

We introduce the Macaulay2 package HomologicalShiftIdeals. It allows to compute the homological shift ideals of a monomial ideal, and to check the homological shift properties, including having linear resolution, having linear quotients, or being polymatroidal. The theory behind these concepts is explained and the main features of the package are p...

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module $ \mathfrak{m}^kM$ is componentwise linear. For large $k$ we describe the pattern of the Betti table of $ \mat...

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear functions in $k$ are determined for $\text{v}(I^k)$. We call $\text{v}(I^k)$ the $\text{v}$-function of $I$. Unde...

Let $\Delta$ be a 1-dimensional simplicial complex. Then $\Delta$ may be identified with a finite simple graph $G$. In this article, we investigate the toric ring $R_G$ of $G$. All graphs $G$ such that $R_G$ is a normal domain are classified. For such a graph, we determine the set $\mathcal{P}_G$ of height one monomial prime ideals of $R_G$. In the...

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\ne(0)$ is called the monomial grade of $I$ and it is denoted by $\nu(I)$. Let $d_k$ be the minimu...

A very well-covered graph is a well-covered graph without isolated vertices such that the size of its minimal vertex covers is half of the number of vertices. If $G$ is a Cohen-Macaulay very well-covered graph, we deeply investigate some algebraic properties of the cover ideal of $G$ via the Rees algebra associated to the ideal.

Let G be a finite simple graph and let I(G) be its edge ideal. In this article, we deeply investigate the squarefree powers of I(G) by means of Betti splittings. When G is a forest, it is shown that the normalized depth function of I(G) is non-increasing. Furthermore, we compute explicitly the regularity function of squarefree powers of I(G) with G...

A very well-covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen-Macaulay very well-covered graphs are splittable. As a consequence, we compute exp...

We consider vector–spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincaré series. A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen–Macaulay vector–spread Borel ideals.

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the d...

Let G be a simple finite graph. A famous theorem of Dirac says that G is chordal if and only if G admits a perfect elimination order. It is known by Fröberg that the edge ideal I(G) of G has a linear resolution if and only if the complementary graph \(G^c\) of G is chordal. In this article, we discuss some algebraic consequences of Dirac’s theorem...

Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\ldots,t_{d-1})\in{\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a ${\bf t}$-spread ideal $I$ in $S$, we associate a unique $f_{\bf t}$-vector and we prove that if $I$ is ${\bf t}$-spread st...

Let $G$ be a finite simple graph with $n$ non isolated vertices, and let $J_G$ its binomial edge ideal. We determine all pairs $(\mbox{projdim}(J_G),\mbox{reg}(J_G))$, where $G$ ranges over all finite simple graphs with $n$ non isolated vertices, for any $n$.

Let S = K[x1,…,xn] be a polynomial ring in n variables with coefficients over a field K. A t-spread lexsegment ideal I of S is a monomial ideal generated by a t-spread lexsegment set. We determine all t-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a c...

We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application we determine the canonical trace $\mbox{tr}(\omega_R)$ of a Cohen-Macaulay ring $R$ of codimension two, which is generically Gorenstein. It is shown that if the defining ideal $I$ of $R$ is generated by $n$...

Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is chordal if and only if $G$ admits a perfect elimination order. It is known by Fr\"oberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only if the complementary graph $G^c$ of $G$ is chordal. In this article, we discuss some algebraic consequences of Dira...

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti...

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\ne(0)$ is called the monomial grade of $I$ and it is denoted by $\nu(I)$. Let $d_k$ be the minimu...

We introduce the class of vector–spread monomial ideals. This notion generalizes that of t–spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector–spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series...

We study the minimal primary decomposition of completely $t$-spread lexsegment ideals via simplicial complexes. We determine some algebraic invariants of such a class of $t$-spread ideals. Hence, we classify all $t$-spread lexsegment ideals which are Cohen-Macaulay.

We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincar\'e series. A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen-Macaulay vector-spread Borel ideals.

We study the homological shifts of polymatroidal ideals. It is shown that the first homological shift ideal of any polymatroidal ideal is again polymatroidal, supporting a conjecture of Bandari, Bayati and Herzog that predicts that all homological shift ideals of a polymatroidal ideal are polymatroidal. We also study the "socle ideal" $\text{soc}(I...

In this paper, we study some algebraic invariants of t-spread ideals, [Formula: see text], such as the projective dimension and the Castelnuovo–Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of t-spread ideals for which such bounds are optimal.

We study some algebraic invariants of $t$-spread ideals, $t\ge 1$, such as the projective dimension and the Castelnuovo-Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of t-spread ideals for which such bounds are optimal.

We introduce the class of vector-spread monomial ideals. This notion generalizes that of $t$-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincar\'e ser...

In this article we introduce the concepts of arbitrary t–spread lexsegments and of arbitrary t–spread lexsegment ideals with t a positive integer. These concepts are a natural generalization of arbitrary lexsegments and arbitrary lexsegment ideals. An ideal generated by an arbitrary t–spread lexsegment is called completely t–spread lexsegment if it...

In this paper we introduce the concepts of arbitrary $t$-spread lexsegments and of arbitrary $t$-spread lexsegment ideals with $t$ a positive integer. These concepts are a natural generalization of arbitrary lexsegments and arbitrary lexsegment ideals. An ideal generated by an arbitrary $t$-spread lexsegment is called completely $t$-spread lexsegme...

Let K be a field and let S=K[x1,…,xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=K[x_1,\ldots ,x_n]$$\end{document} be a standard polynomial ring over a field K....

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 constructive. Indeed, given some positive integers $a_1,\dots,a_r$ and some pairs of positive integers $(k_1,\ell_1),\dots...

We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $t\in \{1,2\}$.