arXiv subjects

Francesco Strazzanti

Publications and source records attributed to Francesco Strazzanti.

27 records · Page 2Linked to original sources

arXiv2017

Dilatations of numerical semigroups

This paper is focused on numerical semigroups and presents a simple construction, that we call dilatation, which, from a starting semigroup $S$, permits to get an infinite family of semigroups which share several properties with $S$. The invariants of each semigroup $T$ of this family are given in terms of the corresponding invariants of $S$ and the Apéry set and the minimal generators of $T$ are also described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that $S$ satisfies one of these properties if and only if each dilatation of $S$ satisfies the corresponding one.

math.AC↗

arXiv2017

Binomial edge ideals of bipartite graphs

We classify the bipartite graphs $G$ whose binomial edge ideal $J_G$ is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. We study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with $J_G$ unmixed and not Cohen-Macaulay.

math.AC↗

arXiv2016

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level $h$; moreover we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included.

math.AC↗

arXiv2015

Families of Gorenstein and almost Gorenstein rings

Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

math.AC↗

arXiv2015

Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

Given two numerical semigroups $S$ and $T$ and a positive integer $d$, $S$ is said to be one over $d$ of $T$ if $S=\{s \in \mathbb{N} \ | \ ds \in T \}$ and in this case $T$ is called a $d$-fold of $S$. We prove that the minimal genus of the $d$-folds of $S$ is $g + \lceil \frac{(d-1)f}{2} \rceil$, where $g$ and $f$ denote the genus and the Frobenius number of $S$. The case $d=2$ is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of $S$ and study the particular case when $S$ is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

math.GR↗

arXiv2015

A rigidity property of local cohomology modules

The relationships between the invariants and the homological properties of $I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves.

math.AC↗

arXiv2014

One half of almost symmetric numerical semigroups

Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T) \leq 2t(S)+1$. On the other hand, if $t(T)$ is even, there exists such $T$ if and only if $S$ is almost symmetric and different from $\mathbb{N}$; in this case the type of $S$ is the number of even pseudo-Frobenius numbers of $T$. Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal.

math.AC↗

arXiv2014

A family of quotients of the Rees algebra

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.

math.AC↗

arXiv2012

The numerical duplication of a numerical semigroup

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup $S$ and a semigroup ideal $E\subseteq S$, produces a new numerical semigroup, denoted by $S\Join^b\E$ (where $b$ is any odd integer belonging to $S$), such that $S=(S\Join^b\E)/2$. In particular, we characterize the ideals $E$ such that $S\Join^bE$ is almost symmetric and we determine its type.

math.AC↗