Lorenzo Guerrieri

Lorenzo Guerrieri
Jagiellonian University | UJ · Institute of Mathematics

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35
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96
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Publications

Publications (35)
Preprint
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It is well-known that an integrally closed domain $D$ can be express as the intersection of its valuation overrings but, if $D$ is not a Pr\"{u}fer domain, the most of valuation overrings of $D$ cannot be seen as localizations of $D$. The Kronecker function ring of $D$ is a classical construction of a Pr\"{u}fer domain which is an overring of $D[t]...
Preprint
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Let $M$ be a perfect module of projective dimension 3 in a Gorenstein, local or graded ring $R$. We denote by $\FF$ the minimal free resolution of $M$. Using the generic ring associated to the format of $\FF$ we define higher structure maps, according to the theory developed by Weyman in "Generic free resolutions and root systems" (Annales de l'Ins...
Preprint
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Let $I$ be a perfect ideal of height 3 in a Gorenstein local ring $R$. Let $\mathbb{F}$ be the minimal free resolution of $I$. A sequence of linear maps, which generalize the multiplicative structure of $\mathbb{F}$, can be defined using the generic ring associated to the format of $\mathbb{F}$. Let $J$ be an ideal linked to $I$. We provide formula...
Article
We produce some interesting families of resolutions of length three by describing certain open subsets of the spectrum of the generic ring for such resolutions constructed in [6].
Article
In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of conta...
Preprint
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Working over a field of characteristic zero, we give structure theorems for all grade three licci ideals and their minimal free resolutions. In particular, we completely classify such ideals up to deformation. The descriptions of their resolutions extend earlier results by Buchsbaum-Eisenbud, Brown, and Sanchez. Our primary tool is the theory of hi...
Preprint
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Given an integral domain $D$ with quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\ldots+\frac{1}{d_n} $ for $d_1, \ldots, d_n$ nonzero elements of $D$. In this article we study problems related with prime ideals, localizations and Krull dimen...
Preprint
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We present a constructive procedure, based on the notion of Ap\'ery set, to obtain the value semigroup of a plane curve singularity from the value semigroup of its blow-up and viceversa. In particular we give a blow-down process that allows to reconstruct a plane algebroid curve form its blow-up, even if it is not local. Then we characterize numeri...
Preprint
The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$ variables over a field. In particular, $R(D)$ is an $n$-dimensional, local, non-Noetherian, non-integrally closed,...
Preprint
Using the theory of "higher structure maps" from generic rings for free resolutions of length three, we give a classification of grade 3 perfect ideals with small type and deviation in local rings of equicharacteristic zero, extending the Buchsbaum-Eisenbud structure theorem on Gorenstein ideals and realizing it as the type D case of an ADE corresp...
Article
Full-text available
It is well-known that an integrally closed domain D can be expressed as the intersection of its valuation overrings but, if D is not a Prüfer domain, most of the valuation overrings of D cannot be seen as localizations of D . The Kronecker function ring of D is a classical construction of a Prüfer domain which is an overring of D [ t ], and its loc...
Article
Full-text available
An Egyptian fraction is a finite sum of distinct rational numbers of the form 1 m , where m is a nonzero integer. It is well-known that every rational number can be expressed as an Egyptian fraction. The purpose of this note is to explore natural analogs of this concept for commutative integral domains.
Article
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In this article, we discuss some applications of the construction of the Apéry set of a good semigroup in Nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}^...
Preprint
In this article we discuss some applications of the construction of the Ap\'ery set of a good semigroup in $\mathbb{N}^d$ given in the previous paper [Partition of the complement of good semigroup ideals and Ap\'ery sets, Communications in Algebra, 49, No. 10, 4136-4158 (2021))]. In particular we study: the duality of a symmetric and almost symmetr...
Article
In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. Our approach is entirely algebraic, and allows us to recover and reinterpret previously known results proved using combinatorial methods.
Preprint
Full-text available
In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star configurations of height two, we give a full description of the defining ideal of the Rees algebra, by explicitly iden...
Article
Good semigroups form a class of submonoids of Nd containing the value semigroups of curve singularities. In this article, we describe a partition of the complement of good semigroup ideals. As main application, we describe the Apéry sets of good semigroups with respect to arbitrary elements. This generalizes to any d≥2 the results in the study of D...
Preprint
Full-text available
We produce some interesting families of resolutions of length three by describing certain open subsets of the spectrum of the generic ring for such resolutions constructed in a recent paper by Weyman.
Article
A quasi-equigenerated monomial ideal I in the polynomial ring R=k[x1,…,xn] is a Freiman ideal if μ(I2)=l(I)μ(I)−(l(I)2) where l(I) is the analytic spread of I and μ(I) is the number of minimal generators of I. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this wo...
Chapter
We study the Apéry Set of good subsemigroups of \(\mathbb N^2\), a class of semigroups containing the value semigroups of curve singularities with two branches. Even if this set in infinite, we show that, for the Apéry Set of such semigroups, we can define a partition in “levels” that allows to generalize many properties of the Apéry Set of numeric...
Preprint
Full-text available
Good semigroups form a class of submonoids of $\mathbb{N}^d$ containing the value semigroups of curve singularities. In this article, we describe a partition of the complements of good semigroup ideals, having as main application the description of the Ap\'{e}ry sets of good semigroups. This generalizes to any $d \geq 2$ the results of a recent pap...
Preprint
Full-text available
In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of conta...
Article
Full-text available
We study the type and the almost symmetric condition for good subsemigoups of \({\mathbb {N}}^2\), a class of semigroups containing the value semigroups of curve singularities with two branches. We define the type in term of a partition of a specific set associated to the semigroup and we show that this definition generalizes the well-known notion...
Preprint
A quasi-equigenerated monomial ideal $I$ in the polynomial ring $R= k[x_1, \ldots, x_n]$ is a Freiman ideal if $\mu(I^2) = l(I)\mu(I)- \binom{l(I)}{2}$ where $l(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal generators of $I$. Freiman ideals are special since there exists an exact formula computing the minimal number of gen...
Preprint
Full-text available
We study the type and the almost symmetric condition for good subsemigoups of N^2 , a class of semigroups containing the value semigroups of curve singularities with two branches. We define the type in term of a partition of a specific set associated to the semigroup and we show that this definition generalizes the well known notion of type of a nu...
Article
Let (R, ) be a regular local ring of dimension d ≥ 2. A local monoidal transform of R is a ring of the form R1 = R[ x]1, where x is a regular parameter, is a regular prime ideal of R and 1 is a maximal ideal of R[ x] lying over . In this paper, we study some features of the rings S = n≥0∞R n obtained as infinite directed union of iterated local mon...
Preprint
Full-text available
We study the Ap\'ery set of good subsemigoups of $\mathbb N^2$, a class of semigroups containing the value semigroups of curve singularities with two branches. Even if this set in infinite, we show that, for the Ap\'ery set of such semigroups, we can define a partition in "levels" that allows to generalize many properties of the Ap\'ery set of nume...
Preprint
Full-text available
Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A local monoidal transform of $R$ is a ring of the form $R_1= R[\frac{\mathfrak{p}}{x}]_{\mathfrak{m}_1}$ where $x \in \mathfrak{p}$ is a regular parameter, $\mathfrak{p}$ is a regular prime ideal of $R$ and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{p}}{x}]...
Article
Full-text available
Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certa...
Preprint
Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certa...
Article
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Ap\'ery set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
Preprint
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Ap\'ery set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
Chapter
Let \(\{R_{n},\mathfrak{m}_{n}\}_{n\geq 0}\) be an infinite sequence of regular local rings with Rn+1 birationally dominating Rn and \(\mathfrak{m}_{n}R_{n+1}\) a principal ideal of Rn+1 for each n. We examine properties of the integrally closed local domain \(S =\bigcup _{n\geq 0}R_{n}\).
Article
Full-text available
Let $\{ R_n, {\mathfrak m}_n \}_{n \ge 0}$ be an infinite sequence of regular local rings with $R_{n+1}$ birationally dominating $R_n$ and ${\mathfrak m}_nR_{n+1}$ a principal ideal of $R_{n+1}$ for each $n$. We examine properties of the integrally closed local domain $S = \bigcup_{n \ge 0}R_n$.

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