We introduce the Macaulay2 package LinearTruncations for finding and studying the truncations of ... more We introduce the Macaulay2 package LinearTruncations for finding and studying the truncations of a multigraded module over a standard multigraded ring that have linear resolutions.
In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of... more In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space P n−1 ֒→ P r−1 determined by generators of a linearly presented m-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most n. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree n. Showing that this ideal has a linear resolution would imply that the conjecture in [8] holds. Furthermore, if this ideal of minors coincides with the one of the image in degree n-what we hope to be true-the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in [8].
We study the WLP and SLP of artinian monomial ideals in S = K[x 1 ,. .. , x n ] via studying thei... more We study the WLP and SLP of artinian monomial ideals in S = K[x 1 ,. .. , x n ] via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n − 2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
La régularité de Castelnuovo-Mumford est l'un des principaux invariants numériques permettant... more La régularité de Castelnuovo-Mumford est l'un des principaux invariants numériques permettant de mesurer la complexité de la structure des modules gradués de type fini sur des anneaux polynomiaux. Il mesure le degré maximal des générateurs des modules de syzygies. Dans cette thèse, nous étudions la régularité de Castelnuovo-Mumford avec différents points de vue et, dans certaines parties, nous nous concentrons principalement sur les syzygies linéaires. Dans le chapitre 2, nous étudions la régularité des homologies de Koszul et des cycles de Koszul de quotients unidimensionnels. Dans le chapitre 3, nous étudions les propriétés de Lefschetz faibles et fortes d'une classe d'idéaux monomiaux artiniens. Nous donnons, dans certains cas, une réponse affirmative à une conjecture d'Eisenbud, Huneke et Ulrich. Dans les chapitres 4 et 5, nous étudions deux comportements asymptotiques différents de la régularité de Castelnuovo-Mumford. Dans le chapitre 4, nous travaillons sur un...
We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with re... more We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.
V is a complete intersection scheme in a multiprojective space if it can be defined by an ideal I... more V is a complete intersection scheme in a multiprojective space if it can be defined by an ideal I with as many generators as codim(V ). We investigate the multigraded regularity of complete intersections scheme in Pn×Pm. We explicitly compute many values of the Hilbert functions of 0-dimensional complete intersections. We show that these values only depend upon n,m, and the bidegrees of the generators of I. As a result, we provide a sharp upper bound for the multigraded regularity of 0-dimensional complete intersections.
In this paper we study the equations of the elimination ideal associated with n + 1 generic multi... more In this paper we study the equations of the elimination ideal associated with n + 1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.
We study the WLP and SLP of artinian monomial ideals in S ¼ K½x 1 , :::, x n via studying their m... more We study the WLP and SLP of artinian monomial ideals in S ¼ K½x 1 , :::, x n via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n-2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial algebras with almost linear resolution.
Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell... more Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.
We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of th... more We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.
Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this p... more Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.
We introduce the Macaulay2 package LinearTruncations for finding and studying the truncations of ... more We introduce the Macaulay2 package LinearTruncations for finding and studying the truncations of a multigraded module over a standard multigraded ring that have linear resolutions.
In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of... more In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space P n−1 ֒→ P r−1 determined by generators of a linearly presented m-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most n. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree n. Showing that this ideal has a linear resolution would imply that the conjecture in [8] holds. Furthermore, if this ideal of minors coincides with the one of the image in degree n-what we hope to be true-the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in [8].
We study the WLP and SLP of artinian monomial ideals in S = K[x 1 ,. .. , x n ] via studying thei... more We study the WLP and SLP of artinian monomial ideals in S = K[x 1 ,. .. , x n ] via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n − 2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
La régularité de Castelnuovo-Mumford est l'un des principaux invariants numériques permettant... more La régularité de Castelnuovo-Mumford est l'un des principaux invariants numériques permettant de mesurer la complexité de la structure des modules gradués de type fini sur des anneaux polynomiaux. Il mesure le degré maximal des générateurs des modules de syzygies. Dans cette thèse, nous étudions la régularité de Castelnuovo-Mumford avec différents points de vue et, dans certaines parties, nous nous concentrons principalement sur les syzygies linéaires. Dans le chapitre 2, nous étudions la régularité des homologies de Koszul et des cycles de Koszul de quotients unidimensionnels. Dans le chapitre 3, nous étudions les propriétés de Lefschetz faibles et fortes d'une classe d'idéaux monomiaux artiniens. Nous donnons, dans certains cas, une réponse affirmative à une conjecture d'Eisenbud, Huneke et Ulrich. Dans les chapitres 4 et 5, nous étudions deux comportements asymptotiques différents de la régularité de Castelnuovo-Mumford. Dans le chapitre 4, nous travaillons sur un...
We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with re... more We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.
V is a complete intersection scheme in a multiprojective space if it can be defined by an ideal I... more V is a complete intersection scheme in a multiprojective space if it can be defined by an ideal I with as many generators as codim(V ). We investigate the multigraded regularity of complete intersections scheme in Pn×Pm. We explicitly compute many values of the Hilbert functions of 0-dimensional complete intersections. We show that these values only depend upon n,m, and the bidegrees of the generators of I. As a result, we provide a sharp upper bound for the multigraded regularity of 0-dimensional complete intersections.
In this paper we study the equations of the elimination ideal associated with n + 1 generic multi... more In this paper we study the equations of the elimination ideal associated with n + 1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.
We study the WLP and SLP of artinian monomial ideals in S ¼ K½x 1 , :::, x n via studying their m... more We study the WLP and SLP of artinian monomial ideals in S ¼ K½x 1 , :::, x n via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n-2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial algebras with almost linear resolution.
Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell... more Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.
We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of th... more We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.
Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this p... more Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.
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