Minimal Free Resolutions of Zero-dimensional Schemes in ℙ1 × ℙ1

2015 ◽  
Vol 22 (01) ◽  
pp. 97-108 ◽  
Author(s):  
Paola Bonacini ◽  
Lucia Marino
Keyword(s):  
Free Resolution ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Minimal Set ◽  
Minimal Free Resolutions ◽  
The Ideal

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

scholarly journals Determinantal ideals without minimal free resolutions

1990 ◽  
Vol 118 ◽  
pp. 203-216 ◽  
Author(s):  
Mitsuyasu Hashimoto
Keyword(s):  
Free Resolution ◽  
Unit Element ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Determinantal Ideals ◽  
Arbitrary Base ◽  
Generic Matrix ◽  
Base Ring ◽  
Minimal Free Resolutions ◽  
The Ideal

Let R be a Noetherian commutative ring with, unit element, and Xij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let S = R[xij] be the polynomial ring over R, and It be the ideal in S, generated by the t × t minors of the generic matrix (xij) ∈ Mm, n(S). For many years there has been considerable interest in finding a minimal free resolution of S/It, over arbitrary base ring R. If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗z P. is a resolution of S/It over the base ring R′.

scholarly journals Minimal free resolutions of 2 × n domino tilings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950118
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Domino Tilings ◽  
Minimal Free Resolutions ◽  
The Ideal

We introduce a squarefree monomial ideal associated to the set of domino tilings of a [Formula: see text] rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

Multi-graded Betti numbers of path ideals of trees

2017 ◽  
Vol 16 (01) ◽  
pp. 1750018 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Explicit Description ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Generating Set ◽  
Graded Betti Numbers ◽  
Minimal Free Resolutions ◽  
Vertex Sets ◽  
Minimal Generating Set

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for [Formula: see text]-ary trees.

scholarly journals Projective covers and minimal free resolutions

1996 ◽  
Vol 19 (1) ◽  
pp. 185-192
Author(s):  
Mark A. Goddard
Keyword(s):  
Sufficient Conditions ◽  
Free Resolution ◽  
Projective Cover ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Direct Sum ◽  
Minimal Free Resolutions ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Projective Covers

Using a generalization of the definition of the projective cover of a module, a special type of surjective free resolution, known as the projective cover of a complex, may be defined. The projective cover is shown to be a direct summand of every surjective free resolution and to be the direct sum of the minimal free resolution and an exact complex. Necessary and sufficient conditions for the projective cover and minimal free resolution to be identical are discussed.

scholarly journals Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 605
Author(s):  
Lukas Katthän
Keyword(s):  
Simplicial Complex ◽  
Linear Part ◽  
Monomial Ideal ◽  
Short Note ◽  
Free Resolution ◽  
Free Resolutions ◽  
Restriction Maps ◽  
Minimal Free Resolution ◽  
Linear Maps ◽  
Minimal Free Resolutions

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of Δ . Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only ± 1 coefficients.

scholarly journals Gluing semigroups and strongly indispensable free resolutions

2019 ◽  
Vol 29 (02) ◽  
pp. 263-278
Author(s):  
Mesut Şahi̇n ◽  
Leah Gold Stella
Keyword(s):  
Complete Intersection ◽  
Special Property ◽  
Free Resolution ◽  
Numerical Semigroups ◽  
Free Resolutions ◽  
Semigroup Rings ◽  
Minimal Free Resolution ◽  
Minimal Free Resolutions ◽  
Symmetric Numerical Semigroups

We study strong indispensability of minimal free resolutions of semigroup rings focusing on the operation of gluing used in the literature to take examples with a special property and produce new ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine simple gluings of [Formula: see text]-generated non-symmetric, [Formula: see text]-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many new complete intersection semigroups of any embedding dimensions, having strongly indispensable minimal free resolutions.

scholarly journals Regularity and Free Resolution of Ideals Which Are Minimal To $d$-Linearity

2016 ◽  
Vol 118 (2) ◽  
pp. 161 ◽  
Author(s):  
M. Morales ◽  
A. A. Yazdan Pour ◽  
R. Zaare-Nahandi
Keyword(s):  
Polynomial Ring ◽  
Free Resolution ◽  
Free Resolutions ◽  
Edge Ideals ◽  
Homology Manifold ◽  
Linear Resolution ◽  
Vertex Set ◽  
Minimal Free Resolutions ◽  
Positive Integers ◽  
The Ideal

For given positive integers $n\geq d$, a $d$-uniform clutter on a vertex set $[n]=\{1,\dots,n\}$ is a collection of distinct $d$-subsets of $[n]$. Let $\mathscr{C}$ be a $d$-uniform clutter on $[n]$. We may naturally associate an ideal $I(\mathscr{C})$ in the polynomial ring $S=k[x_1,\dots,x_n]$ generated by all square-free monomials \smash{$x_{i_1}\cdots x_{i_d}$} for $\{i_1,\dots,i_d\}\in\mathscr{C}$. We say a clutter $\mathscr{C}$ has a $d$-linear resolution if the ideal \smash{$I(\overline{\mathscr{\mathscr{C}}})$} has a $d$-linear resolution, where \smash{$\overline{\mathscr{C}}$} is the complement of $\mathscr{C}$ (the set of $d$-subsets of $[n]$ which are not in $\mathscr C$). In this paper, we introduce some classes of $d$-uniform clutters which do not have a linear resolution, but every proper subclutter of them has a $d$-linear resolution. It is proved that for any two $d$-uniform clutters $\mathscr{C}_1$, $\mathscr{C}_2$ the regularity of the ideal $I(\overline{\mathscr{C}_1 \cup \mathscr{C}_2})$, under some restrictions on their intersection, is equal to the maximum of the regularities of $I(\overline{\mathscr{C}}_1)$ and $I(\overline{\mathscr{C}}_2)$. As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideals of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

scholarly journals Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

Fat Points in ℙ1× ℙ1and Their Hilbert Functions

2004 ◽  
Vol 56 (4) ◽  
pp. 716-741 ◽  
Author(s):  
Elena Guardo ◽  
Adam Van Tuyl
Keyword(s):  
Finite Number ◽  
Hilbert Function ◽  
Free Resolution ◽  
Hilbert Functions ◽  
Minimal Free Resolution ◽  
Fat Points ◽  
Fat Point ◽  
Point Scheme

AbstractWe study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

scholarly journals Stably free resolutions of lattices over finite groups

1990 ◽  
Vol 49 (3) ◽  
pp. 364-385
Author(s):  
K. W. Gruenberg
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Class Group ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Residue Classes ◽  
Projective Class

AbstractFor a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

Sign in / Sign up

Export Citation Format

Share Document