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.

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′.

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.

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 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 On the Betti numbers and Rees algebras of ideals with linear powers

2021 ◽  
Vol 53 (2) ◽  
pp. 575-592
Author(s):  
Lisa Nicklasson
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Linear Relations ◽  
Rees Algebras ◽  
Low Dimension ◽  
The Ideal

AbstractAn ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

On multigraded resolutions

1995 ◽  
Vol 118 (2) ◽  
pp. 245-257 ◽  
Author(s):  
Winfried Bruns ◽  
Jürgen Herzog
Keyword(s):  
Monomial Ideal ◽  
Free Resolution ◽  
Suitable Choice ◽  
Minimal Free Resolution ◽  
Common Multiple ◽  
Least Common Multiple ◽  
The Ideal

This paper was initiated by a question of Eisenbud who asked whether the entries of the matrices in a minimal free resolution of a monomial ideal (which, after a suitable choice of bases, are monomials) divide the least common multiple of the generators of the ideal. We will see that this is indeed the case, and prove it by lifting the multigraded resolution of an ideal, or more generally of a multigraded module, keeping track of how the shifts ‘deform’' in such a lifting; see Theorem 2·1 and Corollary 2·2.

scholarly journals Betti Numbers of Transversal Monomial Ideals

2011 ◽  
Vol 18 (spec01) ◽  
pp. 925-936
Author(s):  
Rahim Zaare-Nahandi
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Free Resolution ◽  
Circulant Matrix ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Ground Field ◽  
Initial Ideal ◽  
The Matrix ◽  
Generic Singularities

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

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 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.

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