scholarly journals Betti Numbers of Monomial Ideals and Shifted Skew Shapes

10.37236/69 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Uwe Nagel ◽  
Victor Reiner
Keyword(s):  
Lower Bound ◽  
Betti Numbers ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Fixed Degree ◽  
Minimal Free Resolution ◽  
Edge Ideals ◽  
Combinatorial Interpretations ◽  
Ferrers Graphs ◽  
Strongly Stable Ideals

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.

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.

Projective curves of degree=codimension+2 II

2016 ◽  
Vol 26 (01) ◽  
pp. 95-104 ◽  
Author(s):  
Wanseok Lee ◽  
Euisung Park
Keyword(s):  
Integral Curve ◽  
Betti Numbers ◽  
Free Resolution ◽  
Relative Location ◽  
Normal Curve ◽  
Minimal Free Resolution ◽  
Graded Betti Numbers ◽  
Projective Curves ◽  
Rational Normal Curve

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

scholarly journals The minimal free resolution of a class of square-free monomial ideals

2004 ◽  
Vol 189 (1-3) ◽  
pp. 263-278 ◽  
Author(s):  
Rahim Zaare-Nahandi ◽  
Rashid Zaare-Nahandi
Keyword(s):  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

scholarly journals From grids to pseudo-grids of lines: Resolution and seminormality

2020 ◽  
pp. 2150213
Author(s):  
Francesca Cioffi ◽  
Margherita Guida ◽  
Luciana Ramella
Keyword(s):  
Analogous Result ◽  
Betti Numbers ◽  
Free Resolution ◽  
Infinite Field ◽  
Minimal Free Resolution ◽  
Mapping Cone ◽  
Configurations Of Lines

Over an infinite field [Formula: see text], we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.

On the Betti numbers of edge ideals of skew Ferrers graphs

2019 ◽  
Vol 30 (01) ◽  
pp. 125-139
Author(s):  
Do Trong Hoang
Keyword(s):  
Explicit Formula ◽  
Betti Number ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Ferrers Graph ◽  
Ferrers Graphs ◽  
Extremal Betti Number

We prove that [Formula: see text] for any staircase skew Ferrers graph [Formula: see text], where [Formula: see text] and [Formula: see text]. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

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.

scholarly journals Optimal Betti Numbers of Forest Ideals

10.37236/125 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael Goff
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Graded Betti Numbers

We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.

scholarly journals Certain homological invariants of bipartite kneser graphs

2021 ◽  
pp. 2250206
Author(s):  
Ajay Kumar ◽  
Pavinder Singh ◽  
Rohit Verma
Keyword(s):  
Lower Bound ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Upper Bounds ◽  
Combinatorial Formula ◽  
Lower And Upper Bounds ◽  
Edge Ideals ◽  
Homological Invariants ◽  
Kneser Graphs ◽  
Combinatorial Data

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

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