scholarly journals Generic Initial Ideals and Graded Betti Numbers

2018 ◽  
Author(s):  
Jürgen Herzog
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers ◽  
Initial Ideals ◽  
Generic Initial Ideals

scholarly journals Rigidity of Linear Strands and Generic Initial Ideals

2008 ◽  
Vol 190 ◽  
pp. 35-61 ◽  
Author(s):  
Satoshi Murai ◽  
Pooja Singla
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Exterior Algebra ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Initial Ideals ◽  
Finite Set ◽  
Generic Initial Ideals ◽  
The Ideal

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers for some i > 1 and k ≥ 0, then for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if for some i > 1 and k ≥ 0, then for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers for all i ≥ 1 if and only if I(k) and I(k+1) have a linear resolution. Here I(d) is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

Weakly Polymatroidal Ideals

2006 ◽  
Vol 13 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Masako Kokubo ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Partially Ordered Sets ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Fundamental Fact ◽  
Polymatroidal Ideals ◽  
Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

scholarly journals The regularity of Tor and graded Betti Numbers

2006 ◽  
Vol 128 (3) ◽  
pp. 573-605 ◽  
Author(s):  
David Eisenbud ◽  
C. (Craig) Huneke ◽  
Bernd Ulrich
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers

scholarly journals Face Numbers and Nongeneric Initial Ideals

10.37236/1882 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Eric Babson ◽  
Isabella Novik
Keyword(s):  
Cyclic Group ◽  
Simple Proof ◽  
Prime Order ◽  
Necessary Conditions ◽  
Betti Numbers ◽  
Proper Action ◽  
Initial Ideals ◽  
The Face ◽  
Algebraic Shifting

Certain necessary conditions on the face numbers and Betti numbers of simplicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley (necessity) and Björner, Frankl, and Stanley (sufficiency) is given. The necessity portion of their result is generalized to certain conditions on the face numbers and Betti numbers of balanced Buchsbaum complexes.

scholarly journals Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

2019 ◽  
pp. 1-18 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi ◽  
Margherita Roggero
Keyword(s):  
Initial Ideals ◽  
Generic Initial Ideals

On Graded Betti Numbers of a Class of Graphs

2018 ◽  
Vol 25 (02) ◽  
pp. 335-348 ◽  
Author(s):  
Saba Yasmeen ◽  
Tongsuo Wu
Keyword(s):  
Betti Numbers ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Graded Betti Numbers

In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.

scholarly journals Path ideals of rooted trees and their graded Betti numbers

2011 ◽  
Vol 118 (8) ◽  
pp. 2411-2425 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Huy Tài Hà ◽  
Augustine OʼKeefe
Keyword(s):  
Betti Numbers ◽  
Rooted Trees ◽  
Graded Betti Numbers
Sign in / Sign up

Export Citation Format

Share Document