scholarly journals Hypergraphs with high projective dimension and 1-dimensional hypergraphs

2017 ◽  
Vol 27 (06) ◽  
pp. 591-617 ◽  
Author(s):  
K.-N. Lin ◽  
P. Mantero
Keyword(s):  
Large Class ◽  
Projective Dimension ◽  
Monomial Ideals ◽  
Connected Component ◽  
Number Of Generators ◽  
Ferrers Graph ◽  
Explicit Procedure

(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals [Formula: see text] with [Formula: see text], i.e. whose projective dimension equals the minimal number of generators of [Formula: see text] minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for [Formula: see text]. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs [Formula: see text] (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.

scholarly journals The stable property of projective dimension

2019 ◽  
Vol 18 (12) ◽  
pp. 1950224
Author(s):  
Somayeh Bandari ◽  
Raheleh Jafari
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Property ◽  
Polymatroidal Ideals

We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen–Macaulay property. Indeed, we study the class of monomial ideals [Formula: see text], whose projective dimension is stable under monomial localizations at monomial prime ideals [Formula: see text], with [Formula: see text]. We study the relations between this property and other sorts of Cohen–Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.

On the Arithmetical Rank of the Edge Ideals of Some Graphs

2012 ◽  
Vol 19 (spec01) ◽  
pp. 797-806 ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Dariush Kiani
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Common Vertex ◽  
Edge Ideals ◽  
Arithmetical Rank

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

scholarly journals Stanley depth of monomial ideals with small number of generators

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Mircea Cimpoeaş
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth ◽  
Number Of Generators ◽  
Stanley Conjecture

AbstractFor a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].

scholarly journals Normally Torsion-free Lexsegment Ideals

2015 ◽  
Vol 22 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Anda Olteanu
Keyword(s):  
Large Class ◽  
Monomial Ideals ◽  
Depth Function ◽  
Lexsegment Ideals ◽  
Torsion Free

In this paper we characterize all the lexsegment ideals which are normally torsion-free. This will provide a large class of normally torsion-free monomial ideals which are not square-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that, for lexsegment ideals, the property being normally torsion-free is equivalent to the property of the depth function being constant.

scholarly journals Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

2017 ◽  
Vol 10 (03) ◽  
pp. 1750061
Author(s):  
Somayeh Moradi
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Simplicial Complexes ◽  
Monomial Ideals ◽  
Edge Ideals ◽  
Graded Betti Numbers ◽  
Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

scholarly journals Morse resolutions of powers of square-free monomial ideals of projective dimension one

2021 ◽  
Author(s):  
Susan M. Cooper ◽  
Sabine El Khoury ◽  
Sara Faridi ◽  
Sarah Mayes-Tang ◽  
Susan Morey ◽  
...  
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Dimension One

scholarly journals Resolutions of monomial ideals of projective dimension 1

2017 ◽  
Vol 45 (12) ◽  
pp. 5453-5464 ◽  
Author(s):  
Sara Faridi ◽  
Ben Hersey
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

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