Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

Jared Culbertson; Anton Dochtermann; Dan P. Guralnik; Peter F. Stiller

doi:10.37236/9120

Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

The Electronic Journal of Combinatorics ◽

10.37236/9120 ◽

2020 ◽

Vol 27 (3) ◽

Author(s):

Jared Culbertson ◽

Anton Dochtermann ◽

Dan P. Guralnik ◽

Peter F. Stiller

Keyword(s):

Monomial Ideals ◽

Chordal Graphs ◽

Linear Quotients

We prove that for all $d \geq 1$, a shellable $d$-dimensional complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients of quadratic monomial ideals.

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k-Decomposable Monomial Ideals

Algebra Colloquium ◽

10.1142/s1005386715000656 ◽

2015 ◽

Vol 22 (spec01) ◽

pp. 745-756 ◽

Cited By ~ 4

Author(s):

Rahim Rahmati-Asghar ◽

Siamak Yassemi

Keyword(s):

Simplicial Complex ◽

Monomial Ideals ◽

Alexander Dual ◽

Linear Quotients ◽

Dimensional Simplicial Complex

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

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Free resolution of powers of monomial ideals and Golod rings

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-25504 ◽

2017 ◽

Vol 120 (1) ◽

pp. 59 ◽

Cited By ~ 1

Author(s):

N. Altafi ◽

N. Nemati ◽

S. A. Seyed Fakhari ◽

S. Yassemi

Keyword(s):

Polynomial Ring ◽

Monomial Ideal ◽

Free Resolution ◽

Monomial Ideals ◽

Monomial Order ◽

Golod Ring ◽

Linear Quotients ◽

The Ideal

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.

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Monomial Ideals of Graphs with Loops

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/aicu-2013-0010 ◽

2014 ◽

Vol 60 (2) ◽

pp. 321-336 ◽

Cited By ~ 1

Author(s):

Maurizio Imbesi ◽

Monica La Barbiera

Keyword(s):

Polynomial Ring ◽

Monomial Ideals ◽

Linear Type ◽

Connected Graphs ◽

Edge Ideals ◽

Linear Resolution ◽

Algebraic Invariants ◽

Linear Quotients

Abstract We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the polynomial ring related to these graphs modulo such ideals. Moreover we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to the previous classes of graphs which can have loops on any vertex. Lastly, it is shown that these ideals are of linear type.

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On Linear Quotients of Squarefree Monomial Ideals

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-019-00737-5 ◽

2019 ◽

Vol 43 (2) ◽

pp. 1213-1221

Author(s):

Erfan Manouchehri ◽

Ali Soleyman Jahan

Keyword(s):

Monomial Ideals ◽

Linear Quotients

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Generalized Newton complementary duals of monomial ideals

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500213 ◽

2020 ◽

pp. 2150021

Author(s):

Katie Ansaldi ◽

Kuei-Nuan Lin ◽

Yi-Huang Shen

Keyword(s):

Polynomial Ring ◽

Monomial Ideal ◽

Free Resolution ◽

Monomial Ideals ◽

Free Resolutions ◽

Stable Ideal ◽

Generalized Newton ◽

Strongly Stable Ideals ◽

The Given ◽

Linear Quotients

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

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