Algebraically rigid simplicial complexes and graphs

This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a K-algebra R are parameterized by the elements of cotangent module T 1 (R) of R. In this article we focus on deformations of toric rings, and give an explicit description of T 1 (R) in the case that R is a toric ring. In particular, we are interested in unobstructed deformations which preserve the toric structure. Such deformations we call separations. Toric rings which do not admit any separation are called inseparable. We apply the theory to the edge ring of a finite graph. The coordinate ring of a convex polyomino may be viewed as the edge ring of a special class of bipartite graphs. It is shown that the coordinate ring of any convex polyomino is inseparable. We introduce the concept of semi-rigidity, and give a combinatorial description of the graphs whose edge ring is semi-rigid. The results are applied to show that for m − k = k = 3, G k,m−k is not rigid while for m − k ≥ k ≥ 4, G k,m−k is rigid. Here G k,m−k is the complete bipartite graph K m−k,k with one edge removed.

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“…and so a 1 − ak 1 = a 2 − ak 2 = a 3 − ak 3 . It follows that (a 1 − ak 1 )w 1 + (a 2 − ak 2 )w 2 + (a 3 − ak 3 )w 3 = 0.…”

Section: Separable and Inseparable Saturated Latticesmentioning

confidence: 99%

Toric Rings, Inseparability and Rigidity

Bigdeli

Herzog

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…It is therefore of interest to classify all inseparable graphs. An attempt for this classification is given in [1].…”

Section: Inseparable Models Of Bi-cm Graphsmentioning

confidence: 99%

“…A graph which does not allow any separation is called inseparable, and a inseparable graph which is obtained by a finite number of separation steps from G is called a separable model of G. Any graph admits separable models and the number of separable models of a graph is finite. Separable and inseparable graphs from the view point of deformation theory have been studied in [1].…”

Section: Introductionmentioning

confidence: 99%

Bi-Cohen-Macaulay graphs

Herzog

Rahimi

2016

Electron. J. Combin.

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In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General bi-Cohen-Macaulay graphs are classified up to separation. The inseparable bi-Cohen-Macaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.2010 Mathematics Subject Classification. 05E40, 13C14.

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“…Recently [6] shows that the coordinate rings of Grassmannians for the Plücker embedding are rigid ideals. As mentioned above [1] also gives classes of rigid monomial ideals. With the present article we therefore believe we make a substantial contribution to the known classes of rigid ideals.…”

Section: Introductionmentioning

confidence: 97%

“…Here, for quadratic letterplace ideals, we find that the base space both is smooth and global and that we can give explicit equations for the whole family of deformations. Recently [1] applied the theory developed by Christophersen and Altman to investigate when monomial ideals are rigid. For edge ideals they develop a number of results for when this holds.…”

Section: Introductionmentioning

confidence: 99%

Rigid ideals by deforming quadratic letterplace ideals

Fløystad

Nematbakhsh

2018

Journal of Algebra

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We compute the deformation space of quadratic letterplace ideals L(2, P ) of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2, P ) defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases J(2, P ) is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.Date: September 21, 2018.

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Algebraically rigid simplicial complexes and graphs

Cited by 12 publications

References 8 publications

Toric Rings, Inseparability and Rigidity

Toric Rings, Inseparability and Rigidity

Bi-Cohen-Macaulay graphs

Rigid ideals by deforming quadratic letterplace ideals

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