Depth of edge rings arising from finite graphs

Hibi, Takayuki; Higashitani, Akihiro; Kimura, Kohei; O'Keefe, Augustine B.

doi:10.1090/s0002-9939-2011-11083-9

Cited by 14 publications

(21 citation statements)

References 4 publications

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“…Lemma 1.1. The set of binomials corresponding to primitive even closed walks (1), (2), (3) and (4) is a Gröbner basis of I G with respect to < rev .…”

Section: Preliminariesmentioning

confidence: 99%

“…Case 3: Again, we set that f is the same as above. Let g be of type (3). First, let g = x p x k+2 x 2k+4 − x k+1 x k+1+p x 2k+3 .…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus S(f, g) reduce to 0 by generators of type (3). Case 7: We assume that both f and g are of type (3). First, we consider the case where…”

Section: Preliminariesmentioning

confidence: 99%

“…The study on edge rings [4] of finite graphs together with their toric ideals [5] have been achieved from viewpoints of both commutative algebra and combinatorics. Following the previous paper [3], which investigated a question about depth of edge rings, the topic of the present paper is depth of initial ideals of normal edge rings.…”

Section: Introductionmentioning

confidence: 99%

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Depth of Initial Ideals of Normal Edge Rings

Hibi

Higashitani

Kimura

et al. 2014

Communications in Algebra

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Abstract. Let G be a finite graph on the vertex set [d] = {1, . . . , d} with the edges e1, . . . , en and K[t] = K[t1, . . . , t d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t e = titj such that e = {i, j} is an edge of G.. . , xn] be the polynomial ring in n variables over K and define the surjective homomorphism π :by setting π(xi) = t e i for i = 1, . . . , n. The toric ideal IG of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exist a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order

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“…Lemma 1.1. The set of binomials corresponding to primitive even closed walks (1), (2), (3) and (4) is a Gröbner basis of I G with respect to < rev .…”

Section: Preliminariesmentioning

confidence: 99%

“…Case 3: Again, we set that f is the same as above. Let g be of type (3). First, let g = x p x k+2 x 2k+4 − x k+1 x k+1+p x 2k+3 .…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus S(f, g) reduce to 0 by generators of type (3). Case 7: We assume that both f and g are of type (3). First, we consider the case where…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Depth of Initial Ideals of Normal Edge Rings

Hibi

Higashitani

Kimura

et al. 2014

Communications in Algebra

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“…p F is an associated prime of K{Q}/K[Q]. Hence (7) holds if and only if all families of holes have *dimension d − 1.…”

Section: Local Cohomology and Holesmentioning

confidence: 99%

Non-normal affine monoid algebras

Katthän

2014

manuscripta math.

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We give a geometric description of the set of holes in a non-normal affine monoid Q. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of K [Q]. From this, we see how various properties of K[Q] like local normality and Serre's conditions (R 1 ) and (S 2 ) are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra K[Q] is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.

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Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

Ballard¹

2021

Association for Women in Mathematics Series

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Depth of edge rings arising from finite graphs

Cited by 14 publications

References 4 publications

Depth of Initial Ideals of Normal Edge Rings

Depth of Initial Ideals of Normal Edge Rings

Non-normal affine monoid algebras

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

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