Dependence of Betti Numbers on Characteristic

Dalili, Kia; Kummini, Manoj

doi:10.1080/00927872.2012.718821

Cited by 18 publications

(18 citation statements)

References 12 publications

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“…In the next section, we will utilize the link and deletion complexes and apply results of Dalili and Kummini [6] to show the graded Betti numbers of I n are independent of the characteristic of the field k.…”

Section: 8mentioning

confidence: 99%

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Minimal free resolutions of 2 × n domino tilings

Bouchat

Brown

2019

J. Algebra Appl.

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We introduce a squarefree monomial ideal associated to the set of domino tilings of a 2 × n rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo-Mumford regularity of the ideal.1991 Mathematics Subject Classification. 05E40, 13A15.

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Section: 8mentioning

confidence: 99%

“…Identifying horizontal pairs, x i and x n+i−1 , we have We now wish to apply the results of Dalili and Kummini [6] to show independence of characteristic.…”

Section: 2mentioning

confidence: 99%

Minimal free resolutions of 2 × n domino tilings

Bouchat

Brown

2019

J. Algebra Appl.

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“…Also see the paper of Dalili and Kummini [46] which also studies how the graded Betti numbers of an edge ideal depend upon the characteristic of the field. Moreover, this graph is the smallest such example, where by smallest we mean that no graph on ten or less vertices has this feature.…”

Section: Splitting Monomial Ideals and Fröberg's Theoremmentioning

confidence: 99%

Monomial Ideals, Computations and Applications

Bigatti¹,

Gimenez²,

Sáenz-de-Cabezón³

2013

Lecture Notes in Mathematics

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“…(See [32], [35] for some results about the linearity defect of such ideals.) Second and furthermore, the linearity defect generally cannot be read off from the Betti table: for example (see [20,Example 2.8]), the ideals I 1 = (x 4 1 , x 3 1 x 2 , x 2 1 x 2 2 , x 1 x 3 2 , x 4 2 , x 3 1 x 3 , x 2 1 x 2 x 2 3 , x 2 1 x 3 3 , x 1 x 2 2 x 2 3 ) and 3 ] have the same graded Betti numbers, but the first one has linearity defect 0 while the second one has positive linearity defect (equal to 1). Third, the quest of finding the aforementioned characterization yields interesting new insights even to the more classical topics concerning Castelnuovo-Mumford regularity or the projective dimension.…”

Section: Introductionmentioning

confidence: 99%

Linearity defect of edge ideals and Fröberg’s theorem

Nguyen

2016

J Algebr Comb

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Abstract. Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals. IntroductionLet (R, m) be a standard graded algebra over a field k with the graded maximal ideal m. Let M be a finitely generated graded R-module. For integers i, j,The regularity is an important invariant of graded modules over R. When R is a polynomial ring and M is a monomial ideal of R, its regularity exposes many combinatorial flavors. This fact has been exploited and proved to be very useful for studying the regularity; for recent surveys, see [13], [28], [38]. A classical and instructive example is Fröberg's theorem. Recall that if G is a graph on the vertex set {x 1 , . . . , x n } (where n ≥ 1), and by abuse of notation, R is the polynomial ring k[x 1 , . . . , x n ], then the edge ideal of G is I(G) = (x i x j : {x i , x j } is an edge of G). Unless otherwise stated, whenever we talk about an invariant of I(G) (including the regularity and the linearity defect, to be defined below), it is understood that the base ring is the polynomial ring R. For m ≥ 3, the cycle C m is the graph on vertices x 1 , . . . , x m with edges x 1 x 2 , . . . , x m−1 x m , x m x 1 . We say that a graph G is weakly chordal (or weakly triangulated) if for every m ≥ 5, neither G, nor its complement contains C m as an induced subgraph. G is chordal if for any m ≥ 4, C m is not an induced subgraph of G. Fröberg's theorem [10] says that I(G) has regularity 2 if and only if the complement graph of G is chordal. It is of interest to find generalizations of this important result; see, for instance, [7], [8]. Theorem 4.1] extended Fröberg's theorem by providing a combinatorial characterization of connected bipartite graphs G such that reg I(G) = 3. In general, the regularity 3 condition on edge ideals is not 2010 Mathematics Subject Classification. 05C25, 13D02, 13H99.

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Dependence of Betti Numbers on Characteristic

Cited by 18 publications

References 12 publications

Minimal free resolutions of 2 × n domino tilings

Minimal free resolutions of 2 × n domino tilings

Monomial Ideals, Computations and Applications

Linearity defect of edge ideals and Fröberg’s theorem

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