Linearity defects of face rings

Okazaki, Ryota; Yanagawa, Kohji

doi:10.1016/j.jalgebra.2007.02.049

Cited by 7 publications

(9 citation statements)

References 17 publications

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“…But a computer experiment suggests that the bound (1) could be very far from sharp. For example, if I ⊂ E is a monomial ideal then we have ld E (E/I ) ≤ max{n − 2, 1} ( [15]). This does not hold for general graded ideals.…”

Section: · · · −→ A(−i) β I (K) −→ · · · −→ A(−2) β 2 (K)mentioning

confidence: 99%

Linearity defect and regularity over a Koszul algebra

Yanagawa¹

2009

MATH. SCAND.

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Let A = i∈ N A i be a Koszul algebra over a field K = A 0 , and * mod A the category of finitely generated graded left A-modules. The linearity defect ld A (M) of M ∈ * mod A is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that ld E (M) < ∞ for all M ∈ * mod E. Improving this, we show that the Koszul dual A ! of a Koszul commutative algebra A satisfies the following.• If E = y 1 , . . . , y n is an exterior algebra, then ld E (M) ≤ c n! 2 (n−1)! for M ∈ * mod E with c := max{ dim K M i | i ∈ Z }.

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Section: · · · −→ A(−i) β I (K) −→ · · · −→ A(−2) β 2 (K)mentioning

confidence: 99%

Linearity defect and regularity over a Koszul algebra

Yanagawa¹

2009

MATH. SCAND.

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“…The linearity defect was studied by many authors, see for example [1], [6], [15], [19], [21], [22], [23]. Nevertheless, it is still an elusive invariant.…”

Section: Introductionmentioning

confidence: 99%

On the Asymptotic Behavior of the Linearity Defect

Nguyen

Vu²

2017

Nagoya Math. J.

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This work concerns the linearity defect of a module M over a noetherian local ring R, introduced by Herzog and Iyengar in 2005, and denoted by ld R M . Roughly speaking, ld R M is the homological degree beyond which the minimal free resolution of M is linear. In the paper, it is proved that for any ideal I in a regular local ring R and for any finitely generated Rmodule M , each of the sequences (ld R (I n M )) n and (ld R (M/I n M )) n is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence (ld R C n ) n where C is a finitely generated graded module over a standard graded algebra over R. The second statement follows from the first together with a result of Avramov on small homomorphisms.2010 Mathematics Subject Classification. 13D02, 13D05, 13H99.

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“…First, while the Hochster's formula can give much information about the regularity of Stanley-Reisner ideals (see for example Dochtermann and Engström [5]), up to now there is no combinatorial interpretation of the linearity defect of Stanley-Reisner ideals. (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).…”

Section: Introductionmentioning

confidence: 99%

“…Section 6 concerns with the computation of linearity defect for the simplest non-weakly-chordal graphs, namely cycles of length ≥ 5. (For complements of cycles, the computation was done in [32,Theorem 5.1].) Besides applications to the theory of regularity and projective dimension of edge ideals, in Section 7, we prove Theorem 1.1.…”

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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Linearity defects of face rings

Cited by 7 publications

References 17 publications

Linearity defect and regularity over a Koszul algebra

Linearity defect and regularity over a Koszul algebra

On the Asymptotic Behavior of the Linearity Defect

Linearity defect of edge ideals and Fröberg’s theorem

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