Betti Diagrams with Special Shape

Let S = K[x 1 ,. .. , x n ] denote the polynomial ring in n variables over a field K with each deg x i = 1 and I ⊂ S a homogeneous ideal. Let F S/I : 0 → j≥1 S (−(p + j)) βp,p+j (S/I) → • • • → j≥1 S (−(1 + j)) β1,1+j (S/I) → S → S/I → 0 be the minimal graded free resolution of S/I over S, where p = proj dim(S/I) is the projective dimension of S/I and β i,i+j (S/I) is the (i, i + j)-th graded Betti number of S/I. The (Castelnuovo-Mumford) regularity of S/I is reg (S/I) = max{j : β i,i+j (S/I) = 0}. A graded Betti number β i,i+j (S/I) = 0 is said to be extremal [4, Definition 4.3.13] if β k,k+ (S/I) = 0 for all pairs (k,) = (i, j) with k ≥ i and ≥ j. Extremal Betti numbers of graded algebras have been studied, for example, in [1,6]. In general, S/I has the unique extremal Betti number if and only if β p,p+r (S/I) = 0, where r = reg(S/I). These equivalent properties hold if S/I is Cohen-Macaulay [2, Lemma 3]. Moreover a Cohen-Macaulay graded algebra S/I is Gorenstein if and only if j β p,p+j (S/I) = β p,p+r (S/I) = 1. Let G be a finite simple graph (i.e. a graph with no loop and no multiple edge) on the vertex set V (G) = {x 1 , x 2 ,. .. , x n } with E(G) its edge set. The edge ideal of G is 150 T. Hibi et al. Arch. Math.

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Breaking up simplicial homology and subadditivity of syzygies

Faridi

Shahada

2021

J Algebr Comb

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Betti Diagrams with Special Shape

Cited by 8 publications

References 19 publications

Subadditivity of Syzygies of Ideals and Related Problems

Subadditivity of Syzygies of Ideals and Related Problems

Extremal Betti numbers of edge ideals

Breaking up simplicial homology and subadditivity of syzygies

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