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Componentwise linearity of ideals arising from graphs

Dipartimento di Matematica e Informatica
Publication Date
  • Linear Resolution
  • Componentwise Linearity
  • Chordal Graphs
  • 13C05
  • 13A30
  • 20M14


LE MATEMATICHE Vol. LXIII (2008) – Fasc. II, pp. 185–189 COMPONENTWISE LINEARITY OF IDEALS ARISING FROM GRAPHS VERONICA CRISPIN - ERIC EMTANDER Let G be a simple undirected graph on n vertices. Francisco and Van Tuyl have shown that if G is chordal, then ⋂ {xi,x j}∈EG〈xi,x j〉 is componen- twise linear. A natural question that arises is for which ti j > 1 the inter- section ideal ⋂ {xi,x j}∈EG〈xi,x j〉ti j is componentwise linear, if G is chordal. In this report we show that ⋂ {xi,x j}∈EG〈xi,x j〉n−1 is componentwise linear for all n≥ 3, if G is a complete graph. We give also an example where G is chordal, but the intersection ideal is not componentwise linear for any t > 1. 1. Introduction Let G be a simple graph on n vertices, EG the edge set of G and VG the vertex set of G. Let R = k[x1, . . . ,xn] be the polynomial ring over a field k. The edge ideal of G is the quadratic squarefree monomial ideal I (G) = 〈{xix j} | {xi,x j} ∈ EG〉 ⊂ R. Then we define the squarefree Alexander dual of I (G) as I (G)∨ = ∩{xi,x j}∈EG〈xi,x j〉. To call I (G)∨ the squarefree Alexander dual of I (G) is natural since it is the Stanley–Reisner ideal of the simplicial complex ∆∨ that is the Alexander dual simplicial complex of ∆, where ∆ in turn is the simplicial complex whose Stanley–Reisner ideal is I (G). Entrato in redazione: 2 gennaio 2009 AMS 2000 Subject Classification: Primary 13C05, 13D40; Secondary 13A30, 20M14 Keywords: Linear resolution, componentwise linearity, chordal graphs The first author was partly sponosered by The Royal Swedish Academy of Sciences 186 VERONICA CRISPIN - ERIC EMTANDER In [4] Herzog and Hibi give the following definition. Given a graded ideal I ⊂ R, we denote by I〈d〉 the ideal generated by the elements of degree d that belong to I. Then we say that a (graded) ideal I ⊂ R is componentwise linear if I〈d〉 has a linear resolution for all d. If the graph G is chordal, that is, every cycle of length m ≥ 3 in G has a chord, then it is proved by Francisco and V

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