Cohen-Macaulayness for symbolic power ideals of edge ideals

Giancarlo Rinaldo, Naoki Terai, Ken ichi Yoshida

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


Let S=K[x1,..., xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G) is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].

Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalJournal of Algebra
Issue number1
Publication statusPublished - Dec 1 2011
Externally publishedYes


  • Cohen-Macaulay
  • Complete intersection
  • Edge ideal
  • FLC
  • Polarization
  • Simplicial complex
  • Symbolic powers

ASJC Scopus subject areas

  • Algebra and Number Theory


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