TY - JOUR
T1 - Hypergraphs with high projective dimension and 1-dimensional hypergraphs
AU - Lin, K. N.
AU - Mantero, P.
N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - (Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals J with pd(R/J) ≤ μ(J) - 1, i.e. whose projective dimension equals the minimal number of generators of J minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for pd(R/J) ≤ μ(J) - 2. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.
AB - (Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals J with pd(R/J) ≤ μ(J) - 1, i.e. whose projective dimension equals the minimal number of generators of J minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for pd(R/J) ≤ μ(J) - 2. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.
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U2 - 10.1142/S0218196717500291
DO - 10.1142/S0218196717500291
M3 - Article
AN - SCOPUS:85028330850
VL - 27
SP - 591
EP - 617
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
SN - 0218-1967
IS - 6
ER -