Narrow Results

(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals $J$ with $\mathrm{pd}(R/J)\le \mu (J)-1$, 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 $\mathrm{pd}(R/J)\le \mu (J)-2$. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs ${\mathscr{H}}$ (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.

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.