Regularity and Free Resolution of Ideals Which Are Minimal To $d$-Linearity

Abstract

Toward a partial classification of monomial ideals with d-linear resolution, in this paper, some classes of d-uniform clutters which do not have linear resolution, but every proper subclutter of them has a d-linear resolution, are introduced and the regularity and Betti numbers of circuit ideals of such clutters are computed. Also, it is proved that for given two d-uniform clutters C1, C2, the CastelnuovoMumford regularity of the ideal I(C1 ∪ C2) is equal to the maximum of regularities of I(C1) and I(C2), whenever V (C1) ∩ V (C2) is a clique or SC(C1) ∩ SC(C2) = ∅.As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideal of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the circuit ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

Stability of Betti numbers under reduction processes: Towards chordality of clutters

“…This successful combinatorial characterization of ideals generated in degree 2 with linear resolution (quotients), motivated many mathematicians to generalize this result for squarefree monomial ideals generated in degree d > 2. Some partial results on this subject are given in [2,3,4,7,11,18,20,23,24]. The goal of this section is to introduce a class of d-uniform clutters (which we call chordal clutters) which extends the definition of the class of chordal graphs, and the corresponding circuit ideal has a linear resolution over any field.…”

A Criterion for a Monomial Ideal to have a Linear Resolution in Characteristic 2

“…There exists a similar notion to the clique complex in higher dimensions. The d-closure of Γ is also called the complex of Γ [4] and the clique complex of Γ [9]. We use the term d-closure to keep track of the dimension at which the operation is applied.…”

“…Recently there has been interest in finding a characterization of square-free monomial ideals with linear resolutions in terms of the combinatorics of their associated simplicial complexes or hypergraphs. See, for example, [4], [7], [8], [9], and [10]. This exploration was motivated by a theorem of Fröberg from [6] in which he gives the following combinatorial classification of the square-free monomial ideals generated in degree two which have linear resolutions.…”