Linear strands of edge ideals of multipartite uniform clutters

Abstract

We construct the first linear strands of the minimal free resolutions of edge ideals of d-partite d-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing the known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the cover ideal of such clutters are Betti numbers of certain simplicial complexes. Furthermore, we restate a characterization for edge ideals of d-partite d-uniform clutters which have linear resolutions based on the recent characterization of arithmetically Cohen-Macaulay sets of points in a multiprojective space.

The ACM property for unions of lines in P1×P2

“…These include, just to cite some of them, hyperplane sections, basic double G-linkage, liaison addition, and liaison. On the other hand, the study of varieties in multiprojective spaces plays an important role in several branches of mathematics, and it finds an application in different contexts; these include the study of monomial ideals (see for instance [2,20]), scrolls ( [6]), symbolic powers ( [5,12,13]), tensor analysis ( [4]) and virtual resolutions ( [3,10]), just to give a partial list.…”

Polarizations of powers of graded maximal ideals

“…In [20] A. Nematbakhsh gives a precise description of when an ideal generated by rainbow monomials has linear resolution. His terminology for rainbow monomials of d colors is edge monomials of d-partite d-uniform clutters.…”

“…He is able to give a characterization through a remarkable connection to the article [8] where they give a characterization of when a point set in the multiprojective space (P 1 ) n is arithmetically Cohen-Macaulay (meaning that the associated multihomogeneous coordinate ring is a Cohen-Macaulay ring). The characterization given in [20] by translating the one in [8] is the following. So this says that a subset A of X1 • X2 • • • • • Xd gives an ideal with d-linear resolution iff both A and its complement are in some sense convex.…”

“…The class of ideals generated by rainbow monomials and with m-linear resolution is precisely the class which is Alexander dual to the class of polarizations of Artinian monomial ideals in m variables, Proposition 2.1. A concise criterion for ideals generated by rainbow monomials to have linear resolution is given by A.Nematbakhsh in [20], and we recall it in Theorem 2.8.…”