Golodness and polyhedral products of simplicial complexes with minimal Taylor resolutions

Abstract

Let K be a simplicial complex such that the Taylor resolution for its Stanley-Reisner ring is minimal. We prove that the following conditions are equivalent: (1) K is Golod; (2) any two minimal non-faces of K are not disjoint; (3) the moment-angle complex for K is homotopy equivalent to a wedge of spheres; (4) the decomposition of the suspension of the polyhedral product Z K (CX, X) due to Bahri, Bendersky, Cohen and Gitler desuspends. K.I. is supported by JSPS KAKENHI (No. 26400094), and D.K. is supported by JSPS KAKENHI (No. 25400087).

Two-dimensional Golod complexes

“…As in Theorems 1.3 and 1.7 as well as [4,5,7,8,9], Golodness over any field of several important classes of simplicial complexes has been proved to be a consequence of the corresponding moment-angle complexes being suspensions. Then it is natural to ask whether or not there is a simplicial complex K such that K is Golod over any field and Z K is not a suspension.…”

“…For several important Golod complexes such as the Alexander dual of sequentially Cohen-Macaulay complexes, the fat-wedge filtration of RZ K has been proved to be trivial [7,8,9]. In particular, by describing a condition for the fat-wedge filtration of RZ K combinatorially, combinatorial characterizations for Golodness of 1-dimensional complexes and triangulations of closed surfaces have been obtained in [7,9].…”

On Monomial Golod Ideals

“…For instance, we do not know if the product of any two homogeneous ideals in Q = k[x, y, z] is Golod. Another reason for the increasing interest is their connection to moment-angle complexes (for example, see [7,10,14]).…”

On monomial Golod ideals

“…For instance, we do not know if the product of any two homogeneous ideals in Q = k[x, y, z] is Golod. Another reason for the increasing interest is their connection to moment-angle complexes, for example see [DS07,IK18,GPTW16].…”