Cellular resolutions from mapping cones

Abstract

One can iteratively obtain a free resolution of any monomial ideal $I$ by considering the mapping cone of the map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal resolution if $I$ has linear quotients, in which case the mapping cone in each step cones a Koszul complex onto the previously constructed resolution. Here we consider cellular realizations of these resolutions. Extending a construction of Mermin we describe a regular CW-complex that supports the resolutions of Herzog and Takayama in the case that $I$ has a `regular decomposition function'. By varying the choice of chain map we recover other known cellular resolutions, including the `box of complexes' resolutions of Corso, Nagel, and Reiner and the related `homomorphism complex' resolutions of Dochtermann and Engstr\"om. Other choices yield combinatorially distinct complexes with interesting structure, and suggests a notion of a `space of cellular resolutions'.