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 thebox 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 aspace of cellular resolutions'.