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Cellular resolutions from mapping cones. (English) Zbl 1301.05379
Summary: 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 J. Herzog and Y. Takayama [Homology Homotopy Appl. 4, No. 2(2), 277–294 (2002; Zbl 1028.13008)] 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 A. Corso and U. Nagel [Trans. Am. Math. Soc. 361, No. 3, 1371–1395 (2009; Zbl 1228.05068)], and U. Nagel and V. Reiner [Electron. J. Comb. 16, No. 2, Research Paper R3, 59 p. (2009; Zbl 1186.13022)] and the related ‘homomorphism complex’ resolutions of A. Dochtermann and A. Engström [Math. Z. 270, No. 1–2, 145–163 (2012; Zbl 1246.13015)]. Other choices yield combinatorially distinct complexes with interesting structure, and suggests a notion of a ‘space of cellular resolutions’.

##### MSC:
 05E40 Combinatorial aspects of commutative algebra 13D02 Syzygies, resolutions, complexes and commutative rings
##### Keywords:
cellular resolution; mapping cone; linear quotients
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##### References:
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