The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the ${r}^{th}$ power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called “extremal ideals”, and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.

Revised:

Accepted:

Published online:

Keywords: powers of ideals, simplicial complex, Betti numbers, free resolutions, monomial ideals, extremal ideals

^{1}; El Khoury, Sabine

^{2}; Faridi, Sara

^{3}; Mayes-Tang, Sarah

^{4}; Morey, Susan

^{5}; Şega, Liana M.

^{6}; Spiroff, Sandra

^{7}

@article{ALCO_2024__7_1_77_0, author = {Cooper, Susan M. and El Khoury, Sabine and Faridi, Sara and Mayes-Tang, Sarah and Morey, Susan and \c{S}ega, Liana M. and Spiroff, Sandra}, title = {Simplicial resolutions of powers of square-free monomial ideals}, journal = {Algebraic Combinatorics}, pages = {77--107}, publisher = {The Combinatorics Consortium}, volume = {7}, number = {1}, year = {2024}, doi = {10.5802/alco.325}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.325/} }

TY - JOUR AU - Cooper, Susan M. AU - El Khoury, Sabine AU - Faridi, Sara AU - Mayes-Tang, Sarah AU - Morey, Susan AU - Şega, Liana M. AU - Spiroff, Sandra TI - Simplicial resolutions of powers of square-free monomial ideals JO - Algebraic Combinatorics PY - 2024 SP - 77 EP - 107 VL - 7 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.325/ DO - 10.5802/alco.325 LA - en ID - ALCO_2024__7_1_77_0 ER -

%0 Journal Article %A Cooper, Susan M. %A El Khoury, Sabine %A Faridi, Sara %A Mayes-Tang, Sarah %A Morey, Susan %A Şega, Liana M. %A Spiroff, Sandra %T Simplicial resolutions of powers of square-free monomial ideals %J Algebraic Combinatorics %D 2024 %P 77-107 %V 7 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.325/ %R 10.5802/alco.325 %G en %F ALCO_2024__7_1_77_0

Cooper, Susan M.; El Khoury, Sabine; Faridi, Sara; Mayes-Tang, Sarah; Morey, Susan; Şega, Liana M.; Spiroff, Sandra. Simplicial resolutions of powers of square-free monomial ideals. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 77-107. doi : 10.5802/alco.325. https://alco.centre-mersenne.org/articles/10.5802/alco.325/

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