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

Cooper, Susan M. ^{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/

[1] Monomial resolutions, Math. Res. Lett., Volume 5 (1998) no. 1-2, pp. 31-46 | DOI | MR | Zbl

[2] Cellular resolutions of monomial modules, J. Reine Angew. Math., Volume 502 (1998), pp. 123-140 | DOI | MR | Zbl

[3] Simplicial resolutions for the second power of square-free monomial ideals, Women in commutative algebra (Assoc. Women Math. Ser.), Volume 29, Springer, Cham, [2021] ©2021, pp. 193-205 | DOI | MR | Zbl

[4] Morse resolutions of powers of square-free monomial ideals of projective dimension one, J. Algebraic Combin., Volume 55 (2022) no. 4, pp. 1085-1122 | DOI | MR | Zbl

[5] Powers of graphs & applications to resolutions of powers of monomial ideals, Res. Math. Sci., Volume 9 (2022) no. 2, Paper no. 31, 25 pages | DOI | MR | Zbl

[6] The geometry of syzygies: a second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, 229, Springer-Verlag, New York, 2005, xvi+243 pages | MR

[7] The scarf complex and Betti numbers of powers of extremal ideals, J. Pure Appl. Algebra, Volume 228 (2024) no. 6, Paper no. 107577, 32 pages | DOI | MR

[8] The facet ideal of a simplicial complex, Manuscripta Math., Volume 109 (2002) no. 2, pp. 159-174 | DOI | MR | Zbl

[9] Monomial resolutions supported by simplicial trees, J. Commut. Algebra, Volume 6 (2014) no. 3, pp. 347-361 | DOI | MR | Zbl

[10] Resolutions of monomial ideals of projective dimension 1, Comm. Algebra, Volume 45 (2017) no. 12, pp. 5453-5464 | DOI | MR | Zbl

[11] Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/

[12] A new explicit finite free resolution of ideals generated by monomials in an $R$-sequence, J. Pure Appl. Algebra, Volume 51 (1988) no. 1-2, pp. 193-195 | DOI | MR | Zbl

[13] Three simplicial resolutions, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 127-141 | DOI | MR | Zbl

[14] Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR

[15] Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin., Volume 16 (2009) no. 2, Paper no. 3, 59 pages | DOI | MR | Zbl

[16] Minimal Resolutions of Monomial Ideals, Ph. D. Thesis, Duke University (2020), 135 pages

[17] Algebraic combinatorics, Universitext, Springer, Berlin, 2007, viii+177 pages (lectures from the Summer School held in Nordfjordeid, June 2003) | MR

[18] Frames and degenerations of monomial resolutions, Trans. Amer. Math. Soc., Volume 363 (2011) no. 4, pp. 2029-2046 | DOI | MR | Zbl

[19] Ideals generated by monomials in an R-sequence, Ph. D. Thesis, the University of Chicago (1966)

[20] Dynamical systems on chain complexes and canonical minimal resolutions, 2019 | arXiv

[21] Rees algebras of edge ideals, Comm. Algebra, Volume 23 (1995) no. 9, pp. 3513-3524 | DOI | MR | Zbl

[22] Resolutions of facet ideals, Comm. Algebra, Volume 32 (2004) no. 6, pp. 2301-2324 | DOI | MR | Zbl

*Cited by Sources: *