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no. 4
Polarizations of Artin monomial ideals
Fløystad, Gunnar1; Gabrielsen, Ine1; Mafi, Amir2
1Universitetet i Bergen, Matematisk institutt, Postboks 7803, 5020 Bergen, Norway
2Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran
Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 993-1035

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This settles a conjecture of A. Almousa, H. Lohne and the first author [3].

Geometrically, polarizations of ideals strictly containing $(x_1^{a_1}, \ldots , x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots , a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible.

Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots , a_n-1$. We prove that this cell complex gives a cellular minimal free resolution of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range of examples all polarizations of the Artin monomial ideal.

We also show that the squeezed balls of G. Kalai [32] derive from polarizations of Artin monomial ideals.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.500
Classification: 13F55, 05E40, 05E45
Keywords: triangulation, ball, sphere, simplicial complex, cell complex, polarization, Artin monomial ideal, linkage, squeezed ball, Bier ball, letterplace ideal

Fløystad, Gunnar  1 ; Gabrielsen, Ine  1 ; Mafi, Amir  2

1 Universitetet i Bergen, Matematisk institutt, Postboks 7803, 5020 Bergen, Norway
2 Department of Mathematics, University Of Kurdistan, P.O. Box: 416, Sanandaj, Iran
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Fløystad, Gunnar; Gabrielsen, Ine; Mafi, Amir. Polarizations of Artin monomial ideals. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 993-1035. doi: 10.5802/alco.500
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[1] Aguiar, Marcelo; Ardila, Federico Hopf monoids and generalized permutahedra, Mem. Amer. Math. Soc., Volume 289 (2023) no. 1437, p. vi+119 | DOI | MR | Zbl

[2] Alilooee, Ali; Banerjee, Arindam Symbolic powers of multipartite hypergraphs and Waldschmidt constant, 2021 | arXiv | Zbl

[3] Almousa, Ayah; Fløystad, Gunnar; Lohne, Henning Polarizations of powers of graded maximal ideals, J. Pure Appl. Algebra, Volume 226 (2022) no. 5, Paper no. 106924, 33 pages | DOI | MR | Zbl

[4] Almousa, Ayah; VandeBogert, Keller Polarizations and hook partitions, J. Pure Appl. Algebra, Volume 226 (2022) no. 9, Paper no. 107056, 20 pages | DOI | MR | Zbl

[5] Beckenbach, Isabel; Scheidweiler, Robert Perfect $f$-matchings and $f$-factors in hypergraphs—a combinatorial approach, Discrete Math., Volume 340 (2017) no. 10, pp. 2499-2506 | DOI | MR | Zbl

[6] Beckenbach, Isabel Leonie Matchings and flows in hypergraphs, Ph. D. Thesis, FU-Berlin (2019)

[7] Berkesch, Christine; Erman, Daniel; Smith, Gregory G. Virtual resolutions for a product of projective spaces, Algebr. Geom., Volume 7 (2020) no. 4, pp. 460-481 | DOI | MR | Zbl

[8] Berkesch, Christine; Klein, Patricia; Loper, Michael C.; Yang, Jay Homological and combinatorial aspects of virtually Cohen-Macaulay sheaves, Trans. London Math. Soc., Volume 8 (2021) no. 1, pp. 413-434 | DOI | MR | Zbl

[9] Bier, Thomas A remark on Alexander duality and the disjunct join, 1992 (preprint, 8 pages)

[10] Björner, A. Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819-1872 | Zbl

[11] Björner, Anders; Paffenholz, Andreas; Sjöstrand, Jonas; Ziegler, Günter M. Bier spheres and posets, Discrete Comput. Geom., Volume 34 (2005) no. 1, pp. 71-86 | DOI | MR | Zbl

[12] Bruns, Winfried; Gubeladze, Joseph Polytopes, rings, and ${K}$-theory, Springer Monographs in Mathematics, Springer, Dordrecht, 2009, xiv+461 pages | DOI | MR | Zbl

[13] Bruns, Winfried; Herzog, Jürgen Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993, xii+403 pages | MR | Zbl

[14] D’Alì, Alessio; Fløystad, Gunnar; Nematbakhsh, Amin Resolutions of co-letterplace ideals and generalizations of Bier spheres, Trans. Amer. Math. Soc., Volume 371 (2019) no. 12, pp. 8733-8753 | DOI | MR | Zbl

[15] De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco Triangulations: Structures for algorithms and applications, Algorithms and Computation in Mathematics, 25, Springer-Verlag, Berlin, 2010, xiv+535 pages | DOI | MR | Zbl

[16] Dufossé, Fanny; Kaya, Kamer; Panagiotas, Ioannis; Uçar, Bora Effective heuristics for matchings in hypergraphs, Analysis of experimental algorithms (Lecture Notes in Comput. Sci.), Volume 11544, Springer, Cham, 2019, pp. 248-264 | DOI | MR

[17] Eagon, John A.; Reiner, Victor Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra, Volume 130 (1998) no. 3, pp. 265-275 | DOI | MR | Zbl

[18] Eisenbud, David Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages | DOI | MR | Zbl

[19] Ene, Viviana; Herzog, Jürgen; Mohammadi, Fatemeh Monomial ideals and toric rings of Hibi type arising from a finite poset, European J. Combin., Volume 32 (2011) no. 3, pp. 404-421 | DOI | MR | Zbl

[20] Fløystad, Gunnar Partitions of vertices and facets in trees and stacked simplicial complexes, Graphs Combin., Volume 40 (2024) no. 4, Paper no. 79, 22 pages | DOI | MR | Zbl

[21] Fløystad, Gunnar; Møller Greve, Bjørn; Herzog, Jürgen Letterplace and co-letterplace ideals of posets, J. Pure Appl. Algebra, Volume 221 (2017) no. 5, pp. 1218-1241 | DOI | MR | Zbl

[22] Fløystad, Gunnar; Orlich, Milo Triangulations of polygons and stacked simplicial complexes: separating their Stanley-Reisner ideals, J. Algebraic Combin., Volume 57 (2023) no. 3, pp. 659-686 | DOI | MR | Zbl

[23] Gangopadhyay, Rahul; Shannigrahi, Saswata Rectilinear crossings in complete balanced $d$-partite $d$-uniform hypergraphs, Graphs Combin., Volume 36 (2020) no. 3, pp. 905-911 | DOI | MR | Zbl

[24] Gasharov, Vesselin; Hibi, Takayuki; Peeva, Irena Resolutions of $\mathbf{a}$-stable ideals, J. Algebra, Volume 254 (2002) no. 2, pp. 375-394 | DOI | MR | Zbl

[25] Hall, Huntington Tracy Counterexamples in discrete geometry, Ph. D. Thesis, University of California, Berkeley (2004) | Zbl

[26] Hao, Hanson; Luo, Yuyuan; Saint Rain, Sterling Polarizations of equigenerated strongly stable ideals, 2023 talk at Joint Mathematics Meetings (JMM 2023)

[27] Hatcher, Allen Algebraic topology, Cambridge University Press, Cambridge, 2002, xii+544 pages | MR

[28] Herzog, Jürgen; Hibi, Takayuki Monomial ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011, xvi+305 pages | DOI | MR

[29] Herzog, Jürgen; Takayama, Yukihide Resolutions by mapping cones, Homology Homotopy Appl., Volume 4 (2002) no. 2, pp. 277-294 | DOI | MR

[30] Huneke, Craig; Ulrich, Bernd The structure of linkage, Ann. of Math. (2), Volume 126 (1987) no. 2, pp. 277-334 | DOI | MR

[31] Jevtić, Filip D.; Timotijević, Marinko; Živaljević, Rade T. Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles, Discrete Comput. Geom., Volume 65 (2021) no. 4, pp. 1275-1286 | DOI | MR | Zbl

[32] Kalai, Gil Many triangulated spheres, Discrete Comput. Geom., Volume 3 (1988) no. 1, pp. 1-14 | DOI | MR | Zbl

[33] Lee, Carl W. Subdivisions and triangulations of polytopes, Handbook of discrete and computational geometry (CRC Press Ser. Discrete Math. Appl.), CRC, Boca Raton, FL, 1997, pp. 271-290 | MR | Zbl

[34] Lu, Dancheng; Wang, Zexin The resolutions of generalized co-letterplace ideals and their powers, J. Algebra, Volume 673 (2025), pp. 321-350 | DOI | MR | Zbl

[35] Matoušek, Jiří Using the Borsuk-Ulam theorem: Lectures on topological methods in combinatorics and geometry, Universitext, Springer-Verlag, Berlin, 2003, xii+196 pages | MR | Zbl

[36] Migliore, Juan C. Introduction to liaison theory and deficiency modules, Progress in Mathematics, 165, Birkhäuser Boston, Inc., Boston, MA, 1998, xiv+215 pages | DOI | MR | Zbl

[37] Miller, Ezra Alexander duality for monomial ideals and their resolutions, 1998 | arXiv | Zbl

[38] Miller, Ezra The Alexander duality functors and local duality with monomial support, J. Algebra, Volume 231 (2000) no. 1, pp. 180-234 | DOI | MR | Zbl

[39] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR | Zbl

[40] Mohammadi, Fatemeh; Pascual-Ortigosa, Patricia; Sáenz-de-Cabezón, Eduardo; Wynn, Henry P. Polarization and depolarization of monomial ideals with application to multi-state system reliability, J. Algebraic Combin., Volume 51 (2020) no. 4, pp. 617-639 | DOI | MR | Zbl

[41] Moradi, Somayeh; Khosh-Ahang, Fahimeh On vertex decomposable simplicial complexes and their Alexander duals, Math. Scand., Volume 118 (2016) no. 1, pp. 43-56 | DOI | MR | Zbl

[42] Munkres, James R. Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, xvi+537 pages | MR | Zbl

[43] Murai, Satoshi Spheres arising from multicomplexes, J. Combin. Theory Ser. A, Volume 118 (2011) no. 8, pp. 2167-2184 | DOI | MR | Zbl

[44] Nagel, Uwe; Reiner, Victor 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

[45] Nematbakhsh, Amin Linear strands of edge ideals of multipartite uniform clutters, J. Pure Appl. Algebra, Volume 225 (2021) no. 10, Paper no. 106690, 21 pages | DOI | MR | Zbl

[46] Olteanu, Anda Constructible ideals, Comm. Algebra, Volume 37 (2009) no. 5, pp. 1656-1669 | DOI | MR | Zbl

[47] Ovchinnikov, Sergei Graphs and cubes, Universitext, Springer, New York, 2011, xiv+287 pages | DOI | MR | Zbl

[48] Peskine, C.; Szpiro, L. Liaison des variétés algébriques. I, Invent. Math., Volume 26 (1974), pp. 271-302 | DOI | MR | Zbl

[49] Postnikov, Alexander Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl

[50] Sturmfels, Bernd Gröbner bases and convex polytopes, University Lecture Series, 8, American Mathematical Society, Providence, RI, 1996, xii+162 pages | DOI | MR | Zbl

[51] Yanagawa, Kohji Alexander duality for Stanley-Reisner rings and squarefree $\mathbf{N}^n$-graded modules, J. Algebra, Volume 225 (2000) no. 2, pp. 630-645 | DOI | MR

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