## Minimal free resolutions of monomial ideals and of toric rings are supported on posets

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- by Timothy B. P. Clark and Alexandre B. Tchernev PDF
- Trans. Amer. Math. Soc.
**371**(2019), 3995-4027 Request permission

## Abstract:

We introduce the notion of a*resolution supported on a poset*. When the poset is a CW-poset, i.e., the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a

*homology CW-poset*that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and, generalizing results of Miller and Sturmfels, we prove a fundamental relationship between Artinianizations and Alexander duality for monomial ideals.

## References

- Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - E. Batzies and V. Welker,
*Discrete Morse theory for cellular resolutions*, J. Reine Angew. Math.**543**(2002), 147–168. MR**1887881**, DOI 10.1515/crll.2002.012 - Dave Bayer, Sorin Popescu, and Bernd Sturmfels,
*Syzygies of unimodular Lawrence ideals*, J. Reine Angew. Math.**534**(2001), 169–186. MR**1831636**, DOI 10.1515/crll.2001.040 - Dave Bayer and Bernd Sturmfels,
*Cellular resolutions of monomial modules*, J. Reine Angew. Math.**502**(1998), 123–140. MR**1647559**, DOI 10.1515/crll.1998.083 - Dave Bayer, Irena Peeva, and Bernd Sturmfels,
*Monomial resolutions*, Math. Res. Lett.**5**(1998), no. 1-2, 31–46. MR**1618363**, DOI 10.4310/MRL.1998.v5.n1.a3 - A. Björner,
*Posets, regular CW complexes and Bruhat order*, European J. Combin.**5**(1984), no. 1, 7–16. MR**746039**, DOI 10.1016/S0195-6698(84)80012-8 - Florian Block and Josephine Yu,
*Tropical convexity via cellular resolutions*, J. Algebraic Combin.**24**(2006), no. 1, 103–114. MR**2245783**, DOI 10.1007/s10801-006-9104-9 - Benjamin Braun, Jonathan Browder, and Steven Klee,
*Cellular resolutions of ideals defined by nondegenerate simplicial homomorphisms*, Israel J. Math.**196**(2013), no. 1, 321–344. MR**3096594**, DOI 10.1007/s11856-012-0149-2 - Hara Charalambous and Alexandre Tchernev,
*Free resolutions for multigraded modules: a generalization of Taylor’s construction*, Math. Res. Lett.**10**(2003), no. 4, 535–550. MR**1995792**, DOI 10.4310/MRL.2003.v10.n4.a12 - Hara Charalambous and Apostolos Thoma,
*On the generalized Scarf complex of lattice ideals*, J. Algebra**323**(2010), no. 5, 1197–1211. MR**2584952**, DOI 10.1016/j.jalgebra.2009.12.019 - Hara Charalambous and Apostolos Thoma,
*On simple $\scr A$-multigraded minimal resolutions*, Combinatorial aspects of commutative algebra, Contemp. Math., vol. 502, Amer. Math. Soc., Providence, RI, 2009, pp. 33–44. MR**2583272**, DOI 10.1090/conm/502/09855 - Timothy B. P. Clark,
*Poset resolutions and lattice-linear monomial ideals*, J. Algebra**323**(2010), no. 4, 899–919. MR**2578585**, DOI 10.1016/j.jalgebra.2009.11.029 - Timothy B. P. Clark,
*A minimal poset resolution of stable ideals*, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 143–166. MR**2932584** - Timothy B. P. Clark and Sonja Mapes,
*Rigid monomial ideals*, J. Commut. Algebra**6**(2014), no. 1, 33–52. MR**3215560**, DOI 10.1216/JCA-2014-6-1-33 - T. B. P. Clark and S. Mapes,
*The Betti poset in monomial resolutions*, http://arxiv.org/abs/1407.5702 (2014). - Timothy B. P. Clark and Alexandre Tchernev,
*Regular CW-complexes and poset resolutions of monomial ideals*, Comm. Algebra**44**(2016), no. 6, 2707–2718. MR**3492183**, DOI 10.1080/00927872.2015.1034545 - Alberto Corso and Uwe Nagel,
*Specializations of Ferrers ideals*, J. Algebraic Combin.**28**(2008), no. 3, 425–437. MR**2438922**, DOI 10.1007/s10801-007-0111-2 - Alberto Corso and Uwe Nagel,
*Monomial and toric ideals associated to Ferrers graphs*, Trans. Amer. Math. Soc.**361**(2009), no. 3, 1371–1395. MR**2457403**, DOI 10.1090/S0002-9947-08-04636-9 - Mike Develin and Josephine Yu,
*Tropical polytopes and cellular resolutions*, Experiment. Math.**16**(2007), no. 3, 277–291. MR**2367318** - Anton Dochtermann and Alexander Engström,
*Cellular resolutions of cointerval ideals*, Math. Z.**270**(2012), no. 1-2, 145–163. MR**2875826**, DOI 10.1007/s00209-010-0789-z - Anton Dochtermann, Michael Joswig, and Raman Sanyal,
*Tropical types and associated cellular resolutions*, J. Algebra**356**(2012), 304–324. MR**2891135**, DOI 10.1016/j.jalgebra.2011.12.028 - Anton Dochtermann and Fatemeh Mohammadi,
*Cellular resolutions from mapping cones*, J. Combin. Theory Ser. A**128**(2014), 180–206. MR**3265923**, DOI 10.1016/j.jcta.2014.08.007 - Anton Dochtermann and Raman Sanyal,
*Laplacian ideals, arrangements, and resolutions*, J. Algebraic Combin.**40**(2014), no. 3, 805–822. MR**3265234**, DOI 10.1007/s10801-014-0508-7 - Gunnar Fløystad,
*Cellular resolutions of Cohen-Macaulay monomial ideals*, J. Commut. Algebra**1**(2009), no. 1, 57–89. MR**2462382**, DOI 10.1216/JCA-2009-1-1-57 - Vesselin Gasharov, Irena Peeva, and Volkmar Welker,
*The lcm-lattice in monomial resolutions*, Math. Res. Lett.**6**(1999), no. 5-6, 521–532. MR**1739211**, DOI 10.4310/MRL.1999.v6.n5.a5 - A. V. Geramita, B. Harbourne, J. Migliore, and U. Nagel,
*Matroid configurations and symbolic powers of their ideals*, Trans. Amer. Math. Soc.**369**(2017), no. 10, 7049–7066. MR**3683102**, DOI 10.1090/tran/6874 - D. R. Grayson and M. E. Stillman,
*Macaulay2, a software system for research in algebraic geometry*, http://www.math.uiuc.edu/Macaulay2/. - Jürgen Herzog and Takayuki Hibi,
*Monomial ideals*, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR**2724673**, DOI 10.1007/978-0-85729-106-6 - Michael Jöllenbeck and Volkmar Welker,
*Minimal resolutions via algebraic discrete Morse theory*, Mem. Amer. Math. Soc.**197**(2009), no. 923, vi+74. MR**2488864**, DOI 10.1090/memo/0923 - Shin-Yao Jow and Ezra Miller,
*Multiplier ideals of sums via cellular resolutions*, Math. Res. Lett.**15**(2008), no. 2, 359–373. MR**2385647**, DOI 10.4310/MRL.2008.v15.n2.a13 - Ashok Kumar and Chanchal Kumar,
*Multigraded Betti numbers of multipermutohedron ideals*, J. Ramanujan Math. Soc.**28**(2013), no. 1, 1–18. MR**3060297** - Jeffrey Mermin,
*The Eliahou-Kervaire resolution is cellular*, J. Commut. Algebra**2**(2010), no. 1, 55–78. MR**2607101**, DOI 10.1216/JCA-2010-2-1-55 - Ezra Miller and Bernd Sturmfels,
*Combinatorial commutative algebra*, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR**2110098** - Uwe Nagel and Victor Reiner,
*Betti numbers of monomial ideals and shifted skew shapes*, Electron. J. Combin.**16**(2009), no. 2, Special volume in honor of Anders Björner, Research Paper 3, 59. MR**2515766** - Isabella Novik,
*Lyubeznik’s resolution and rooted complexes*, J. Algebraic Combin.**16**(2002), no. 1, 97–101. MR**1941987**, DOI 10.1023/A:1020838732281 - Ryota Okazaki and Kohji Yanagawa,
*On CW complexes supporting Eliahou-Kervaire type resolutions of Borel fixed ideals*, Collect. Math.**66**(2015), no. 1, 125–147. MR**3295068**, DOI 10.1007/s13348-014-0104-0 - Peter Orlik and Volkmar Welker,
*Algebraic combinatorics*, Universitext, Springer, Berlin, 2007. Lectures from the Summer School held in Nordfjordeid, June 2003. MR**2322081** - Irena Peeva and Mauricio Velasco,
*Frames and degenerations of monomial resolutions*, Trans. Amer. Math. Soc.**363**(2011), no. 4, 2029–2046. MR**2746674**, DOI 10.1090/S0002-9947-2010-04980-3 - V. Reiner and V. Welker,
*Linear syzygies of Stanley-Reisner ideals*, Math. Scand.**89**(2001), no. 1, 117–132. MR**1856984**, DOI 10.7146/math.scand.a-14333 - Achilleas Sinefakopoulos,
*On Borel fixed ideals generated in one degree*, J. Algebra**319**(2008), no. 7, 2739–2760. MR**2397405**, DOI 10.1016/j.jalgebra.2008.01.017 - Emil Sköldberg,
*Morse theory from an algebraic viewpoint*, Trans. Amer. Math. Soc.**358**(2006), no. 1, 115–129. MR**2171225**, DOI 10.1090/S0002-9947-05-04079-1 - Diana Kahn Taylor,
*IDEALS GENERATED BY MONOMIALS IN AN R-SEQUENCE*, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)–The University of Chicago. MR**2611561** - Alexandre B. Tchernev,
*Representations of matroids and free resolutions for multigraded modules*, Adv. Math.**208**(2007), no. 1, 75–134. MR**2304312**, DOI 10.1016/j.aim.2006.02.002 - Alexandre Tchernev and Marco Varisco,
*Modules over categories and Betti posets of monomial ideals*, Proc. Amer. Math. Soc.**143**(2015), no. 12, 5113–5128. MR**3411130**, DOI 10.1090/proc/12643 - Mauricio Velasco,
*Minimal free resolutions that are not supported by a CW-complex*, J. Algebra**319**(2008), no. 1, 102–114. MR**2378063**, DOI 10.1016/j.jalgebra.2007.10.011 - Charles A. Weibel,
*An introduction to homological algebra*, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR**1269324**, DOI 10.1017/CBO9781139644136 - Sergey Yuzvinsky,
*Taylor and minimal resolutions of homogeneous polynomial ideals*, Math. Res. Lett.**6**(1999), no. 5-6, 779–793. MR**1739231**, DOI 10.4310/MRL.1999.v6.n6.a14

## Additional Information

**Timothy B. P. Clark**- Affiliation: Department of Mathematics and Statistics, Loyola University Maryland, 4501 North Charles Street, Baltimore, Maryland 21210
- MR Author ID: 890222
- ORCID: 0000-0002-1196-8645
- Email: tbclark@loyola.edu
**Alexandre B. Tchernev**- Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, 1400 Washington Avenue, Albany, New York 12222
- MR Author ID: 610821
- Email: atchernev@albany.edu
- Received by editor(s): October 9, 2015
- Received by editor(s) in revised form: July 14, 2017, and November 9, 2017
- Published electronically: October 23, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 3995-4027 - MSC (2010): Primary 13D02, 06A11, 14M25; Secondary 05E40
- DOI: https://doi.org/10.1090/tran/7614
- MathSciNet review: 3917215