Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T18:08:03.856Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear syzygies, hyperbolic Coxeter groups and regularity

Part of: Arithmetic rings and other special rings Special aspects of infinite or finite groups Commutative algebra: Homological methods

Published online by Cambridge University Press:  20 May 2019

Alexandru Constantinescu ,
Thomas Kahle  and
Matteo Varbaro
Alexandru Constantinescu
Affiliation:
Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany email [email protected]
Thomas Kahle
Affiliation:
Fakultät für Mathematik, Otto-von-Guericke Universität, Universitätsplatz 2, D-39106 Magdeburg, Germany email [email protected]
Matteo Varbaro
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, Genova 16146, Italy email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.

Keywords

Stanley–Reisner ringsimplicial complexsyzygyhyperbolic Coxeter groupflag-no-square complex

MSC classification

Primary: 13F55: Stanley-Reisner face rings; simplicial complexes 20F55: Reflection and Coxeter groups
Secondary: 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 6 , June 2019 , pp. 1076 - 1097
Copyright
© The Authors 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Large parts of this research were carried out at Mathematisches Forschungsinstitut Oberwolfach within the Research in Pairs program. The work was continued with support from the MIUR-DAAD Joint Mobility Program (project no. 57267452).

References

Avramov, L. L., Conca, A. and Iyengar, S. B., Subadditivity of syzygies of Koszul algebras , Math. Ann. 361 (2013), 511–534.Google Scholar
Ananyan, T. and Hochster, M., Small subalgebras of polynomial rings and Stillman’s conjecture, Preprint (2016), arXiv:1610.09268.Google Scholar
Bayer, D. and Stillman, M., On the complexity of computing syzygies , J. Symbolic. Comput. 6 (1988), 135–147.Google Scholar
Brodmann, M. P. and Sharp, R. Y., Local cohomology, Cambridge Studies in Advanced Mathematics, vol. 136, second edition (Cambridge University Press, Cambridge, 2013).Google Scholar
Caviglia, G., Chardin, M., McCullough, J., Peeva, I. and Varbaro, M., Regularity of prime ideals , Math. Z. 291 (2019), 421–435.Google Scholar
Caviglia, G. and Sbarra, E., Characteristic-free bounds for the Castelnuovo–Mumford regularity , Compositio Math. 141 (2005), 1365–1373.Google Scholar
Constantinescu, A., Kahle, T. and Varbaro, M., Linear syzygies, flag complexes, and regularity , Collect. Math. 67 (2016), 357–362.Google Scholar
Coxeter, H. S. M., The complete enumeration of finite groups of the form R i 2 = (R i R j ) k ij = 1 , J. Lond. Math. Soc. (2) s1‐10 (1935), 21–25.Google Scholar
Dao, H., Huneke, C. and Schweig, J., Bounds on the regularity and projective dimension of ideals associated to graphs , J. Algebraic Combin. 38 (2013), 37–55.Google Scholar
Davis, M., The geometry and topology of Coxeter groups, London Mathematical Society Monographs (new series), vol. 32 (Princeton University Press, Princeton, NJ, 2008).Google Scholar
Davis, M. and Januszkiewicz, T., Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. 62 (1991), 417–451.Google Scholar
Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol 150 (Springer, New York, 1995).Google Scholar
Eisenbud, D. and Goto, S., Linear free resolutions and minimal multiplicity , J. Algebra 88 (1984), 89–133.Google Scholar
Eisenbud, D., Green, M., Hulek, K. and Popescu, S., Restricting linear syzygies: algebra and geometry , Compositio Math. 141 (2005), 1460–1478.Google Scholar
Eisenbud, D. and Schreyer, F.-O., Betti numbers of graded modules and cohomology of vector bundles , J. Amer. Math. Soc. 22 (2009), 859–888.Google Scholar
Gelfand, S. I. and Manin, Y. I., Methods of homological algebra, Springer Monographs in Mathematics, second edition (Springer, Berlin, 2003).Google Scholar
Green, M. and Lazarsfeld, R., On the projective normality of complete linear series on an algebraic curve , Invent. Math. 83 (1986), 73–90.Google Scholar
Januszkiewicz, T. and Świątkowski, J., Hyperbolic Coxeter groups of large dimension , Comment. Math. Helv. 78 (2003), 555–583.Google Scholar
Januszkiewicz, T. and Świątkowski, J., Simplicial nonpositive curvature , Publ. Math. Inst. Hautes Études Sci. 104 (2006), 1–85.Google Scholar
Januszkiewicz, T. and Świątkowski, J., Non-positively curved developments of billiards , J. Topol. 3 (2010), 63–80.Google Scholar
Knuth, D. E., Mathematics and computer science: coping with finiteness , Science 194 (1976), 1235–1242.Google Scholar
Lyubeznik, G., On the local cohomology modules H A i (R) for ideals A generated by monomials in an R-sequence , in Complete intersections (Acireale 1983), Lecture Notes in Mathematics, vol. 1092 (Springer, Berlin, 1984), 214–220.Google Scholar
Ma, L., Schwede, K. and Shimomoto, K., Local cohomology of Du Bois singularities and applications to families , Compositio Math. 153 (2017), 2147–2170.Google Scholar
Mayr, E. W. and Meyer, A. A., The complexity of the word problems for commutative semigroups and polynomial ideals , Adv. Math. 46 (1982), 305–329.Google Scholar
McCullough, J. and Peeva, I., Counterexamples to the Eisenbud–Goto regularity conjecture , J. Amer. Math. Soc. 31 (2018), 473–496.Google Scholar
McCullough, J. and Seceleanu, A., Bounding projective dimension , in Commutative algebra (Springer, New York, 2013), 551–576.Google Scholar
Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, Berlin, 2005).Google Scholar
Munkres, J. R., Elements of algebraic topology, Vol. 2 (Addison-Wesley, Menlo Park, CA, 1984).Google Scholar
Nevo, E. and Peeva, I., C 4 -free edge ideals , J. Algebraic Combin. 37 (2013), 243–248.Google Scholar
Osajda, D., A combinatorial non-positive curvature I: Weak systolicity, Preprint (2013),arXiv:1305.4661.Google Scholar
Osajda, D., A construction of hyperbolic Coxeter groups , Comment. Math. Helv. 88 (2013), 353–367.Google Scholar
Peeva, I. and Stillman, M., Open problems on syzygies and Hilbert functions , J. Commut. Algebra 1 (2009), 159–195.Google Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.3), Sage Development Team, https://www.sagemath.org, 2013.Google Scholar
Schwede, K., F-injective singularities are Du Bois , Amer. J. Math. 131 (2009), 445–473.Google Scholar
Ullery, B., Designer ideals with high Castelnuovo–Mumford regularity , Math. Res. Lett. 21 (2014), 1215–1225.Google Scholar