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Linear syzygies, hyperbolic Coxeter groups and regularity

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

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]
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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 

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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).

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