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Linear syzygies, hyperbolic Coxeter groups and regularity

  • Alexandru Constantinescu (a1), Thomas Kahle (a2) and Matteo Varbaro (a3)

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.

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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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Linear syzygies, hyperbolic Coxeter groups and regularity

  • Alexandru Constantinescu (a1), Thomas Kahle (a2) and Matteo Varbaro (a3)

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