Betti diagrams from graphs

Engström, Alexander; Stamps, Matthew

doi:10.2140/ant.2013.7.1725

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Betti diagrams from graphs

Alexander Engström and Matthew T. Stamps

Vol. 7 (2013), No. 7, 1725–1742

DOI: 10.2140/ant.2013.7.1725

Abstract

The emergence of Boij–Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a $k$ -linear minimal resolution arises from that of the Stanley–Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of $2$ -linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with $2$ -linear resolutions, threshold graphs, and anti-lecture-hall compositions. Moreover, we prove that any Betti diagram of a module with a $2$ -linear resolution is realized by a direct sum of Stanley–Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.

Keywords

linear resolutions, Boij–Söderberg theory, threshold graphs

Mathematical Subject Classification 2010

Primary: 13D02

Secondary: 05C25

Milestones

Received: 6 November 2012

Revised: 25 January 2013

Accepted: 12 March 2013

Published: 12 October 2013

Authors

Alexander Engström
Department of Mathematics Aalto University P.O. Box 11100 FI-00076 Aalto Finland

Matthew T. Stamps
Department of Mathematics Aalto University P.O. Box 11100 FI-00076 Aalto Finland

Vol. 7, No. 7, 2013