Vol. 7, No. 7, 2013

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Betti diagrams from graphs

Alexander Engström and Matthew T. Stamps

Vol. 7 (2013), No. 7, 1725–1742
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
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