Open Access
2013 Betti diagrams from graphs
Alexander Engström, Matthew Stamps
Algebra Number Theory 7(7): 1725-1742 (2013). DOI: 10.2140/ant.2013.7.1725


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.


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Alexander Engström. Matthew Stamps. "Betti diagrams from graphs." Algebra Number Theory 7 (7) 1725 - 1742, 2013.


Received: 6 November 2012; Revised: 25 January 2013; Accepted: 12 March 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1300.13013
MathSciNet: MR3117505
Digital Object Identifier: 10.2140/ant.2013.7.1725

Primary: 13D02
Secondary: 05C25

Keywords: Boij–Söderberg theory , linear resolutions , threshold graphs

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 7 • 2013
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