Vol. 7, No. 7, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Other MSP Journals
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
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