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arXiv · 1210.8069

Betti diagrams from graphs

Alexander Engström·Matthew T. Stamps
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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.

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Alexander Engström, Matthew T. Stamps. 2013-03-13. Betti diagrams from graphs. https://doi.org/10.2140/ant.2013.7.1725

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