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Divisors on graphs, binomial and monomial ideals, and cellular resolutions

Divisors on graphs, binomial and monomial ideals, and cellular resolutions We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $${\mathbb {Z}}$$ Z -graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

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References (82)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-015-1589-2
Publisher site
See Article on Publisher Site

Abstract

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $${\mathbb {Z}}$$ Z -graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Nov 21, 2015

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