DGA structures on minimal free resolutions of binomial edge ideals

We provide a characterisation of all graphs whose parity binomial edge ideals have pure resolutions. Specifically, we show that the minimal free resolution of a parity binomial edge ideal is pure if and only if the corresponding graph is a complete bipartite graph, or a disjoint union of paths and odd cycles. We then prove the existence of a differential graded algebra (DGA) structure on the minimal free resolution of the initial ideal of the binomial edge ideal of a complete graph using algebraic Morse theory, and provide code to compute the products explicitly. Finally, we prove the non-existence of a DGA structure on the minimal free resolution of the binomial edge ideal of a complete graph with 5 or more vertices, in cases where the coefficient field of the ground ring has characteristic 2 or 3.

Publisher

University of Galway

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CC BY-NC-ND

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University of Galway Theses (PhD Theses)