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arXiv · 2404.10672

Betti numbers of normal edge rings (\bf{I})

Zexin Wang·Dancheng Lu
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Abstract

We introduce a novel approach named the {\it induced-subgraph approach} for investigating the Betti numbers of normal edge rings. Utilizing this approach, we compute all the multigraded Betti numbers of the edge rings associated with two-ear graphs (Definition~4.1) and compact graphs (Definition~5.1). In particular, we show that for two-ear graphs and compact graphs of type 1 or 2, their multigraded Betti numbers are always equal to the top multigraded Betti numbers of some of their induced subgraphs. In contrast, some of the multigraded Betti numbers of compact graphs of type 3 are not the top multigraded Betti numbers of any of their induced subgraphs. We expect that our approach may apply to many other normal edge rings.

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Zexin Wang, Dancheng Lu. 2026-08-22. Betti numbers of normal edge rings (\bf{I}). https://arxiv.org/abs/2404.10672

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