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arXiv · 1703.04687

Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings

Thomas Huettemann·Zuhong Zhang
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Abstract

Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Kohno and Pajitnov as a special case.

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Thomas Huettemann, Zuhong Zhang. 2017-03-14. Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings. https://doi.org/10.1016/j.jpaa.2019.01.015

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