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Let $K$ be a non-empty set of ideals of the commutative ring $R$, closedunder 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 elementsof $X$ is in $K$. For $X$ not a $K$-set we investigate the set of thosehomomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a$K$-set. We also consider corresponding notions of rank of matrices and Bettinumbers of chain complexes; this includes an analysis of the case of McCoyrank. Our setup also recovers results on jump loci obtained by Kohno andPajitnov as a special case.

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