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

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 $p:{\mathbb{Z}}^s\to {\mathbb{Z}}^t$ such that $p_{\ast }(X)$ 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.

arXiv comment12 pages; v2: minor changes and corrections; v3: final version, 13 pages, minor changes

References in corpus (1)

• Examples of topological spaces with arbitrary cohomology jump loci