Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings (Q2001400)

Let \(R\) be a commutative integral domain with unit. A group homomorphism \(p:\mathbb{}{Z}^s\rightarrow \mathbb{Z}^t\) determines a homomorphism \(p_*:R[\mathbb{Z}^s]\rightarrow R[\mathbb{Z}^t]\) of a group rings. Given a bounded complex \(C\) of finitely generated free \(R[\mathbb{Z}^s]\)-modules we obtain the induced chain complex \(p_!(C)=C\bigotimes_{R[\mathbb{Z}^s]}R[\mathbb{Z}^t]\) of finitely generated free \(R[\mathbb{Z}^t]\)-modules. \textit{T. Kohno} and \textit{A. Pajitnov} [Cent. Eur. J. Math. 12, No. 9, 1285--1304 (2014; Zbl 1308.55003)] proved the result characterizing the jump loci of the Betti numbers with respect to varying the group homomorphism \(p\). The authors of the paper under this review examine the behaviour of ranks of matrices and Betti numbers of chain complexes over Laurent polynomial rings in several indeterminates under a linear change of variables. They prove that the Betti numbers jump loci (Theorem 5.5). The authors generalise a notion Betti numbers and work with non-unital commutative rings.