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

Let C be a chain complex of finitely generated free modulesover a commutative Laurent polynomial ring Ls in sindeterminates. Given a group homomorphism ➝p:Zs➝Zt we let p!(C)=C⊗LsLt denote the resulting induced complex over the Laurent polynomial ring Lt in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that bk(p!(C))>bk(C), have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.