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

Let C be a chain complex of finitely generated free modules over a commutative LAURENT polynomial ring L _s in s indeterminates. Given a group homomorphism p:Z ^s ➝Z ^t we let p _! (C)=C⊗ _{L s} L _t denote the resulting induced complex over the LAURENT polynomial ring L _t in t indeterminates. We prove that the BETTI number jump loci, that is, the sets of those homomorphisms p such that b _k (p _! (C))>b _k (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.