Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings |
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Authors: | Thomas Hüttemann Zuhong Zhang |
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Affiliation: | 1. Mathematical Sciences Research Centre, School of Mathematics and Physics, Queen''s University Belfast, Belfast BT7 1NN, Northern Ireland, UK;2. School of Mathematics, Beijing Institute Of Technology, 5 South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China |
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Abstract: | Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring in s indeterminates. Given a group homomorphism we let denote the resulting induced complex over the Laurent polynomial ring in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that , 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. |
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Keywords: | Primary 13B25 secondary 13C99 Jump loci McCoy rank of matrices Betti number Laurent polynomial ring |
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