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NON-COMMUTATIVE CHARACTERISTIC POLYNOMIALS AND COHN LOCALIZATION

Published online by Cambridge University Press:  24 August 2001

DESMOND SHEIHAM
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ; des@sheiham.com
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Abstract

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : PP of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : PP of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.

The paper gives an example of a non-commutative ring A and an endomorphism α : PP for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

Type
Research Article
Copyright
The London Mathematical Society 2001

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