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Journal of the London Mathematical Society 2001 64(1):13-28; doi:10.1017/S0024610701002307
© 2001 by London Mathematical Society
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© The London Mathematical Society

Non-Commutative Characteristic Polynomials and Cohn Localization

Desmond Sheiham

Department of Mathematics and Statistics, University of Edinburgh King's Buildings, Edinburgh EH9 3JZ, des{at}sheiham.com

Received 12 June 2000.

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism {alpha} : P -> P of a finitely generated projective A-module determines (P, {alpha}) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism {alpha} : P -> P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 – x{alpha}] isin 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 {alpha} : P -> P for which the generalized characteristic polynomial does not determine (P, {alpha}) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map {sum}–1A[x] -> A[[x]] where {sum}–1 A[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 ↦ 0.


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