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Homotopy classification and the generalized Swan homomorphism

Published online by Cambridge University Press:  07 January 2009

F.E.A. Johnson
F.E.A. Johnson
Affiliation:
feaj@math.ucl.ac.ukDepartment of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K.
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Abstract

In his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

Keywords

Derived module categorycoprojective moduleSwan homomorphismalgebraic n-complex

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Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 3 , December 2009 , pp. 491 - 536
Copyright
Copyright © ISOPP 2009

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