The Swan Mapping

  • F. E. A. JohnsonEmail author
Part of the Algebra and Applications book series (AA, volume 17)


In his fundamental paper on group cohomology Swan (Ann. Math. 72:267–291, 1960) defined, for any finite group G, a homomorphism \((\mathbf{Z}/\vert G \vert)^{*} \rightarrow\widetilde{K_{0}}(\mathbf{Z}[G]) \) which, in this restricted context, has since been used extensively both in the classification of projective modules (Swan in J. Reine Angew. Math. 342:66–172, 1983) and the algebraic homotopy theory of finite complexes (Johnson in Stable modules and D(2)-problem. LMS lecture notes in mathematics, vol. 301. Cambridge University Press, Cambridge, 2003). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form \(S_{J}: \mathrm{Aut}_{\mathcal{D}\mathrm{er}}(J)\rightarrow\widetilde{K_{0}}(\varLambda)\).


Exact Sequence Commutative Diagram Group Cohomology Finite Type Projective Module 
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    Johnson, F.E.A.: Stable Modules and the D(2)-Problem. LMS Lecture Notes in Mathematics, vol. 301. Cambridge University Press, Cambridge (2003) zbMATHCrossRefGoogle Scholar
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    Swan, R.G.: Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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