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The Swan Mapping

Part of the Algebra and Applications book series (AA, volume 17)

Abstract

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 \(S_{J}: \mathrm{Aut}_{\mathcal{D}\mathrm{er}}(J)\rightarrow\widetilde{K_{0}}(\varLambda)\).

Keywords

Exact Sequence Commutative Diagram Group Cohomology Finite Type Projective Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 52.
    Johnson, F.E.A.: Stable Modules and the D(2)-Problem. LMS Lecture Notes in Mathematics, vol. 301. Cambridge University Press, Cambridge (2003) MATHCrossRefGoogle Scholar
  2. 91.
    Swan, R.G.: Periodic resolutions for finite groups. Ann. Math. 72, 267–291 (1960) MathSciNetMATHCrossRefGoogle Scholar
  3. 94.
    Swan, R.G.: Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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