Euler Characteristics

  • Kenneth S. Brown
Part of the Graduate Texts in Mathematics book series (GTM, volume 87)


Let Γ be a finite group and P a finitely generated projective ℤΓ-module. There are several ways that one might try to associate a “rank” to P. One candidate for this, which we will denote by ε(P) or εΓ(P), is defined via extension of scalars with respect to the augmentation map ℤΓ → ℤ. Namely, we consider PΓ = ℤ ⊗ℤΓ P, which is a finitely generated projective ℤ-module; since projective ℤ-modules are free, we can set
$$\varepsilon (P) = r{k_z}({P_\Gamma })$$
the right hand side being the rank in the naive sense (i.e., the cardinality of a basis). Note that 1.1 makes sense even if Γ is infinite.


Conjugacy Class Spectral Sequence Direct Summand Chain Complex Euler Characteristic 


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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Kenneth S. Brown
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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