Parametrizing Ω1(Z):G=C×Φ

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


Both first and second minimality criteria fail in a case of particular interest, namely when G is a direct product G=F×Φ in which F is a free group and Φ is finite. As we observed in Proposition 13.2, when Φ is the trivial group, the conclusion also fails. Nevertheless, using a rather more intricate argument, we are still able to show that the conclusion is sustained when the finite factor Φ is nontrivial; that is we shall show:
Third minimality criterion:

\(\mathcal{I}\) lies at the minimal level of Ω 1(Z) when G is a direct product F m ×Φ where F m is a free group of rank m≥1 and Φ is finite and nontrivial.

The results of this section first appeared in Johnson (J. Algebra 337:181–194, 2011). The proof requires a knowledge of all the syzygies Ω r (Z) over Z[F n ×C m ] so we begin by giving a complete resolution of Z in this case.


Exact Sequence Direct Product Complete Resolution Group Ring Minimality Criterion 
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  1. 1.
    Bass, H.: Projective modules over free groups are free. J. Algebra 1, 367–373 (1964) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bass, H.: Algebraic K-Theory. Benjamin, Elmsford (1968) zbMATHGoogle Scholar
  3. 29.
    Farrell, F.T.: An extension of Tate cohomology to a class of infinite groups. J. Pure Appl. Algebra. 10 (1977) Google Scholar
  4. 58.
    Johnson, F.E.A.: Infinite branching in the first syzygy. J. Algebra 337, 181–194 (2011) MathSciNetCrossRefGoogle Scholar
  5. 75.
    Montgomery, M.S.: Left and right inverses in group algebras. Bull. Am. Meteorol. Soc. 75, 539–540 (1969) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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