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Parametrizing Ω1(Z):G=C×Φ

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

Abstract

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.

Keywords

Exact Sequence Direct Product Complete Resolution Group Ring Minimality Criterion 
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

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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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