Parametrizing Ω1(Z): Generic Case

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


In this chapter we will take G to be a finitely generated group and we denote by \(\mathcal{SF}_{+}\) the isomorphism classes of finitely generated nonzero stably free modules over Z[G]. As before we denote by Ω 1(Z) the first syzygy of Z over Z[G]; that is, the stable class [J] of any module J which occurs in an exact sequence of Λ-modules
$$0 \rightarrow J \rightarrow \varLambda^m \rightarrow \mathbf{Z}\rightarrow 0$$
where Λ=Z[G]. We have previously seen that both Ω 1(Z) and \(\mathcal{SF}_{+}\) have the structure of trees in which the roots do not extend infinitely downwards and, for certain G at least, we considered the structure of \(\mathcal{SF}_{+}\) in some detail. Here we seek to parametrize Ω 1(Z) by \(\mathcal{SF}_{+}\); that is, the ‘unknown’ by the ‘known’. We will show that, under suitable conditions, there is a height preserving mapping of trees \(\kappa: \mathcal{SF}_{+} \rightarrow\varOmega_{1}(\mathbf{Z})\); compare also Johnson (K-Theory 34:141–150, 2005). We then proceed to establish conditions under which κ is injective and/or surjective. The problem divides naturally into two cases, according to whether Ext1(Z,Λ) is zero or not.
Generic Case:


Singular Case:


The Generic Case admits of a reasonably complete conclusion, which we detail below. The Singular Case, however, is more intricate and is considered in Chaps.  14 and  15.


Height Preserving Syzygies Complete Decision Isomorphism Classes Stable Class 
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© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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