Syzygies and Homotopy Theory pp 213-220 | Cite as

# Parametrizing *Ω*_{1}(**Z**): Generic Case

Chapter

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## Abstract

In this chapter we will take where 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.

*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$$0 \rightarrow J \rightarrow \varLambda^m \rightarrow \mathbf{Z}\rightarrow 0$$

*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 Ext^{1}(**Z**,*Λ*) is zero or not.-
**Generic Case:** -
Ext

^{1}(**Z**,*Λ*)=0 -
**Singular Case:** -
Ext

^{1}(**Z**,*Λ*)≠0

## Keywords

Height Preserving Syzygies Complete Decision Isomorphism Classes Stable Class
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## References

- 18.Curtis, C.W., Reiner, I.: Methods of Representation Theory, vols. I & II. Wiley-Interscience, New York (1981/1987) Google Scholar
- 34.Gruenberg, K.W.: Cohomological Topics in Group Theory. Lecture Notes in Mathematics, vol. 143. Springer, Berlin (1970) zbMATHGoogle Scholar
- 46.Jacobinski, H.: Genera and decompositions of lattices over orders. Acta Math.
**121**, 1–29 (1968) MathSciNetzbMATHCrossRefGoogle Scholar - 50.Johnson, F.E.A.: Minimal 2-complexes and the D(2)-problem. Proc. Am. Math. Soc.
**132**, 579–586 (2003) CrossRefGoogle Scholar - 54.Johnson, F.E.A.: The stable class of the augmentation ideal. K-Theory
**34**, 141–150 (2005) MathSciNetzbMATHCrossRefGoogle Scholar - 93.Swan, R.G.: K-Theory of Finite Groups and Orders (notes by E.G. Evans). Lecture Notes in Mathematics, vol. 149. Springer, Berlin (1970) CrossRefGoogle Scholar

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