Advertisement

Generalized Swan Modules

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

Abstract

Let G=C ×Φ where Φ is a nontrivial finite group. In this chapter, subject to conditions on Φ, we begin the task of parametrizing the first syzygy Ω 1(Z) over Z[G]. We do so via a generalization, introduced by Edwards in his thesis (Ph.D. Thesis, University College London, 2006), of a type of module first studied by Swan. The original Swan modules arise as follows; let Φ be a finite group and let ϵ:Z[Φ]→Z be the augmentation homomorphism. For nZ we take n:ZZ to be the homomorphism xnx. The eponymous Swan module \((\mathcal{I}, n)\) is defined as \((\mathcal{I}, n) = \varprojlim(\epsilon, \mathbf{n})\). From the commutative diagram
$$\begin{array}{clccclc}0 \rightarrow &\mathcal{I} & \longrightarrow &(\mathcal{I}, n)&\stackrel{\epsilon}{\longrightarrow} & \mathbf{Z}&\rightarrow 0\\&\downarrow\scriptstyle\mathrm{Id}&&\cap&&\downarrow\scriptstyle \mathbf{n}&\\0 \rightarrow &\mathcal{I}& \longrightarrow &\mathbf{Z}[\varPhi]&\stackrel{\epsilon}{\longrightarrow} &\mathbf{Z}&\rightarrow 0\end{array}$$
we see that \((\mathcal{I}, n)\) imbeds in Z[Φ] as a Z-sublattice of index n. The main properties of the \((\mathcal{I}, n)\) were established by Swan (Ann. Math. 72:267–291, 1960). Indeed, this was the original context for the projectivity criterion of Chap.  5; Swan showed that \((\mathcal{I}, n)\) is projective over Z[Φ] if and only if n is a unit mod |Φ|. This and other aspects generalize in a manner which we now proceed to describe.

Keywords

Swan Modules Nontrivial Finite Group Augmentation Homomorphism Syzygies Projectivity 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

  1. 25.
    Edwards, T.M.: Algebraic 2-complexes over certain infinite abelian groups. Ph.D. Thesis, University College London (2006) Google Scholar
  2. 68.
    MacLane, S.: Homology. Springer, Berlin (1963) zbMATHGoogle Scholar
  3. 91.
    Swan, R.G.: Periodic resolutions for finite groups. Ann. Math. 72, 267–291 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 94.
    Swan, R.G.: Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

Personalised recommendations