Generalized Swan Modules

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


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.


Swan Modules Nontrivial Finite Group Augmentation Homomorphism Syzygies Projectivity Criterion 
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© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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