Abstract
In this chapter we begin the detailed study of the SFC property for group rings of the form Z[F n ×Φ] where F n is the free group of rank n≥1 and Φ is finite. In the first instance we consider the rings Z[F n ×C m ] where C m is the cyclic group of order m.
As Z[Φ] is a retract of Z[F n ×Φ] it follows that a prior condition for Z[F n ×Φ] to have stably free cancellation is that Z[Φ] should also have this property. The question of stably free cancellation for Z[Φ] and related rings has been studied extensively by Swan (Ann. Math. 76:55–61, 1962; K-Theory of finite groups and order (notes by E.G. Evans), Lecture notes in mathematics, vol. 149, Springer, Berlin, 1970; J. Reine Angew. Math. 342:66–172, 1983) and Jacobinski (Acta Math. 121:1–29, 1968), building upon earlier work of Eichler (Math. Z. 43:102–109, 1938).
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Johnson, F.E.A. (2012). Group Rings of Cyclic Groups. In: Syzygies and Homotopy Theory. Algebra and Applications, vol 17. Springer, London. https://doi.org/10.1007/978-1-4471-2294-4_10
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DOI: https://doi.org/10.1007/978-1-4471-2294-4_10
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