Abstract
In this chapter we continue the study of stably free cancellation over the integral group rings Z[F n ×Φ] in the case where Φ is the dihedral group of order 2m defined by the presentation
Our main result, first proved in Johnson (Q. J. Math., 2011, doi:10.1093/qmath/har006), is that Z[F n ×D 2p ] has SFC when p is an odd prime. This breaks down for p=2. Although Z[C ∞×D 4] still has SFC (the case n=1) when n≥2 a result of O’Shea shows that Z[F n ×D 4] has infinitely many isomorphically distinct stably free modules of rank 1.
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Notes
- 1.
See Appendix C.
- 2.
When n=1 this can be regarded as saying R[D 2p ] has stably free cancellation where R=Z[t,t −1] is the ring of Laurent polynomials over Z. The corresponding result over the ring Z[t] of genuine polynomials was established by Strouthos using Quillen patching [89].
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Johnson, F.E.A. (2012). Group Rings of Dihedral Groups. In: Syzygies and Homotopy Theory. Algebra and Applications, vol 17. Springer, London. https://doi.org/10.1007/978-1-4471-2294-4_11
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DOI: https://doi.org/10.1007/978-1-4471-2294-4_11
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