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Rings with Stably Free Cancellation

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

Abstract

The ring Λ has the stably free cancellation property (abbreviated to SFC) when for any Λ-module S and any positive integers m, k
$$S \oplus \varLambda^m \cong \varLambda^{m+k} \quad \Longrightarrow\quad S \cong \varLambda^k.$$
(SFC)
As noted in Chap.  1, the theorem of Gabel ensures that every stably free Λ-module is free precisely when Λ has the SFC property; thus we may concentrate our discussion on finitely generated modules.
There is a stronger property than stably free cancellation; the ring Λ is projective free when every finitely generated projective Λ-module is free. As a fundamental notion projective freeness is inconveniently restrictive; even so, it is a property too useful to be ignored and we make use of it at a number of places. For rings of Laurent polynomials the two notions are connected via the theorem of Grothendieck (Bass et al. in Publ. Math. Inst. Ht. Etudes Sci. 22:61–79, 1964) that \(\widetilde{K_{0}}(A[t, t^{-1}]) \cong\widetilde{K_{0}}(A)\) if A is a coherent ring of finite global dimension. Under this hypothesis, if A[t,t −1] has property SFC then
$$A[t, t^{-1}]\ \mbox{is projective free}\quad \Longleftrightarrow\quad \widetilde {K_0}(A) = 0.$$

Keywords

Finite Global Dimension Coherent Ring Laurent Polynomial Free Projection Free Group Algebras 
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.

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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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