Rings with Stably Free Cancellation

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


The ring Λ has the stably free cancellation property (abbreviated to SFC) when for any Λ-module S and any positive integers m, k
$$S \oplus \varLambda^m \cong \varLambda^{m+k} \quad \Longrightarrow\quad S \cong \varLambda^k.$$
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
$$A[t, t^{-1}]\ \mbox{is projective free}\quad \Longleftrightarrow\quad \widetilde {K_0}(A) = 0.$$


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.


  1. 2.
    Bass, H.: Algebraic K-Theory. Benjamin, Elmsford (1968) zbMATHGoogle Scholar
  2. 4.
    Bass, H., Heller, A., Swan, R.G.: The Whitehead group of a polynomial extension. Publ. Math. IHÉS 22, 61–79 (1964) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 11.
    Bourbaki, N.: Commutative Algebra. Hermann/Addison Wesley, Paris/Reading (1972) zbMATHGoogle Scholar
  4. 19.
    Dicks, W.: Free algebras over Bezout domains are Sylvester domains. J. Pure Appl. Algebra 27, 15–28 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 20.
    Dicks, W., Sontag, E.D.: Sylvester domains. J. Pure Appl. Algebra 13, 243–275 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 66.
    Lam, T.Y.: A First Course in Noncommutative Rings. Springer, Berlin (2001) zbMATHCrossRefGoogle Scholar
  7. 67.
    Lam, T.Y.: Serre’s Problem on Projective Modules. Springer, Berlin (2006) CrossRefGoogle Scholar
  8. 76.
    Ojanguran, M., Sridharan, R.: Cancellation of Azumaya algebras. J. Algebra 18, 501–505 (1971) MathSciNetCrossRefGoogle Scholar
  9. 79.
    Parimala, S., Sridharan, R.: Projective modules over polynomial rings over division rings. J. Math. Kyoto Univ. 15, 129–148 (1975) MathSciNetzbMATHGoogle Scholar
  10. 81.
    Quillen, D.G.: Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 85.
    Sheshadri, C.S.: Triviality of vector bundles over the affine space K 2. Proc. Natl. Acad. Sci. USA 44, 456–458 (1958) CrossRefGoogle Scholar
  12. 87.
    Steinitz, E.: Rechteckige Systeme und Moduln in algebraischen Zahlkörpen. I. Math. Ann. 71, 328–354 (1911) MathSciNetCrossRefGoogle Scholar
  13. 88.
    Steinitz, E.: Rechteckige Systeme und Moduln in algebraischen Zahlkörpen. II. Math. Ann. 72, 297–345 (1912) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 90.
    Suslin, A.A.: Projective modules over polynomial rings are free. Sov. Math. Dokl. 17, 1160–1164 (1976) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

Personalised recommendations