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The Calculus of Corners and Squares

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

Abstract

This chapter is a systematic study of projective modules via the decomposition of rings into fibre squares. The genesis of this approach was geometric, namely ‘Mayer-Vietoris patching’ for vector bundles. Its translation into pure algebra was effected by Milnor in (Introduction to algebraic K-theory, Annals of mathematics studies, vol. 72. Princeton University Press, Princeton, 1971) and subsequently extended, notably by Karoubi (Ann. Sci. Éc. Norm. Super. (4) 7:359–404, 1974). In its original formulation, the method aimed to describe the stable structure of projective modules. However, our treatment is heavily influenced by that of Swan (J. Reine Angew. Math. 342:66–172, 1983) wherein the method is adapted to the dual ‘unstable’ theory; that is, the study of what is lost on stabilization.

Keywords

Fibre Square Karoubi Finitely Generated Ring Homomorphism Patch Conditions 
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.

References

  1. 63.
    Karoubi, M.: Localization des formes quadratiques. I. Ann. Sci. Éc. Norm. Super. (4) 7, 359–404 (1974) MathSciNetzbMATHGoogle Scholar
  2. 73.
    Milnor, J.: Introduction to Algebraic K-Theory. Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971) zbMATHGoogle Scholar
  3. 74.
    Mitchell, B.: Theory of Categories. Academic Press, San Diego (1965) zbMATHGoogle Scholar
  4. 94.
    Swan, R.G.: Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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