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.
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References
Karoubi, M.: Localization des formes quadratiques. I. Ann. Sci. Éc. Norm. Super. (4) 7, 359–404 (1974)
Milnor, J.: Introduction to Algebraic K-Theory. Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)
Mitchell, B.: Theory of Categories. Academic Press, San Diego (1965)
Swan, R.G.: Projective modules over binary polyhedral groups. J. Reine Angew. Math. 342, 66–172 (1983)
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Johnson, F.E.A. (2012). The Calculus of Corners and Squares. In: Syzygies and Homotopy Theory. Algebra and Applications, vol 17. Springer, London. https://doi.org/10.1007/978-1-4471-2294-4_3
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