The Calculus of Corners and Squares
- 945 Downloads
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.
KeywordsFibre Square Karoubi Finitely Generated Ring Homomorphism Patch Conditions