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• W2506318178 abstract "1. In the beginning. Like the topological Z-theory of Atiyah and Hirzebruch [2], algebraic ^-theory started from Grothendieck's proof of his generalized Riemann-Roch theorem in 1957 [12]. There first appears the Grothendieck group K(X) of vector bundles on a scheme X, which Grothendieck intended as a vehicle for the global intersection theory later developed in S.G.A. 6 [21]. Atiyah and Hirzebruch meanwhile took X to be a finite CW complex and defined K~(X) = K(SX) for n ^ 1, where S denotes suspension. If, in Grothendieck's case, X ?= spec (A) is affine, or, in the case of Atiyah-Hirzebruch, A is the ring C(X) of continuous functions on X, then the category of vector bundles on Zis equivalent to the category 0>(A) of finitely generated projective ^-modules (cf. [37] and [43]). Whence a definition of K(A) (or K0(A))9 as the Grothendieck group of &(A), which makes sense for an arbitrary ring A. This K0(A) is, if nothing more, a useful tool for investigating the structure of projective ^-modules P. Consider, for example, a problem that has greatly influenced our subject, posed by Serre in 1955 [37]: Is P free if A = F[t9 •••, t„]9 a polynomial ring over a field Fl Seshadri affirmed the first nontrivial case, n = 2 [39]. In 1958 Serre proved the following fundamental results [38] : (1) K0(A) -» K0(A[t]) is an isomorphism for regular rings A. 1" @default.
• W2506318178 created "2016-08-23" @default.
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• W2506318178 date "1975-01-01" @default.
• W2506318178 modified "2023-09-24" @default.
• W2506318178 title "Algebraic ^-Theory: A Historical Survey" @default.
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