The Homotopy Theory of Projective Modules

Serre [8; 9] has established the rudiments of a dictionary for translating the language of projective modules into that of vector bundles. With this point of departure we have attempted to adapt some of the results and methods of homotopy theory to certain purely arithmetic and even noncommutative settings. Detailed proofs of our results will appear elsewhere. The authors are very grateful to Albrecht Dold who has educated them on the relevant parts of bundle theory. We first recall briefly the topological setting which our theorems parody. [X, Y] denotes the homotopy classes of maps of X to F. If ƒ: A—>B induces homotopy isomorphisms in dimensions Sr and if X is a finite CW complex of dimension ^ w , then one shows easily (see [10, Appendix]) that [X, A]—>[X, B] is surjective for r^m and injective for r>m. As special cases we have I. [X, Bow]—*[X, Bo(r+i)] is surjective torrent and injective for r>m\ and II . [X, 0(r)]—>[X, 0 (r + 1)] is surjective for r^m + 1 and injective for r>m + l. Here Bo(r) is the classifying space of the orthogonal group 0(r) . The classification theorem [ l l , §19] says that [X, B0(r)] represents the functor, [(equivalence classes of) real r-plane bundles over X ] , and then the map in I is obtained by adding a trivial line bundle. Moreover, the argument in [ l l , §18] shows that [X, O(r)]/wo(0(r)) represents the functor [real r-plane bundles over SX] , where SX is the (reduced) suspension of X. Atiyah and Hirzebruch [l ] define a functor K*(X) = K\X) ®K(X) as follows: K°(X) is the Grothendieck ring of vector bundles over X. Choosing a base point gives K° an augmentation, and X(X) is then the augmentation ideal of K°(SX). Thus I and II above define a kind of stable range for K° and K respectively. Now let R be a commutative ring, X its spectrum of maximal ideals, and À an i^-algebra finitely generated as an jR-module. We require only that X be what might be called a "Zariski complex," i.e. that each closed set be a finite union of irreducible closed sets (see [4, §3]). I f P i s a A-module and # G X then Px is the localization of P at x; we say /-rank P^r if Px contains a A^-free direct summand of rank r for all x. Note that P is required to be neither projective nor