By Barry Dayton, Charles Weibel (auth.), J. F. Jardine, V. P. Snaith (eds.)

A NATO complex examine Institute entitled "Algebraic K-theory: Connections with Geometry and Topology" was once held on the Chateau Lake Louise, Lake Louise, Alberta, Canada from December 7 to December eleven of 1987. This assembly used to be together supported through NATO and the ordinary Sciences and Engineering examine Council of Canada, and used to be subsidized partly via the Canadian Mathematical Society. This e-book is the quantity of court cases for that assembly. Algebraic K-theory is largely the examine of homotopy invariants bobbing up from earrings and their linked matrix teams. extra importantly maybe, the topic has turn into principal to the learn of the connection among Topology, Algebraic Geometry and quantity thought. It attracts on all of those fields as a subject matter in its personal correct, however it serves in addition to an efficient translator for the applying of techniques from one box in one other. The papers during this quantity are consultant of the present kingdom of the topic. they're, for the main half, study papers that are essentially of curiosity to researchers within the box and to these meaning to be such. there's a part on difficulties during this quantity which will be of specific curiosity to scholars; it includes a dialogue of the issues from Gersten's recognized record of 1973, in addition to a brief record of recent problems.

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134] ...... ) ~. This gives a GL . n denote of global units acts on SP (X)/A* n vector bundles P on is isomorphic X with trivial. In particular. if P is a rank n vector bundle with trivial B. DAYTON AND C. WEIBEL determinant. then open cover {U} P 25 comes from of X such that 0U-modules together via matrices Proof: For each a € A*. I. 1}. If {guy} -1 the cocycle {ag~ } let P may be obtained by patching free f or a denote the diagonal n-by-n matrix Because ~. GL. ~. let denote {guy} and are they define the same underlying n vector bundle on X.

Where in SP (X) SP (X)/A*. is exactly the orbit set Example: Pic(X} A is the ring of continuous S2 with values in m. By [St. 6J. Z. On the other hand, by [St. 2]. the on SP2 {X) sends n to -n and the image of 26 ON THE NATURALITI OF SP2 (X} is m in P2 (X} ~ PIC. SKO AND SKI mum. For our applications. l. When T: X ~Y is a finite map. 3. n Lemma: Let T: X ~ Y be a fini te map of schemes and vector bundle on X. neighborhood Proof: U of Then for every point y such that Y finite A-module. Let -1 it is a free = spec(A} and y of is free on T P a rank Y there is a -1 (U).

T. Algebra 67, 210-229, 1980. : 'Descent for the K-theory of polynomial rings', Math. Zeit. 191, 405-415, 1986. : 'Localization of the K-theory of polynomial extensions', Math. Ann. 244, 33-53, 1979. : 'Module structures in the K-theory of graded rings', J. Algebra 105, 465-483, 1987. : 'Mayer-Vietoris sequences and module structures on NK*', Lecture Notes in Math. 854, Springer-Verlag, 1981. : 'Fibre spaces in algebraic geometry', [1949c], Oevres Scientifiques, Vol. I, Springer-Verlag, 1979. S.