By Robert Boltje, G.-Martin Cram, V. P. Snaith (auth.), P. G. Goerss, J. F. Jardine (eds.)

A NATO complicated learn Institute entitled "Algebraic K-theory and Algebraic Topology" was once held at Chateau Lake Louise, Lake Louise, Alberta, Canada from December 12 to December sixteen of 1991. This publication is the amount of court cases for this assembly. The papers that seem listed here are consultant of lots of the lectures that got on the convention, and consequently current a "snapshot" of the nation ofthe K-theoretic artwork on the finish of 1991. The underlying target of the assembly was once to debate contemporary paintings concerning the Lichtenbaum-Quillen complicated of conjectures, fro~ either the algebraic and topological issues of view. The papers during this quantity take care of a variety of themes, together with motivic cohomology theories, cyclic homology, intersection homology, better category box idea, and the previous telescope conjecture. This assembly was once together funded through gives you from NATO and the nationwide technology Foun dation within the usa. i want to take this chance to thank those organisations for his or her help. i'd additionally wish to thank the opposite participants of the organizing com mittee, specifically Paul Goerss, Bruno Kahn and Chuck Weibel, for his or her assist in making the convention winning. This used to be the second one NATO complicated examine Institute to be held during this venue; the 1st was once in 1987. The luck of either meetings owes a lot to the professionalism and helpfulness of the management and employees of castle Lake Louise.

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References (1] R. Boltje : Canonical and explicit Brauer induction in the character ring of a finite group and a generalisation for Mackey functors; Augsburg Univ. thesis (1989) . (2] R. 181-182 (1990) 31-59. (3] R. Boltje, V. Snaith and P. Symonds : Algebraicisation af Explicit Brauer Induction; J. Alg. (2) 148 (1992) 504-527. (4] J-L. Brylinski: Theorie du corps de classes de Kato et revetements abeliens de surfaces; Ann. Inst. Fourier 33 (1983) 23-38. (5] 0. Hyodo : Wild ramification in the imperfect residue field case; Adv.

Hftd(X,p,~d)-+ Hd-t(X,Jid+t(p,~d))-+ 0. Combining Poincare duality over F and arithmetic duality for Galois cohomology ofF ( cf. [CT /S/SJ Lemme 5 p. 790) yields an isomorphism Hitd(X,p,~d) ~ Hom(Hlt{X,Z/n),Z/n) ~ 1rfb(X)/n, and the main theorem of unramified class field theory for smooth projective varieties over a finite field precisely says that the composite map is an isomorphism ((K/S1] Theorem 1, (CT/S/S] Theoreme 5 p. 792, (CT/R2]). We thus conclude : Hd-t(X, Jid+t(p,~d)) = O and by going over to the direct limit of powers of a fixed prime l : which proves the degree d- 1 part of Theorems A and B.

1. - Let U be as above. Let i E N and j E Z be integers. Then the groups Ht 1 (U, Q,jZ1(j)) vanish fori 2': d + 2, and they vanish for i = d + 1 and d "/:- 2j, 2j- 1. ( 2 ) Proof: Let n be a positive integer and let j be an arbitrary integer. 1 (U,J1~j) = 0 for q 2': d + 1. 1. The present version was suggested by N. Suwa. 2 in the cases d = 3 and d > 3. 52 is concentrated in the range 0:::; p:::; 1 (since cd(F)=1) and q:::; d. 1), hence also Hjt(U, QtfZI(j)) = 0 fori~ d + 2. This gives the first part of the lemma.