International Congress of Mathematicians

422 L. Hesselholttogether with a multiplicative (upper half-plane) spectral sequenceE 2 }t = H- s (C pn -i,TEE t (Cj)^ 7r,+tlî(C p ,.-i,THH(C))starting from the Tate cohomology of the (trivial) C p »-i-module TEE t .(C). Thelower fiber sequence is the Tate sequence; see Greenlees and May [17] or [20]. Infavorable cases, the maps F and F induce isomorphisms of homotopy groups innon-negative degrees. Indeed, this is true in the case at hand (if k is perfect). Thedifferential structure of the spectral sequenceElt = H- s (C pn^,TEE t (y\K,1/pj)^ 7r s+t (H(G pn - 1 ,THH(F|if)),Z/p)was determined in collaboration with lb Madsen [20] in the case where the residuefield fcis perfect. This is the main calculational result of the work reported here. Thefollowing result, for perfect fc, is a rather immediate consequence. The extension tonon-perfect fc is given in [19].Theorem 4.1 Suppose that ß p v c K. Then the canonical map is an isomorphismof pro-abelian groupsW. 0* (VM) ® z S Z/P .(AV) ^ TR;(F|if;p,Z/p»).We can now state the general version of theorem 3.1 which does not requirethat the residue field fc be separably closed. The second tensor factor on the lefthand side in the statement of theorem 4.1 is the symmetric algebra on the Z/p»-module ß p v, which is free of rank one. Spelling out the statement for the group indegree q, we get an isomorphism of pro-abelian groups0 W. 0\- 2^ ® ßp ^ TR'(F|if ;p, Z/p»).In the case of a separably closed residue field, theorem 3.1 idenfies the Frobeniusfixed set of the common pro-abelian group with K q (K,Z/p v ). In the general case,one has instead a short-exact sequence0 _• 0 (W. 0^M2s® np) F=1 -+ K q (K, Z/p») -+ 0 (W. 0\- 2^ ® ßp) F -U 0valid for all integers q. (There is a similar sequence for the topological cyclic homologygroup TCg(V\K;p,Z/p v ) [20] which includes the summand "s = 0" on theleft.) Comparing with the general version of theorem 2.1, we obtain the followingresult promised earlier [20, 14].Theorem 4.2 Suppose that ß p v c K. Then the canonical mapis an isomorphism.if» M (if) ®z S Z/P .(AV) ^ K*(K,Z/p v )