International Congress of Mathematicians

Algebraic ÜT-theory and Trace Invariants 421We briefly outline the steps in the proof: We proved in [15] that the sequence0 -• if g (fc,Z/p») -• TR g (fc;p,Z/p») i-4 TR g (fc;p,Z/p») -• 0is exact. This uses [4, 16, 18]. Given this, the theorem by McCarthy [28] thatfor nilpotent extensions, relative ÜT-theory and relative topological cyclic homologyagree, and the continuity results of Suslin [32] for ÜT-theory and Madsen and theauthor [21] for TR show that also the sequence0 -+ K q (V,Z/p v ) -+ TR'(F;p,Z/p») i=4 TR'(F;p,Z/p») -+ 0is exact. Theorem 3.1 follows by comparing the localization sequence of Quillen [30]• • • -+ K q (k,Z/p») A K q (V,Z/p») ^ K q (K,Z/p») -+ • • •to the corresponding sequence by Madsen and the author [20]• • • -+ TR^(fc;p,Z/p») 4 TR n q(V ;p,1/p v ) ^ TR g (F|if;p,Z/p») -+••..Again, the assumption in the statement of theorem 3.1 that the residue field k beseparably closed is not essential. The general statement will be given below. It isalso not necessary for theorem 3.1 to assume that V be of geometric type.4. The Tate spectral sequenceIf G is a finite group and X a G-space, it is usually not possible to evaluatethe groups 7r»(X G ) from knowledge of the G-modules 7r»(X). At first glance, thisis the problem that one faces in evaluating the groupsTR n q(C ] p) =Tx q (TEE(C) c^).However, the mapping fiber of the structure map TR"(C;p) —^ TR" _1 (C;p), it turnsout, is given by the Borei construction ML (C p —i, THH(C)) whose homotopy groupsare the abutment of a (first quadrant) spectral sequenceElt = H s (C p „-i,TEE t (Cj)=> 7r J+t H.(C p »-i,THH(C)).This suggests that the groups TR q (C;p) can be evaluated inductively starting fromthe case n = 1. However, it is generally difficult to carry out the induction step.In addition, the absence of a multiplicative structure makes the spectral sequenceabove difficult to solve. The main vehicle to overcome these problems, first employedby Bökstedt-Madsen in [7], is the following diagram of fiber sequencesH.(C p »-i,THH(C)) >TR n (C;p) »TR^^p)rfH. (C p »-i, THH(C)) • B (C p »-i, THH(C)) • É(C p »-i, THH(C))