CONTINUITY OF K-THEORY : AN EXAMPLE IN EQUAL CHARACTERISTICS

If k is a perfect field of characteristic p > 0, we show that the Quillen K-groups Ki(k[[t]]) are uniquely p-divisible for i = 2, 3. In fact, the Milnor K-groups KM n (k((t))) are uniquely p-divisible for all n > 1. This implies that K(A)→ holim←− nK(A/m) is 4-connected after profinite completion for A a complete discrete valuation ring with perfect residue field. Let A be a complete discrete valuation ring with maximal ideal m. Let K(A) = holim ←− n K(A/m) We say that K-theory is continuous (at A) if it commutes with the (inverse) limit, in the sense that K(A)̂→ Ktop(A)̂ is an equivalence, where X → X̂denotes profinite completion. This question of continuity has acquired new relevance since the fibers of K(A/mn)̂→ K(A/m)̂ are now better understood, and have been shown by McCarthy [Mc] to agree with the corresponding fibers in topological cyclic homology. Hence we are in a position where we sometimes can calculate K(A). One situation where we have an affirmative answer is the theorem of Suslin and Panin [Su], [P], which says that if A is a Henselian discrete valuation ring with maximal ideal m, then K(A)̂̀ → holim ←− n K(A/mn)̂̀ is an equivalence for all primes ` different from the characteristic of (the field of fractions of) A. So, if A is of characteristic zero, then K-theory is continuous at A. This theorem was used critically in Bökstedt and Madsen’s calculation [BM] of the K-theory of the p-adic integers in order to get the correspondence with topological cyclic homology (here the situation was a bit special, as a similar statement holds for TC). Received by the editors October 17, 1996. 1991 Mathematics Subject Classification. Primary 11S70; Secondary 13J05, 19D45, 19D50.