International Congress of Mathematicians

418 L. Hesselholt(i) a pro-log differential graded ring (ET, M E) and a map of pro-log rings(ii) a map of pro-log graded ringssuch that ÀF = FA and such thatA: (W.(A),M)->(E?,M E );F: ET -> ET_i,Fdlog„ a = dlog„_i a, for all a £ M,FdA[a]„ = A[a]^Z 1 dA[a]„_i, for all a £ A;(iii) a map of pro-graded modules over the pro-graded ring ET,V: F*ET_i -+ET,such that XV = VX, FV = p and FdV = d.A map of Witt complexes over (A, M) is a map of pro-log differential gradedrings which commutes with the maps A, F and V. Standard category theory showsthat there exists a universal Witt complex over (A,M). This, by defintion, is thede Rham-Witt complex W. W A M) . (The canonical maps W.(A) —t W. 0,9 A M) and01 A M) —t Wi 01 i M) are isomorphisms, so the construction really does combinedifferential forms and Witt vectors.) We lift the logarithmic derivative to a mapiff(if)^I^Q» l/jM)which to the symbol {ai,...,a q } associates dlog n cti...dlog n a q . This trace mapbetter captures the Milnor A'-groups. Indeed, the following result was obtained incollaboration with Thomas Geisser [14]:Theorem 2.1 Suppose that p p » C K and thatk is separably closed. Then the tracemap induces an isomorphism of pro-abelian groupsKM(K)/p«^(WMl ViM) /p«) F=1 .To prove this, we first show that W n ii q , v M) /p has a (non-canonical) k-vectorspace structure and find an explicit basis. The dimension isdim fc {W n ü\ ViM) lp) =n-le Y,Pwhere \k : k p \ = p r and e the ramification index of Ä". It is not difficult to see thatthis is an upper bound for the dimension. The proof that it is also a lower bound ismore involved and uses a formula for the de Rham-Witt complex of a polynomialextension by Madsen and the author [19]. We then evaluate the kernel of 1 — Fand compare with the calculation of Ki M (K)/p by Kato [24, 4]. The assumption