International Congress of Mathematicians

416 L. Hesselholt1. Algebraic üC-theoryThe algebraic if-theory of Quillen [30], inherently, is a multiplicative theory.Trace invariants allow the study of this theory by embedding it in an additivetheory. It is possible, by this approach, to evaluate the if-theory (with coefficients)of henselian discrete valuation fields of mixed characteristic. We first recall theexpected value of the if-groups of a field k.The groups K*(k) form a connected anti-commutative graded ring, there is acanonical isomorphism I: k* ^y Kfik), and £(x) • 1(1 — x) = 0. One defines theMilnor if-groups K^(k) to be the universal example of this algebraic structure [29].The canonical map K^(k) —¥ K q (k) is an isomorphism, if q < 2. Yet us now fix theattention on the if-groups with finite coefficients. (The rational if-groups, while ofgreat interest, are of a rather different nature [11, 12].) The groups if» (k,Z/m) forman anti-commutative graded Z/m-algebra, at least if v 2 (m) ^ 1,2 and vz(m) ^ 1.And if ß m C k, there is a canonical liftingyb /ßm-—-—>Ki(k),K 2 (k,Z/m)which to a primitive roth root of unity ( associates the Bott element 6ç. Hence, inthis case, there is an additional map of graded rings Sz/ m (p m ) —* K*(k,Z/m). TheBeilinson-Licthenbaum conjectures predict that the combined mapiff (k) 8zS z/ro (fa) -+K.(k,Z/m)be an isomorphism of graded rings [1, 26]. The case TO = 2 V follows from thecelebrated proof of the Milnor conjecture by Voevodsky [34]. We here considerthe case of a henselian discrete valuation field of mixed characteristic (0,p) withp odd and m = p v [20, 14]. The groups K^(k)/m typically are non-zero in onlyfinitely many degrees. Hence, above this range, the groups K*(k,Z/m) are twoperiodic.All rings (resp. graded rings, resp. monoids) considered in this paperare assumed commutative (resp. anti-commutative, resp. commutative) and unitalwithout further notice.2. The de Rham-Witt complexLet V be a henselian discrete valuation ring with quotient field K of characteristiczero and residue field k of odd characteristic p. (At this writing, we furtherrequire that V be of geometric type, i.e. that V be the henselian local ring atthe generic point of the special fiber of a smooth scheme over a henselian discretevaluation ring V 0 C V with perfect residue field.) A first example of a trace map isprovided by the logarithmic derivative^MKf(K)^œ {VM)