International Congress of Mathematicians

ICM 2002 • Vol. II • 415-125Algebraic Jf-theory and Trace InvariantsLars Hesselholt*(Dedicated to lb Madsen on his sixtieth birthday)AbstractThe cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject ofBökstedt's lecture at the congress in Kyoto, is a map of pro-abelian groupsK-fiA) ^.TR;(A;p)from Quillen's algebraic A"-theory to a topological refinement of Connes' cyclichomology. Over the last decade, our understanding of the target and itsrelation to A"-theory has been significantly advanced. This and possible futuredevelopment is the topic of my lecture.The cyclotomic trace takes values in the subset fixed by an operator Fcalled the Frobenius. It is known that the induced mapK*(A,Z/p v ) -^ TR;(A;P,Z/P V ) F=1is an isomorphism, for instance, if A is a regular local F p -algebra, or if Ais a henselian discrete valuation ring of mixed characteristic (0,p) with aseparably closed residue field. It is possible to evaluate A"-theory by meansof the cyclotomic trace for a wider class of rings, but the precise connectionbecomes slightly more complicated to spell out.The pro-abelian groups TR*(A;p) are typically very large. But they comeequipped with a number of operators, and the combined algebraic structure isquite rigid. There is a universal example of this structure — the de Rham-Wittcomplex — which was first considered by Bloch-Deligne-Illusie in connectionwith Grothendieck's crystalline cohomology. In general, the canonical mapW.Q q A^TR- q (A-p)is an isomorphism, if q < 1, and the higher groups, too, can often be expressedin terms of the de Rham-Witt groups. This is true, for example, if A is aregular F p -algebra, or if A is a smooth algebra over the ring of integers in alocal number field. The calculation in the latter case verifies the Lichtenbaum-Quillen conjecture for focal number fields, or more generally, for henseliandiscrete valuation fields of geometric type.2000 Mathematics Subject Classification: 19D45, 19D50, 19D55, 11S70,14F30, 55P91.* Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail:larsh@math.mit.edu