On the de Rham-Witt complex in mixed characteristic, by Lars Hesselholt and Ib Madsen

The purpose of this paper is twofold. Firstly, it gives a thorough treatment of the generalization to Z_{(p)}-algebras (with p odd) of the de Rham-Witt complex of Deligne-Illusie. We define this (pro-)complex as the universal example of an algebraic structure, which we call a Witt complex. Another example is the pro-complex TR_*(A;p) given by the homotopy groups of fixed sets of topological Hochschild homology. Among other things, we give an explicit formula for the de Rham-Witt complex of A[x] in terms of that of A. The same formula expresses TR_*(A[x];p) in terms of TR_*(A;p).

Secondly, let A be a smooth algebra over a discrete valuation ring V of mixed characteristic (0,p) with quotient field K and perfect residue field k. We have previously constructed a long-exact sequence of Witt complexes

     ... -> TR_*(A_k;p) -> TR_*(A;p) -> TR_*(A|A_K;p) -> ... ,
which is similar to the localization sequence in K-theory. Assuming that K contains the p^v'th roots of unity, we have also evaluated the groups TR_*(V|K;p,Z/p^v) in terms of the de Rham-Witt complex of V with log poles at the closed point. We now generalize this calculation to a smooth V-algebra A.

Lars Hesselholt <larsh@math.mit.edu>
Ib Madsen <imadsen@imf.au.dk>