### 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>