On the de Rham-Witt complex in mixed characteristic (with 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 Bloch-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*(Ak;p) → TR*(A;p) → TR*(A|AK;p) → ... ,
which is similar to the localization sequence in K-theory. Assuming that K contains the pvth roots of unity, we have also evaluated the groups TR*(V|K;p,Z/pv) in terms of the de Rham-Witt complex of V with log poles at the closed point. We generalize this calculation to a smooth V-algebra A.

Lars Hesselholt <larsh@math.nagoya-u.ac.jp>