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

... &rarr TR_{*}(A_{k};p) &rarr TR_{*}(A;p) &rarr TR_{*}(A|A_{K};p) &rarr ... ,

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 generalize this calculation to a smooth V-algebra A.