The absolute and relative de Rham-Witt complexes
We compare the absolute and relative de Rham-Witt complexes considered
by the author and Madsen and by Langer and Zink, which both generalize
the classical de Rham-Witt complex of Bloch, Deligne, and Illusie from
Fp-schemes to Z(p)-schemes. From
this comparison, we derive a Gauss-Manin connection on the crystalline
cohomology of X/Wn(S) for a smooth family X/S. The
comparison result is based on a formula that expresses the absolute de
Rham-Witt complex of a polynomial algebra in a finite number over a
variables in terms of the absolute de Rham-Witt complex of the ring of
coefficients. The formula in the one-variable case was proved in
earlier joint work with Ib Madsen, and the analogous result for the
relative de Rham-Witt complex was proved by Langer and Zink.
Lars Hesselholt
<larsh@math.nagoya-u.ac.jp>