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.mit.edu>