#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 833-890

** Takeshi Tsuji **
On the Maximal Unramified Quotients of
$p$-Adic Étale Cohomology Groups
and Logarithmic Hodge--Witt Sheaves

Let $O_K$ be a complete discrete valuation ring of mixed characteristic
$(0,p)$ with perfect residue field. From the semi-stable conjecture ($C_{st}$)
and the theory of slopes, we obtain isomorphisms between the maximal unramified
quotients of certain Tate twists of $p$-adic étale cohomology groups
and the cohomology groups of logarithmic Hodge-Witt sheaves for a proper
semi-stable scheme over $O_K$. The object of this paper is to show that
these isomorphisms are compatible with the symbol maps to the $p$-adic
vanishing cycles and the logarithmic Hodge-Witt sheaves, and that they
are compatible with the integral structures under certain restrictions.
We also treats an open case and a proof of $C_{st}$ in such a
case is given for that purpose. The results are used in the work of U. Jannsen
and S. Saito in this volume.

2000 Mathematics Subject Classification: 14F30, 14F20

Keywords and Phrases: p-adic étale cohomology, p-adic vanishing cycles, logarithmic Hodge-Witt
sheaf, p-adic Hodge theory

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