In this volume, the authors construct a theory of weights on the
log crystalline cohomologies of families of open smooth varieties
in characteristic p>0, by defining and constructing four
filtered complexes. Fundamental properties of these filtered
complexes are proved, in particular the p-adic purity, the
functionality of three filtered complexes, the weight-filtered base
change formula, the weight-filtered Kunneth formula, the
weight-filtered Poincare duality, and the E2-degeneration of p-adic
weight spectral sequences. In addition, the authors state some
theorems on the weight filtration and the slope filtration on the
rigid cohomology of a separated scheme of finite type over a
perfect field of characteristic p>0.
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