Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

In this chapter, we give an outline of the construction of the weight filtration and the calculation of the slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic p < 0. They are constructed and calculated by using de Jong’s alteration theorem, Tsuzuki’s proper cohomological descent for rigid cohomology, the comparison theorem between log crystalline cohomology and rigid cohomology by the second-named author and the results in the previous chapters in this book. It is analogous to the fact that the mixed Hodge structure on the cohomology of a separated scheme of finite type over ℂ is constructed from the mixed Hodge complex on open smooth schemes over ℂ via the technique of proper hypercovering. The detailed proof of the results in this chapter is given by another book [70] by the first-named author.