Weight filtrations on log crystalline cohomologies of families of open smooth varieties / Yukiyoshi Nakkajima, Atsushi Shiho

... The first chapter is of independent interest and contains preliminaries on filtered derived categories and (a diagram of) topoi needed in the course of the book. The second chapter (half of the book) discusses the many technical details needed to reach the two main results we have only very roughly described above as well as many compatibilities (functoriality, cohomology with compact support, Poincaré duality, degeneration of spectral sequences, etc.) and relations with previous works. The last chapter is mostly an announcement of results about the same kind of questions (weight structures on cohomology of p-adic varieties) using not crystalline techniques but rigid analytic techniques (divided power envelopes are replaced by tubular neighborhoods, etc.). Assuming a (necessary) basic knowledge of the complex theory, the introduction of the book contains (in (0.0.0.22)) a very helpful table showing what kind of crystalline objects correspond to what kind of complex objects. (MathSciNet)