Weight filtration and slope filtration on the rigid cohomology of a variety in characteristic \(p>0\) (Q2857646)

From the abstract: ``We construct a theory of weights on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic \(p>0\) by using the log crystalline cohomology of a split proper hypercovering of the scheme. We also calculate the slope filtration on the rigid cohomology by using the cohomology of the log de Rham-Witt complex of the hypercovering.''NEWLINENEWLINEThe introduction of this book begins with an exposition of Grothendieck's philosophy of motives, of its known incarnations in \(\ell\)-adic cohomology, in de Rham cohomology and in Betti cohomology, in particular of Deligne's work on the cohomology of proper smooth \({\mathbb C}\)-schemes with strict normal cossings divisor, phrased in the language of mixed Hodge structures.NEWLINENEWLINEThis motivates the search for the correct analog of Deligne's constructions for varieties over fields of characteristic \(p>0\). The goal here is to provide such an analog in full generality and in a context as broad as possible, with rigid resp. log crystalline cohomology taking over the role of de Rham cohomology.NEWLINENEWLINEMore precisely, to quote from the introduction:NEWLINENEWLINE``The purposes of this book are the following:NEWLINENEWLINE1) To construct the theory of weight filtration on the log crystalline cohomology of a family of simplicial open smooth varieties in characteristic \(p>0\).NEWLINENEWLINE2) To construct the theory of slope filtration with the weight filtration of a split simplicial open smooth variety in characteristic \(p>0\).NEWLINENEWLINE3) To define and study the weight filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic \(p\).NEWLINENEWLINE4) To calculate the slope filtration on the rigid cohomology above.NEWLINENEWLINE1) is a straight generalization of results of the author and \textit{A. Shiho} [Weight filtrations on log crystalline cohomologies of families of open smooth varieties. Berlin: Springer (2008; Zbl 1187.14002)]. Let \(N\) be a nonnegative integer. The key point for 2) is a comparison theorem between the split \(N\)-truncated cosimplicial preweight filtered vanishing cycle zariskian complex and the split \(N\)-truncated cosimplicial preweight-filtered log de Rham Witt complex [...]. The existence of the weight filtration in 3) follows from 1), de Jong's alteration theorem [...], Tsuzuki's proper cohomological descent in rigid cohomology [...] and a generalization of Shiho's comparison theorem [...] between the rigid cohomology of the scheme in 3) and the log crystalline cohomology of a certain proper hypercovering of the scheme [...]. 4) is an immediate consequence of 2) and the generalization of Shiho's comparison theorem.''NEWLINENEWLINEThe book is subdivided in three main parts:NEWLINENEWLINEPart I. Weight filtration on the log crystalline cohomology of a simplicial family of open smooth varieties in characteristic \(p>0\)NEWLINENEWLINEPart II. Weight filtration and slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic \(p>0\)NEWLINENEWLINEPart III. Weight filtrations and slope filtrations on the rigid cohomologies with closed support and with compact support