\(p\)-adic periods and derived de Rham cohomology (Q2892815)

The paper under review gives a new proof of Fontaine's de Rham conjecture. B. Bhatt has in the meanwhile extended this to the crystalline conjecture. The problem is to construct (for schemes over \(p\)-adic field) a map from de Rham cohomology to \(p\)-adic étale cohomology. The main new tool is the \(h\)-topology which can be used to compute the étale side. Using this topology, the result of \textit{A. de Jong} [Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)] allows to restrict to semistable models, and even to repeated fibrations by curves. For these higher de Rham cohomology modulo \(p^n\) can be killed by global \(h\)-coverings (Poincaré lemma). For example, for curves one uses pullback of \(p\)-powers on the generalized Jacobian. This allows to define the map. That it induces an isomorphism follows from the usual argument: It respects cup-products and characteristic classes.NEWLINENEWLINE The execution of this program requires heavy machinery from homological algebra. The role of the \(h\)-topology has some similarity to that of the syntomic topology in \textit{T. Tsuji}'s [Invent. Math. 137, No. 2, 233--411 (1999; Zbl 0945.14008)] proof.