Potential semi-stability of \(p\)-adic étale cohomology (Q1601472)

The paper under review shows that for an arbitrary scheme \(X\) of finite type over a \(p\)-adic field \(K\) (completely discretely valued of characteristic zero, with perfect residue-field of characteristic \(p\)) the \(p\)-adic étale cohomology is potentially semistable, that is after a finite extension of \(K\) it is associated (in the sense of Fontaine's functors) to a suitable crystalline object. Previously this had been shown for proper smooth schemes, and the essential idea is to reduce to that case using simplicial techniques, just as it is done in complex Hodge theory. However this makes it necessary to repeat certain local arguments in this context, to first construct a map between étale and crystalline cohomology, and there are also a number of subtleties one has to take care of. One other important ingredient of the proof is of course de Jong's reduction theory which allows to assume semistable reduction.