Cohomological Descent for Faltings' $p$-adic Hodge Theory and Applications

Abstract: Faltings' approach in $p$-adic Hodge theory can be schematically divided into two main steps: firstly, a local reduction of the computation of the $p$-adic \'etale cohomology of a smooth variety over a $p$-adic local field to a Galois cohomology computation and then, the establishment of a link between the latter and differential forms. These relations are organized through Faltings ringed topos whose definition relies on the choice of an integral model of the variety, and whose good properties depend on the (logarithmic) smoothness of this model. Scholze's generalization for rigid analytic varieties has the advantage of depending only on the variety (i.e. the generic fibre). Inspired by Deligne's approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings' approach to any integral model, i.e. without any smoothness assumption. An essential ingredient of our proof is a descent result of perfectoid algebras in the arc-topology due to Bhatt and Scholze. As an application of our cohomological descent, using a variant of de Jong's alteration theorem for morphisms of schemes due to Gabber-Illusie-Temkin, we generalize Faltings' main $p$-adic comparison theorem to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of $\mathbb{Z}_p$ (without any assumption of smoothness) for torsion \'etale sheaves (not necessarily finite locally constant). As a second application, we prove a local version of the relative Hodge-Tate filtration as a consequence of the global version constructed by Abbes-Gros.