Faltings' main p-adic comparison theorems for non-smooth schemes

To understand the p-adic étale cohomology of a proper smooth variety over a p-adic field, Faltings compared it to the cohomology of his ringed topos, by the so-called Faltings' main p-adic comparison theorem, and then deduced various comparisons with p-adic cohomologies originating from differential forms. In this article, we generalize the former to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of (without any smoothness assumption) for torsion abelian étale sheaves (not necessarily finite locally constant). Our proof relies on our cohomological descent for Faltings' ringed topos, using a variant of de Jong's alteration theorem for morphisms of schemes due to Gabber-Illusie-Temkin to reduce to the relative case of proper log-smooth morphisms of log-smooth schemes over a complete discrete valuation ring proved by Abbes-Gros. A by-product of our cohomological descent is a new construction of Faltings' comparison morphism, which does not use Achinger's results on -schemes.