Almost étale coverings

This chapter explains Faltings' theory of almost étale extensions, a tool that has become essential in many questions in arithmetic geometry, even beyond p-adic Hodge theory. It begins with a brief historical overview of almost étale extensions, noting how Faltings developed the “almost purity theorem” and proved the Hodge–Tate decomposition of the étale cohomology of a proper smooth variety. The chapter proceeds by discussing almost isomorphisms, almost finitely generated projective modules, trace, rank and determinant, almost flat modules and almost faithfully flat modules, almost étale coverings, almost faithfully flat descent, and liftings. Finally, it describes group cohomology of discrete A–G-modules and Galois cohomology.