Almost Ring Theory

This book through and complete fundations for the method of almost étale extensions, which is at the basis of Faitings'approach to p-adic Hodge theory. The central notion is that of on "almost ring". Almost rings are thé commutative unitary monoids in a tensor category obtained as a quotient V-Mod/S of thé category V-Mod of modules over a fixed ring V, thé subcategory S consists of ait modules annihilated by a fixed ideal m of V, satisfying certain natural conditions. The reader is assumed to be familiar with general categorical notions, some basic commutative algebra and some advanced homological algebra (derived catégories, simplicial methods). Apart from these general prerequisites, thé text is as self-contained as possible. One novel feature of the book - compared with Faltings' earlier treatment - is the systematic exploitation of the cotangent complex, especially for the study of déformations of almost algebras.