Almost ring theory

TL;DR

This paper develops the theory of almost modules and algebras, introducing an almost homological algebra framework to generalize Faltings' almost etale extensions and study their properties.

Contribution

It constructs the almost cotangent complex, generalizes results on almost etale morphisms, and explores invariance and descent properties in the almost setting.

Findings

Almost trace is almost perfect iff the morphism is almost etale.

Generalization of Faltings' results on lifting almost etale morphisms.

Invariance of almost etale morphisms under Frobenius.

Abstract

The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct the almost version of the cotangent complex and we use it to generalise some results of Faltings on the lifting of almost etale morphisms and almost etale algebras over nilpotent extensions. We also study the "almost trace" of an almost flat and almost finitely presented morphism, in particular we show that the almost trace is (almost) perfect if and only if the morphism is almost etale. Finally we study some cases of non-flat descent for almost rings, and establish the invariance of almost etale morphisms under Frobenius.