RING THEORY
CONTENTS 1 INTRODUCTION 1 1 MOTIVATIONS AND A LITTLE HISTORY 1 2 THE METHOD OF ALMOST ´ETALE EXTENSIONS 1 3 CONTENTS OF THIS BOOK 1 4 THE VIEW FROM ABOVE 1 5 ACKNOWLEDGEMENTS 2 HOMOLOGICAL THEORY 2 1 SOME RING-THEORETIC PRELIMINARIES 2 2 CATEGORIES OF ALMOST MODULES AND ALGEBRAS 2 3 UNIFORM SPACES OF ALMOST MODULES 2 4 ALMOST HOMOLOGICAL ALGEBRA 2 5 ALMOST HOMOTOPICAL ALGEBRA 3 ALMOST RING THEORY 3 1 FLAT, UNRAMI?ED AND ´ETALE MORPHISMS 3 2 NILPOTENT DEFORMATIONS OF ALMOST ALGEBRAS AND MODULES 3 3 NILPOTENT DEFORMATIONS OF TORSORS 3 4 DESCENT 3 5 BEHAVIOUR OF ´ETALE MORPHISMS UNDER FROBENIUS 4 FINE STUDY OF ALMOST PROJECTIVE MODULES 4 1 ALMOST TRACES 4 2 ENDOMORPHISMS OF GM 4 3 MODULES OF ALMOST ?NITE RANK 4 4 LOCALIZATION IN THE ?AT SITE 4 5 CONSTRUCTION OF QUOTIENTS BY ?AT EQUIVALENCE RELATIONS 5 HENSELIZATION AND COMPLETION OF ALMOST ALGEBRAS 5 1 HENSELIAN PAIRS 5 2 CRITERIA FOR UNRAMI?ED MORPHISMS 5 3 TOPOLOGICAL ALGEBRAS AND MODULES 5 4 HENSELIAN APPROXIMATION OF STRUCTURES OVER ADICALLY COMPLETE RINGS 5 5 LIFTING THEOREMS FOR HENSELIAN PAIRS 5 6 SMOOTH LOCUS OF AN AF?NE ALMOST SCHEME 5 7 QUASI-PROJECTIVE ALMOST SCHEMES 5 8 LIFTING AND DESCENT OF TORSORS 6 VALUATION THEORY 6 1 ORDERED GROUPS AND VALUATIONS 6 2 BASIC RAMI?CATION THEORY 6 3 ALGEBRAIC EXTENSIONS 6 4 LOGARITHMIC DIFFERENTIALS 6 5 TRANSCENDENTAL EXTENSIONS 6 6 DEEPLY RAMI?ED EXTENSIONS 7 ANALYTIC GEOMETRY 7 1 DERIVED COMPLETION FUNCTOR 7 2 COTANGENT COMPLEX FOR FORMAL SCHEMES AND ADIC SPACES 7 3 DEFORMATIONS OF FORMAL SCHEMES AND ADIC SPACES 7 4 ANALYTIC GEOMETRY OVER A DEEPLY RAMI?ED BASE 7 5 SEMICONTINUITY OF THE DISCRIMINANT