Almost Ring Theory

by Gabber, Ofer; Ramero, Lorenzo

• ISBN13: 9783540405948
• ISBN10: 3540405941
• Format: Paperback
• Copyright: 2003-10-01
• Publisher: Springer Verlag

Summary

This book develops thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to p-adic Hodge theory. The central notion is that of an "almost ring". Almost rings are the commutative unitary monoids in a tensor category obtained as a quotient V-Mod/S of the category V-Mod of modules over a fixed ring V; the subcategory S consists of all 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 categories, simplicial methods). Apart from these general prerequisites, the 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 deformations of almost algebras.

Table of Contents

Introduction	1	(10)
Motivations and a little history	1	(2)
The method of almost etale extensions	3	(4)
Contents of this book	7	(2)
The view from above	9	(1)
Acknowledgements	10	(1)
Homological theory	11	(39)
Some ring-theoretic preliminaries	11	(5)
Categories of almost modules and algebras	16	(7)
Uniform spaces of almost modules	23	(8)
Almost homological algebra	31	(9)
Almost homotopical algebra	40	(10)
Almost ring theory	50	(42)
Flat, unramified and etale morphisms	51	(2)
Nilpotent deformations of almost algebras and modules	53	(8)
Nilpotent deformations of torsors	61	(10)
Descent	71	(11)
Behaviour of etale morphisms under Frobenius	82	(10)
Fine study of almost projective modules	92	(38)
Almost traces	92	(8)
Endomorphisms of Gm	100	(7)
Modules of almost finite rank	107	(6)
Localization in the flat site	113	(10)
Construction of quotients by flat equivalence relations	123	(7)
Henselization and completion of almost algebras	130	(65)
Henselian pairs	131	(4)
Criteria for unramified morphisms	135	(6)
Topological algebras and modules	141	(7)
Henselian approximation of structures over adically complete rings	148	(14)
Lifting theorems for henselian pairs	162	(9)
Smooth locus of an affine almost scheme	171	(9)
Quasi-projective almost schemes	180	(9)
Lifting and descent of torsors	189	(6)
Valuation theory	195	(47)
Ordered groups and valuations	196	(10)
Basic ramification theory	206	(4)
Algebraic extensions	210	(10)
Logarithmic differentials	220	(6)
Transcendental extensions	226	(8)
Deeply ramified extensions	234	(8)
Analytic geometry	242	(45)
Derived completion functor	243	(8)
Cotangent complex for formal schemes and adic spaces	251	(11)
Deformations of formal schemes and adic spaces	262	(10)
Analytic geometry over a deeply ramified base	272	(6)
Semicontinuity of the discriminant	278	(9)
Appendix	287	(14)
Simplicial almost algebras	287	(5)
Fundamental group of an almost algebra	292	(9)
References	301	(3)
Index	304