Almost Ring Theory Series: Lecture Notes in Mathematics

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springer.com Ofer Gabber, Lorenzo Ramero Almost Ring Theory Series: Lecture Notes in Mathematics 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 ofan "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. 2003, VI, 318 p. Printed book Softcover 59,99 € | £54.99 | $74.99 [1]64,19 € (D) | 65,99 € (A) | CHF 71,00 eBook 50,28 € | £43.99 | $59.99 [2]50,28 € (D) | 50,28 € (A) | CHF 56,50 Available from your library or springer.com/shop MyCopy [3] Printed eBook for just € | $ 24.99 springer.com/mycopy Order online at springer.com / or for the Americas call (toll free) 1-800-SPRINGER / or email us at: [email protected]. / For outside the Americas call +49 (0) 6221-345-4301 / or email us at: [email protected]. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with [1] include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with [2] include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. [3] No discount for MyCopy. Part of .

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Noncommutative Localization in Noncommutative Geometry
Noncommutative localization in noncommutative geometry Zoran Skodaˇ Abstract The aim of these notes is to collect and motivate the basic localiza- tion toolbox for the geometric study of “spaces” locally described by noncommutative rings and their categories of modules. We present the basics of Ore localization of rings and modules in great detail. Common practical techniques are studied as well. We also describe a counterexample to a folklore test principle for Ore sets. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with em- phasis on descent formalism, flatness, abelian categories of quasicoher- ent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Eilenberg-Watts, Deligne and Rosenberg. The Cohn universal localization does not have good flatness prop- erties, but it is determined by the localization map already at the ring level, like the perfect localizations are. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help arXiv:math/0403276v2 [math.QA] 1 Mar 2005 the understanding of both subjects. Contents 1 Introduction 2 2 Noncommutative geometry 6 3 Abstract localization 12 1 2 Noncommutative localization in noncommutative geometry 4 Ore localization for monoids 15 5 Ore localization for rings 22 6 Practical criteria for Ore sets 25 7 Ore localization for modules 30 8 Monads, comonads and gluing 33 9 Distributive laws and compatibility 40 10 Commutative localization 45 11 Ring maps vs.
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Arxiv:1803.03229V2 [Math.AC] 8 Sep 2018 N H Perfection, the and Aia Fr(O Xml,Rcudb -Dclycmlt) T R Map Complete)
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