Categories and Modules with K-Theory in View
This book, first published in 2000, develops aspects of category theory fundamental to the study of algebraic K-theory. Ring and module theory illustrates category theory which provides insight into more advanced topics in module theory. Starting with categories in general, the text then examines categories of K-theory. This leads to the study of tensor products and the Morita theory. The categorical approach to localizations and completions of modules is formulated in terms of direct and inverse limits, prompting a discussion of localization of categories in general. Finally, local-global techniques which supply information about modules from their localizations and completions and underlie some interesting applications of K-theory to number theory and geometry are considered. Many useful exercises, concrete illustrations of abstract concepts placed in their historical settings and an extensive list of references are included. This book will help all who wish to work in K-theory to master its prerequisites.
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abelian category abelian groups additive category adjoint Cauchy sequence chirality cofibration cokernel commutative diagram commutative domain construction contravariant Dedekind domain define definition direct limit direct product direct sum direct system directed set element epimorphism Ex(C example Exercise field of fractions finitely generated module finitely generated projective flat fractional ideal free module functor F G-exact category given Hom(L Hom(M idempotent inclusion induces injective integer inverse K-theory kernel lattice left R-module Lemma Mn(R MoDR module categories monomorphism MooR Mor(L Morita equivalent morphism natural isomorphism natural transformation Noetherian nonunital rings nonzero prime ideal notation O-order p-adic pair phism preadditive category prime ideal projective modules Proof Proposition R-module homomorphism R-R-bimodule rad(R result right category right Ore set right R-module ring homomorphism short exact sequence Show split submodule Suppose surjective tensor product Theorem Let theory unique universal object universal property valuation ring Verify zero object