An Introduction to Homological Algebra (Google eBook)
With a wealth of examples as well as abundant applications to algebra, this is a must-read work: an easy-to-follow, step-by-step guide to homological algebra. The author provides a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added. Applications include the following: * to rings -- Lazard's theorem that flat modules are direct limits of free modules, Hilbert's Syzygy Theorem, Quillen-Suslin's solution of Serre's problem about projectives over polynomial rings, Serre-Auslander-Buchsbaum characterization of regular local rings (and a sketch of unique factorization); * to groups -- Schur-Zassenhaus, Gaschutz's theorem on outer automorphisms of finite p-groups, Schur multiplier, cotorsion groups; * to sheaves -- sheaf cohomology, Cech cohomology, discussion of Riemann-Roch Theorem over compact Riemann surfaces. Learning homological algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples. All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices. Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology.
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abelian category additive functor adjoint assume basis bicomplex bigraded chain map cls(z cohomology coker commutative diagram commutative ring complex composite contravariant functor Corollary coset cotorsion covariant functor define Definition denote derived functors direct limit direct sum direct summand direct system domain element exact rows Example Exercise exists extension finitely presented flat free abelian group free module function given gives group G Hence Hn(G homology groups homomorphism HomR(A hypothesis inclusion injective resolution inverse limits left ideal left noetherian Lemma long exact sequence maximal morphisms multiplication naturally isomorphic nonzero notation obj(C open cover partially ordered presheaf projective resolution Proof Proposition prove R-map right R-module RMod S~lR says semisimple sheaf sheaves short exact sequence shows spectral sequence split subgroup submodule subset surjective tensor product Theorem topological space Torf torsion torsion-free Tot(M unique vector space