Introduction to Homological Algebra, 85
An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author’s attempt to make it lovable.
This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and X; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences.
This book will be of interest to practitioners in the field of pure and applied mathematics.
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abelian group algebra Assume basis bicomplex bidegree bimodule chain map cohomology column commutative diagram complex connecting homomorphisms Consider the diagram contravariant Corollary coset define Definition denote derived functors diagram commute diagram with exact direct limit direct system domain dual element epic epimorphism exact functor exact rows example Exercise filtration finitely related flat modules follows formula free abelian free module function gives GL(n hence Hom(A Hom(B homology hypothesis implies index set induction injective module injective resolution inverse limits isomorphism left exact left ideal left noetherian left R-module Lemma Let F monic morphism multiplication natural equivalence nonzero notation pd(A polynomial projective module projective resolution proof of Theorem prove quotient R-map right exact right R-module semisimple short exact sequence spectral sequence splits ST'R subgroup submodule summand tensor torsion torsion-free Tot(M unique universal mapping problem vector space zero