An Introduction to Homological Algebra
The landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors.
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6-functor abelian category abelian group acyclic bimodule bounded called central extension chain complex chain homotopy chain homotopy equivalence chain map cochain cohomology cokernel colimit composition construction cotriple define Definition denote derived functors diagram differential dim(R direct sum double complex element exact functor exact triangle Example filtration finite flat forgetful functor formula full subcategory functor F graded Hence Hochschild homology Hom(A homomorphism ideal induces injective k-algebra kernel KX Corollary KX Exercise left adjoint left exact left R-module Lemma Let f Lie algebra long exact sequence map f mod—R module monic morphism natural isomorphism natural map noetherian plex profinite projective resolution quasi-isomorphism quotient R-mod R-module resp right exact right R-module semisimple Sheaves(X short exact sequence Show simplicial object simplicial set spectral sequence split exact subgroup Suppose surjection Theorem topological space Torš trivial universal central extension yields zero