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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5-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 converges Corollary cotriple define Definition denote derived functors diagram differential dimension direct sum double complex element exact functor exact triangle Example Exercise fc-algebra fc-module filtration finite flat forgetful functor formula free module full subcategory G-mod g-module graded group G Hence Hochschild homology Hom(A homomorphism ideal induces injective it-module kernel left adjoint left exact left module Lemma Lie algebra lim1 long exact sequence morphism natural isomorphism natural map natural transformation noetherian plex polynomial profinite projective resolution Proof quasi-isomorphism quotient resp right exact semisimple sheaf Sheaves(X short exact sequence Show simplicial object simplicial set spectral sequence split exact subcomplex subgroup Suppose surjection Theorem topological space Torf triangulated category trivial universal central extension yields zero