An Introduction to Homological Algebra
Homological Algebra has grown in the nearly three decades since the rst e- tion of this book appeared in 1979. Two books discussing more recent results are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand– Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfand and Manin divide the history of Homological Algebra into three periods: the rst period ended in the early 1960s, culminating in applications of Ho- logical Algebra to regular local rings. The second period, greatly in uenced by the work of A. Grothendieck and J. -P. Serre, continued through the 1980s; it involves abelian categories and sheaf cohomology. The third period, - volving derived categories and triangulated categories, is still ongoing. Both of these newer books discuss all three periods (see also Kashiwara–Schapira, Categories and Sheaves). The original version of this book discussed the rst period only; this new edition remains at the same introductory level, but it now introduces the second period as well. This change makes sense pe- gogically, for there has been a change in the mathematics population since 1979; today, virtually all mathematics graduate students have learned so- thing about functors and categories, and so I can now take the categorical viewpoint more seriously. When I was a graduate student, Homological Algebra was an unpopular subject. The general attitude was that it was a grotesque formalism, boring to learn, and not very useful once one had learned it.
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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