Springer Grundlehren Der Mathematischen Wissenschaften 114, 1963

Springer Classics in Mathematics, 1995

Homological Algebra

Algebraic Topology

"Our subject starts with homology, homomorphisms, and tensors. Homology provides an algebraic 'picture' of topological spaces, assigning to each space X a family of abelian groups $ H_0(X), \ldots, H_n(X) $, to each continuous map $ f : X \rightarrow Y $ a family of group homomorphisms $ f_n : H_n(X) \rightarrow H_n(Y) $. Properties of the space or the map can often be effectively found from properties of the groups $ H_n $ or the homomorphisms $ f_n $. A similar process associates homology groups to other mathematical objects; for example, to a group $ \Pi $ or to an associative algebra $ \Lambda $. Homology in all such cases is our concern."

Chapter I : Modules, Diagrams, and FunctorsEdit

Section I.1 : The Arrow NotationEdit

Section I.2 : ModulesEdit

  • The symbol $ \varkappa $ is an alternate way of drawing a kappa. (It's \varkappa in TeX).

Section I.3 : DiagramsEdit

Section I.4 : Direct SumsEdit

Section I.5 : Free and Projective ModulesEdit

  • The symbol $ \varrho $ is an alternate way of drawing a rho. (It's \varrho in TeX).

Section I.6 : The Functor HomEdit

Section I.7 : CategoriesEdit

Section I.8 : FunctorsEdit

Chapter II : Homology of ComplexesEdit

Section II.1 : Differential GroupsEdit

  • Example 7: The segments q x I and p x I should be oriented upwards.

Section II.2 : ComplexesEdit

Section II.3 : CohomologyEdit

Section II.4 : The Exact Homology SequenceEdit

Section II.5 : Some Diagram Lemmas Edit

Section II.6 : Additive RelationsEdit

Section II.7 : Singular HomologyEdit

Section II.8 : HomotopyEdit

Section II.9 : Axioms for HomologyEdit

Chapter III : Extensions and ResolutionsEdit

Chapter IV : Cohomology of GroupsEdit

Section IV.1 : The Group RingEdit

  • this means more exactly that $ \mu_0 y $ is that function on $ \Pi $ to Z for which...: i.e., if we regard elements of $ \Pi(Z) $ as functions $ \Pi \rightarrow Z $ sending $ x \mapsto m(x) $.

Section IV.2 : Crossed HomomorphismsEdit

  • The motivation for studying $ f_a = xa - a $ is to look at fixed points of modules.

Chapter V : Tensor and Torsion ProductsEdit

Chapter VI : Types of AlgebrasEdit

Chapter VII : DimensionEdit

Chapter VIII : ProductsEdit

Chapter IX : Relative Homological AlgebraEdit

Chapter X : Cohomology of Algebraic SystemsEdit

Chapter XI : Spectral SequencesEdit

Chapter XII : Derived FunctorsEdit