Homology

a₁ abelian category additive relation B₁ bar resolution bifunctor bimodule boundary chain complex chain transformation coefficients cohomology groups composite congruence connecting homomorphisms construct contracting homotopy contravariant corresponding couniversal covariant functor defined definition degree denote DG-algebra DG-module differential dimension direct sum elements epic epimorphism equivalent extension filtration finite formula free abelian group free module function given gives graded algebra graded K-module hence homo homology homomorphism f identity II-module implies induced injective integers inverse K-algebra kernel left R-module Lemma long exact sequence Mac Lane Math module homomorphism monic monomorphism morphism natural isomorphism object pair phism projective resolution Proof Prop proper projective Proposition prove quotient R-module right R-module ring short exact sequence simplicial spectral sequence subgroup submodule tensor product torsion product unique vector space yields Z(II zero