Homology

a₁ abelian category additive relation B₁ bar resolution bifunctor bimodule boundary chain transformation coefficients cohomology coker commutative diagram composite congruence connecting homomorphisms construction contracting homotopy corresponding couniversal covariant functor defined definition degree denote DG-module DGA-algebra differential dimension elements epic epimorphism equivalence Ext¹ extension filtration finite formula free abelian group function given gives graded algebra graded K-module hence homo homology groups homomorphism f identity II-module implies induced injective integers inverse K-algebra K-split kernel left R-module Lemma long exact sequence module homomorphism monic monomorphism morphism natural isomorphism object pair phism projective resolution Proof Prop proper projective proper short exact Proposition prove quotient R-module right exact ring short exact sequence simplicial singular spectral sequence splits subgroup submodule tensor product torsion product transformation f unique yields Z(II zero