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a₁ abelian category additive relation axioms B₁ bar resolution bifunctor bimodule boundary chain complex chain homotopy chain transformation coefficients cohomology composite congruence connecting homomorphisms construct contracting homotopy corresponding couniversal covariant functor defined definition degree denote DG-algebra differential dimension dual E₁ elements epic epimorphism equivalence Ext¹ extension filtration finite formula free abelian group free module function given gives graded algebra graded K-module hence homo homology groups homomorphism f identity II-module implies induced injective integers inverse K-algebra kernel left R-module Lemma long exact sequence module homomorphism monic monomorphism morphism natural isomorphism object pair phism projective resolution Proof Prop Proposition prove quotient R-module ring short exact sequence simplicial singular spectral sequence splits subgroup submodule tensor product torsion product transformation f unique vector space yields Z(II zero