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Modules Diagrams and Functors
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abelian category additive relation axioms bar resolution bifunctor bigraded bimodule boundary called chain complex chain homotopy chain transformation cochain coefficients cohomology cokernel commutative diagram composite congruence connecting homomorphisms construct contracting homotopy corresponding couniversal covariant functor cycle defined definition degree denote DG-algebra DG-module differential dimension dual elements epic epimorphism equivalent extension filtration finite formula free abelian group free module function given gives graded algebra graded K-module group 77 hence homo homology groups Horn i?-module identity implies induced injective integers inverse K-algebra K-split kernel left exact left R-module Lemma long exact sequence Math module homomorphism monic monomorphism morphism natural isomorphism object pair phism polynomial projective resolution Proof Prop proper projective proper short exact Proposition prove quotient right exact ring short exact sequence simplicial singular spectral sequence splits subgroup submodule tensor product torsion product unique vector space yields zero