= Horseshoe lemma
For a short exact sequence of modules with projective resolutions of its outer terms, there is a projective resolution of the middle term forming a degreewise split short exact sequence of resolutions. Lift the degree-zero projective cover of the quotient into the middle module and add the projective cover of the submodule; this surjects onto the middle module. Its kernel fits into the corresponding short exact sequence of kernels. Repeating this construction proves the claim. Additive contravariant functors preserve the degreewise split exact sequences with their direction reversed.
Back to article page
