Horseshoe lemma

In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects formula_1 and formula_2 to resolutions of
extensions of formula_1 by formula_2. It says that if an object formula_5 is an extension of formula_1 by formula_2, then a resolution of formula_5 can be built up inductively with the "n"th item in the resolution equal to the coproduct of the "n"th items in the resolutions of formula_1 and formula_2. The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis.
Formal statement.
Let formula_11 be an abelian category with enough projectives. If
is a diagram in formula_11 such that the column is exact and the
rows are projective resolutions of formula_1 and formula_2 respectively, then
it can be completed to a commutative diagram
where all columns are exact, the middle row is a projective resolution
of formula_5, and formula_16 for all "n". If formula_11 is an
abelian category with enough injectives, the dual statement also holds.
The lemma can be proved inductively. At each stage of the induction, the properties of projective objects are used to define maps in a projective resolution of formula_5. Then the snake lemma is invoked to show that the simultaneous resolution constructed so far has exact rows.