Horseshoe lemma

In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects

to resolutions of extensions of

can be built up inductively with the nth item in the resolution equal to the coproduct of the nth items in the resolutions of

The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis.

be an abelian category with enough projectives.

such that the column is exact and the rows are projective resolutions of

respectively, then it can be completed to a commutative diagram where all columns are exact, the middle row is a projective resolution of

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

Then the snake lemma is invoked to show that the simultaneous resolution constructed so far has exact rows.

This article incorporates material from horseshoe lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.