Schanuel's lemma


In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting.

Statement

Schanuel's lemma is the following statement:
If 0 → K → P → M → 0 and 0 → K' → P ' → M → 0 are short exact sequences of R-modules and P and P ' are projective, then K ⊕ P ' is isomorphic to K ' ⊕ P.

Proof

Define the following submodule of P ⊕ P ', where φ : P → M and φ' : P ' → M:
The map π : X → P, where π is defined as the projection of the first coordinate of X into P, is surjective. Since φ' is surjective, for any p P, one may find a q P ' such that φ = φ '. This gives X with π = p. Now examine the kernel of the map π :
We may conclude that there is a short exact sequence
Since P is projective this sequence splits, so X ≅ K ' ⊕ P. Similarly, we can write another map π : X → P ', and the same argument as above shows that there is another short exact sequence
and so X ≅ P ' ⊕ K. Combining the two equivalences for X gives the desired result.

Long exact sequences

The above argument may also be generalized to long exact sequences.

Origins

discovered the argument in Irving Kaplansky's homological algebra course at the University of Chicago in Autumn of 1958. Kaplansky writes: