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:Let R be a ring with identity.
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, where and :The map, where is defined as the projection of the first coordinate of into, is surjective. Since is surjective, for any, one may find a such that. This gives with. Now examine the kernel of the map :
We may conclude that there is a short exact sequence
Since is projective this sequence splits, so. Similarly, we can write another map, and the same argument as above shows that there is another short exact sequence
and so. Combining the two equivalences for gives the desired result.