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.

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.

Define the following submodule of ${\displaystyle P\oplus P'}$, where ${\displaystyle \phi \colon P\to M}$ and ${\displaystyle \phi '\colon P'\to M}$:

The map ${\displaystyle \pi \colon X\to P}$, where ${\displaystyle \pi }$ is defined as the projection of the first coordinate of ${\displaystyle X}$ into ${\displaystyle P}$, is surjective. Since ${\displaystyle \phi '}$ is surjective, for any ${\displaystyle p\in P}$, one may find a ${\displaystyle q\in P'}$ such that ${\displaystyle \phi (p)=\phi '(q)}$. This gives ${\displaystyle (p,q)\in X}$ with ${\displaystyle \pi (p,q)=p}$. Now examine the kernel of the map ${\displaystyle \pi }$:

We may conclude that there is a short exact sequence

Since ${\displaystyle P}$ is projective this sequence splits, so ${\displaystyle X\cong K'\oplus P}$. Similarly, we can write another map ${\displaystyle \pi '\colon X\to P'}$, and the same argument as above shows that there is another short exact sequence

and so ${\displaystyle X\cong P'\oplus K}$. Combining the two equivalences for ${\displaystyle X}$ gives the desired result.