Schanuel's lemma

The lemma of Schanuel, named after Stephen Schanuel, is a simple and basic statement from the mathematical subfield of homological algebra.

Formulation of the lemma

There was a ring and be it

Short exact sequences in the category of left - modules with projective. Then, that is, the two direct sums are isomorphic.

Application

Is a projective resolution, so that is projective, so that goes for any projective resolution.

Indeed, if another projective resolution, so consider the short exact sequences

By the lemma of Schanuel, that is is a direct summand of the projective module by assumption and hence also projective.

Formation

Stephen Schanuel Lemma discovered this in 1958 during a lecture held by Irving Kaplansky on homological algebra. It was essentially on the above application. Kaplansky reported: