Tensor product of projective modules.

Question

How do we prove tensor product of two projective modules is projective?

My attempt: Let $P ,P'$ be projective modules. By equivalent conditions we have $K ,K'$ such that $K \oplus P$ and $K' \oplus P'$ are free. And hence by splitting $(K \otimes K') \oplus (P \otimes P')$, free implies projective ?

Is there any other way?

Answer 1

Use the following facts:

1. Tensor products of free modules are free.
2. The tensor product is a functor. Hence, if there are split monomorphisms $M \to M'$ and $N \to N'$, it induces a split monomorphism $M \otimes N \to M' \otimes N'$.
3. A module $P$ is projective iff there is a split monomorphism $P \to F$ into a free module $F$.

The claim follows. This is almost the same proof as yours, but it is a bit simpler since we don't need to distribute the summands and it uses more general easy facts and combines them - making the proof more conceptual.

There are also other proofs. For example this one:

1. A composition of exact functors is exact.
2. $P$ is projective iff $\hom(P,-)$ is exact.
3. We have an isomorphism of functors $\hom(M \otimes N,-) \cong \hom(M,-) \circ \hom(N,-)$.

The claim follows immediately.