[Math] Projective resolution of tensor product

Question

Let $M,N$ are $R$-modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain a projective resolution of $M\otimes N$ using $P^\bullet, Q^\bullet$? If I understand correctly the homology groups of $P^\bullet\otimes Q^\bullet$ are $\mathrm{Tor}(M,N)$ so it's not a projective resolution.

Answer 1

It is not a projective resolution, see Mac Lane, Homology, Theorem 9.3 page 164.