Projective resolution of tensor product
Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology groups of $P^\bullet\otimes Q^\bullet$ are $Tor(M,N)$ so its not a projective resolution.
Solution 1:
It is not a projective resolution, see Mac Lane, Homology, Theorem 9.3 page 164.