(All rings are commutative)
Let $A$ be a noetherian ring. Let $B$ be a noetherian $A$-algebra (not nessecerily f.g!)
Suppose $M$ and $N$ are finitely generated projective $B$-modules (for my application I can assume that $M=N$ are of rank $1$).
Consider the $B\otimes_A B$-module $M\otimes_A N$. Is it projective?
Note that it is flat: Given a $B\otimes_A B$-module $L$, there is a canonical isomorphism $(M\otimes_A N) \otimes_{B\otimes_A B} L \cong M \otimes_B L \otimes_B N$ which clearly shows that if $M$ and $N$ are flat over $B$ then $M\otimes_A N$ is flat over $B\otimes_A B$, but I am not sure about projectivity. In my application, the ring $B\otimes_A B$ is not noetherian.
Any ideas?