Let $M$,$N$ be two modules over commutative ring $R$.
Can we say that $pd (M \otimes N )= $ $pd( M) + pd( N)$?
Thanks!



Projective dimension and tensorial product of two modules
If $M$ and $N$ are Tor-independent (i.e., Tor$\_i^R(M,N)=0$ for $i>0$), then you have some positive results. For instance if $R$ is local complete intersection, the three modules are of finite projective dimension (and finite) then you have an affirmative answer as a consequence of Auslander-Buchschaum formula and Hunecke-Wiegand depth formula. See e.g. the expository paper <http://arxiv.org/pdf/1302.2170.pdf> mainly section 6. This answers less than the above comments, but I cannot comment due to reputation.


