Let ($A,m,k$) and ($B,n,k'$) be noetherian local rings, and let $A\to B$ be a local homomorphsim. Let $M$ be a finite $A$-module and $N$ be a finite $B$-module which is $A$-flat. Then
depth$_B(M\otimes_A N)$=depth$_A M$ + depth$_{B}(N\otimes k)$
In the proof, the case depth$M>0$, while depth$_B(N\otimes k)=0$ was left as an exercise, which I failed to proof. Since $N$ is flat, a free resolution of $M$ induces a free resoluton of $M\otimes N$. So we only need to proof that $Ext^i_A(k,M)=0$ means $Ext^i_B(k',M\otimes N)=0$. Also, we know that depth$_B(N\otimes k)=0$ means $n\in Ass_B(N\otimes k)$. But how does it help? Thanks in advance!