Let A be a finite-dimensional algebra. Let M be a left A-module and N be a right A-module. Choose an injective resolution $E_1^*$ of $M$ in $A$-mod and an injective resolution $E_2^*$ of $N$ in mod-$A$. Then, do we have $E_1^*\otimes E_2^*$ is an injective resolution of $M \otimes N$ in the category of $A$-$A$-bimodules? I could show that $E_1^*\otimes E_2^*$ are injective objects in the category of $A$-$A$-bimodules. But how to show that it gives a resolution?
Is this also true in case of projective resolution?
Thanks in advance!