Let $\phi:A \to B$ be a flat ring homomorphism, $M$ be a $B$-module which is flat when considered as an $A$-module. Is the tensor product $M \otimes\_B M \otimes\_B ... \otimes\_B M$ flat over $A$? If not true in general, is there any known cases of $\phi$ (other than etale morphisms) when it holds true? Furthermore, is the exterior power $\wedge^m M$ flat, wedge taken as $B$ modules (i.e., $M \otimes\_B ...\otimes\_B M$ modulo equivalence relations) for any $m \in \mathbb{Z}$?



Is flatness preserved under exterior power
Let $A=\mathbb{C}[x^2]$, $B=\mathbb{C}[x^2,x^3]$, and $M=\mathbb{C}[x]$.


Then $B$ and $M$ are free as $A$-modules, but $M\otimes\_BM$ (and also $\wedge^2M$) has a one-dimensional $A$-submodule spanned by $x\otimes 1 - 1\otimes x$, and so isn't flat.


