# Local rings and flatness

Let $A \rightarrow B$ be a flat and local homomorphism of commutative local rings.

Let $M,N$ be two $B$-modules which are free of finite rank as $A$-modules.

Consider the product $M \otimes_B N$ as an $A$-module. Is this $A$-module flat?

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In general the tensor product may not be torsion-free even. Let $B=k[[x,y]]/(xy)$ and $A=k[[x-y]]\subset B$. Let $M=B/xB$ and $N=B/yB$.