This is a basic question on tensor products of linear maps. Let $R$ be a commutative ring and let $\varphi: M\to N$ be an injective linear map of finitely generated free $R$-modules.

Question: Are the $k$-th tensor power maps $\varphi^{\otimes k}: M^{\otimes k}\to N^{\otimes k}$ also injective for all $k\geq 0?$

To show this I've tried using the fact that $M$ and $N$ are flat $R$-modules, but I think I also need to assume that if $M$ and $N$ are flat then $M\otimes_R N$ is flat.

Is this the correct approach to take?

Many thanks in advance.

  • 3
    $\begingroup$ While Dietrich Burde's answer is the right one, it is true that the tensor product of two flat modules is flat. If you think about it in terms of exactness of the functors, then the tensor product functor associated to $M\otimes_R N$ is the composition of the tensor product functors associated to $M$ and $N$. Since the composition of exact functors is exact, you get what you want. $\endgroup$ – Alex Youcis Mar 30 '14 at 16:07

Hint: for a commutative ring $R$, the tensor product $M\otimes_RN$ of two free $R$-modules $M$ and $N$ is again a free module, hence flat.


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