Tensor product and direct product

Question

Let $R$ be a commutative ring, let $M$ be an $R$-module, and let $(N_i)_{i\in I}$ be a family of $R$-modules. There is a canonical morphism of $R$-modules $$\varepsilon\colon M\otimes_R\bigl(\prod_{i\in I}N_i\bigr)\rightarrow\prod_{i\in I}(M\otimes_RN_i).$$ It is known that this morphism need neither be a monomorphism nor an epimorphism. On the positive side, we know that it is an epimorphism for every family $(N_i)_{i\in I}$ if and only if $M$ is of finite type, that it is an isomorphism for every family $(N_i)_{i\in I}$ if and only if $M$ is of finite presentation, and that it is a monomorphism for every $(N_i)_{i\in I}$ if and only if $M$ is Mittag-Leffler.

I am interested in $\varepsilon$ being a monomorphism not for every family $(N_i)_{i\in I}$, but only for constant families, i.e., in $$\varepsilon\colon M\otimes_R(N^\kappa)\rightarrow(M\otimes_RN)^\kappa$$ being a monomorphism for any (or just some) (infinite) cardinal $\kappa$ and any (or just some) $R$-module $N$.

Are there any results known about this problem?

(I am mostly interested in case $M$ is a commutative $R$-algebra, so that $M\otimes_R\bullet$ is extension of scalars.)

Answer 1

(This answer is inspired by Andreas Blass' comment to my question.)

Remark: Let $M$ be an $R$-module, and let $(N_i)_{i\in I}$ be a family of $R$-modules. For $i\in I$ we consider the canonical injection $N_i\rightarrowtail\bigoplus_{j\in I}N_j$. Taking the product of all these morphisms we get a canonical monomorphism of $R$-modules $$\Delta_{M,(N_i)_{i\in I}}\colon\prod_{i\in I}N_i\rightarrowtail\prod_{i\in I}\bigoplus_{j\in I}N_j.$$

Claim: Let $M$ be a flat $R$-module and let $I$ be a set. If the canonical morphism $$\varepsilon_{M,N,I}\colon M\otimes_R(N^I)\rightarrow(M\otimes_RN)^I$$ is a monomorphism for every $R$-module $N$, then so is the canonical morphism $$\varepsilon_{M,(N_i)_{i\in I}}\colon M\otimes_R\bigl(\prod_{i\in I}N_i\bigr)\rightarrow\prod_{i\in I}(M\otimes_RN_i)$$ for every family $(N_i)_{i\in I}$ of $R$-modules.

Proof: Identifying $\prod_{i\in I}\bigoplus_{j\in I}(M\otimes_RN_j)$ and $\prod_{i\in I}(M\otimes_R(\bigoplus_{j\in J}N_j))$ by means of the canonical isomorphism it can be checked that the following two morphisms of $R$-modules are equal:

$$M\otimes_R(\prod_{i\in I}N_i)\overset{\varepsilon_{M,(N_i)_{i\in I}}}\longrightarrow\prod_{i\in I}(M\otimes_RN_i)\overset{\Delta_{M,(M\otimes_RN_i)_{i\in I}}}\longrightarrow\prod_{i\in I}\bigoplus_{j\in I}(M\otimes_RN_j),$$

$$M\otimes_R(\prod_{i\in I}N_i)\overset{M\otimes_R\Delta_{M,(N_i)_{i\in I}}}\longrightarrow M\otimes_R(\prod_{i\in I}\bigoplus_{j\in J}N_j)\overset{\varepsilon_{M,\oplus_{i\in I}N_i,I}}\rightarrowtail\prod_{i\in I}(M\otimes_R(\bigoplus_{j\in J}N_j)).$$

As $\Delta_{M,(N_i)_{i\in I}}$ is a monomorphism and $M$ is flat it follows that $M\otimes_R\Delta_{M,(N_i)_{i\in I}}$ is a monomorphism, too. Therefore, $\varepsilon_{M,(N_i)_{i\in I}}$ is a monomorphism, and the claim is proven. $\square$

Question: Can we get rid of the flatness hypothesis?