Does module Hom commute with tensor product in the second variable?

Question

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that $$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$ as $A$-modules?

(Note that there is a natural morphism from the left to the right, I think it's not easy to check it is injective or surjective, but I didn't really do it; also, I think elements of the RHS are hard to decompose, so I don't hope for a (natural) arrow in the opposite direction.)

If this is not true, how about we assume that $A$ is a local ring and $N$ is a flat $A$-module or even a flat local $A$-algebra?

Could anyone give some hint or a proof, or a counterexample?

Other appropriate conditions that guarantee the isomorphism are appreciated.

$\textbf{Edit:}$ My main concern is the case when $A=\mathscr{O}_{\mathbb{C}^n,0}=M, N=\mathscr{E}_{\mathbb{C}^n,0}$, and $L$ is the stalk at $0\in \mathbb{C}^n$ of some coherent $\mathscr{O}_{\mathbb{C}^n}$-module, where $\mathscr{O}_{\mathbb{C}^n}$ and $\mathscr{E}_{\mathbb{C}^n}$ mean the sheaves of holomorphic functions and complex-valued smooth functions on $\mathbb{C}^n$ respectively. The flatness of $\mathscr{E}_{\mathbb{C}^n}$ over $\mathscr{O}_{\mathbb{C}^n}$ can be found here (Theorem 7.2.1), which cites Bernard Malgrange's book 'Ideals of differentiable functions' (page 88, Coro 1.12), and here is another discussion on MO.

Answer 1

The example $A=M=\mathbb{Z}$, $L=N=\mathbb{Q}$ shows that the answer is negative: we have $\text{Hom}(L,M)=0$ so $\text{Hom}_A(L,M)\otimes_AN=0$, but $\text{Hom}_A(L,M\otimes_AN)=\text{Hom}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Q})=\mathbb{Q}\neq 0$

Answer 2

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commutes with limits in the second variable. Dually, you can think about homs as a kind of limit (in the second variable); you're asking the tensor product functor $(-) \otimes_A N$ to commute with this limit, but usually tensor products only commute with colimits.

This sort of reasoning not only suggests that your statement should be false but suggests what extra hypotheses might make it true: namely, some kind of projectivity hypothesis on $L$, or some kind of flatness hypothesis on $N$. In fact the statement is true if either $L$ or $N$ is finitely presented projective; these conditions are equivalent to requiring that $\text{Hom}_A(L, -)$ commutes with all colimits or that $(-) \otimes_A N$ commutes with all limits respectively.

But it's also true if $L$ is finitely presented and $N$ is flat! In this case $\text{Hom}_A(L, M)$ is a finite limit (really an iterated finite limit, but this isn't an issue) in $M$, and $(-) \otimes_A N$ preserves it. Dually, it's also true if $N$ is finitely presented and $L$ is projective: in this case $M \otimes_A N$ is a finite colimit in $M$, and $\text{Hom}_A(L, -)$ preserves it. Note that in Neil Strickland's example neither $L$ nor $N$ is finitely presented.