Return to 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.

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.

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that $$Hom_A (L, M)\otimes_A N \cong Hom_A (L, M\otimes_A N)$$ 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 is 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.

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.