# Localization - isomorphism

#### AlexanderW

Let $$\displaystyle A$$ be a commutative ring and $$\displaystyle S \subset A$$ a multiplicatively closed subset and $$\displaystyle M, N\$$ $$\displaystyle A-$$modules.

Then there exists a uniquely determined isomorphism of $$\displaystyle S^{-1}A-$$modules:

$$\displaystyle \phi: S^{-1}M \otimes _{S^{-1}A}S^{-1}N \to S^{-1}(M \otimes_A N) \\$$

with

$$\displaystyle \phi\left ( \frac{m}{s} \otimes \frac{n}{t} \right ) = \frac{m \otimes n}{st}$$

Is it the right way, to proof that $$\displaystyle \widetilde{\phi}: S^{-1}M \times S^{-1}N \to S^{-1}(M \otimes_A N)$$ is $$\displaystyle S^{-1}A-$$bilinear and induces the homomorphism $$\displaystyle \phi$$? Then I have to proof that $$\displaystyle \phi$$ is injective and surjective.

#### NonCommAlg

MHF Hall of Honor
Let $$\displaystyle A$$ be a commutative ring and $$\displaystyle S \subset A$$ a multiplicatively closed subset and $$\displaystyle M, N\$$ $$\displaystyle A-$$modules.

Then there exists a uniquely determined isomorphism of $$\displaystyle S^{-1}A-$$modules:

$$\displaystyle \phi: S^{-1}M \otimes _{S^{-1}A}S^{-1}N \to S^{-1}(M \otimes_A N) \\$$

with

$$\displaystyle \phi\left ( \frac{m}{s} \otimes \frac{n}{t} \right ) = \frac{m \otimes n}{st}$$

Is it the right way, to proof that $$\displaystyle \widetilde{\phi}: S^{-1}M \times S^{-1}N \to S^{-1}(M \otimes_A N)$$ is $$\displaystyle S^{-1}A-$$bilinear and induces the homomorphism $$\displaystyle \phi$$? Then I have to proof that $$\displaystyle \phi$$ is injective and surjective.
$$\displaystyle S^{-1}M \otimes_{S^{-1}A} S^{-1}N \cong (M \otimes_A S^{-1}A) \otimes_{S^{-1}A} S^{-1}N \cong M \otimes_A (S^{-1}A \otimes_{S^{-1}A} S^{-1}N)$$

$$\displaystyle \cong M \otimes_A S^{-1}N \cong M \otimes_A (N \otimes_A S^{-1}A) \cong (M \otimes_A N) \otimes_A S^{-1}A \cong S^{-1}(M \otimes_A N).$$

AlexanderW