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Thread: Localization - isomorphism

  1. #1
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    Localization - isomorphism

    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.
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  2. #2
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    Quote Originally Posted by AlexanderW View Post
    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.
    $\displaystyle S^{-1}M \otimes_{S^{-1}A} S^{-1}N \cong (M \otimes_A S^{-1}A) \otimes_{S^{-1}A} S^{-1}N \cong M \otimes_A (S^{-1}A \otimes_{S^{-1}A} S^{-1}N)$

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