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Math Help - Localization - isomorphism

  1. #1
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    Localization - isomorphism

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

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

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

    with

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

    Is it the right way, to proof that \widetilde{\phi}: S^{-1}M  \times   S^{-1}N \to S^{-1}(M \otimes_A N) is S^{-1}A-bilinear and induces the homomorphism \phi? Then I have to proof that \phi is injective and surjective.
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  2. #2
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    Quote Originally Posted by AlexanderW View Post
    Let A be a commutative ring and S \subset A a multiplicatively closed subset and M, N\ A-modules.

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

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

    with

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

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

    \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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