Localization - isomorphism

Jan 2010
23
0
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
May 2008
2,295
1,663
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.
\(\displaystyle S^{-1}M \otimes_{S^{-1}A} S^{-1}N \cong (M \otimes_A S^{-1}A) \otimes_{S^{-1}A} S^{-1}N \cong M \otimes_A (S^{-1}A \otimes_{S^{-1}A} S^{-1}N)\)

\(\displaystyle \cong M \otimes_A S^{-1}N \cong M \otimes_A (N \otimes_A S^{-1}A) \cong (M \otimes_A N) \otimes_A S^{-1}A \cong S^{-1}(M \otimes_A N).\)
 
  • Like
Reactions: AlexanderW