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Is it true that if $A$ is a commutative ring, then the module of fractions $S^{-1}M$ of a flat $A$-module $M$ is a flat $S^{-1}A$-module?

This is certainly true for localizations at primes, but I'm unsure if this holds in general. I suspect that there is a simple counterexample, though I'm having trouble coming up with one. Can someone point towards a proof/counterexample?

mathyokan
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 07 '21 at 04:12
  • What proof do you have for localizations at primes, and why wouldn't it work just as well for arbitrary localizations? – Eric Wofsey Oct 07 '21 at 05:02

1 Answers1

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Let $M$ be a flat $A$-module, $S\subset A$ a multiplicative set and $N\to N'$ an injective homomorphism of $A$-modules. We would like to show that the induced homomorphism $$N\otimes_A S^{-1}M \to N' \otimes_A S^{-1}M$$ is injective. Now, recall that

$$S^{-1}M\cong S^{-1}A\otimes_A M.$$

Hence $$N\otimes_A S^{-1}M\cong N\otimes_A M \otimes_A S^{-1}A\to N'\otimes_A M \otimes_A S^{-1}A\cong N'\otimes_A S^{-1}M$$ is injective by flatness of both $S^{-1}A$ and $M$.

Douglas Molin
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