$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{q}}$ is flat over $A_{\mathfrak{p}}$? Bruns seems to be suggesting that, but I don't see how. I can see that $N_{\mathfrak{p}}$ is flat over $A_{\mathfrak{p}}$ and that $N_{\mathfrak{q}}$ is a localization of $N_{\mathfrak{p}}$ as a $B_{\mathfrak{p}}$-module, but I can't apply the usual argument which works for rings. Help?

Take the 2-minute tour
×

MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Commutative Ring Theory, Theorem 7.1, page 46. – Georges Elencwajg Jun 29 '10 at 18:39