In the book "étale cohomology" by Milne, proposition 2.5 at p.9, it said :

Let $B$ be a flat $A-$algebra where $A$ and $B$ are noetherian rings, and consider $b \in B$. If the image of $b$ in $B/mB$ is not a zero-divisor for any maximal ideal $m$ of $A$, then $B/(b)$ is a flat $A-$ algebra.

At the beginning of the proof, he said we can reduce to the case where $\phi : A \rightarrow B$ is a local homomorphism of local noetherian rings. The proof in this case uses the fact that it's a local homomorphsim.

But I think that in order to reduce the general case to the local case, we need the following condition, which I can't get from the original condition.

For any maximal ideal $n$ of $B$, the image of $b$ in $B/pB$ is not a zero-divisor, where $p = \phi^{-1} (n)$.

How do you think?

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I am not sure about Milne's reduction, but your fix is too strong. First off all, I don't understand why you write $B/p B$ with $p = \phi^{-1}(n)$. I assume $\phi$ is the map $A \to B$. But then $\phi(n)$ is an ideal of $A$, not $B$, and $b$ is an element of $B$. I am going to assume you meant "$b$ is not a zero divisor in $B/nB$."
But requiring this for every maximal ideal of $B$ implies that $b$ is a unit! We surely don't want to impose that.
Yes, $\phi : A \rightarrow B$ is the ring homomorphism which defines the $A$-algebra structure of $B$. $p = \phi^{-1} (n)$ is a prime ideal of $A$, but $pB$ is the ideal of $B$ generated by $\phi (p) = \phi \circ \phi^{-1} (n)$, which doesn't equal to $n$ in general. Hence the condition is not as strong as you think. The reason I impose the condition is "flatness is a local property". About the name, I am sorry, is there anyway to change it? – Rothendieck Mar 23 '10 at 13:58