In Milne, Etale Cohomology, Proposition 2.5 (§2) is stated as follows:

(All rings noetherian.)

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

Now consider $A = k[[x_1,x_2]]$, $\mathfrak{m}=(x_1,x_2)$ and $\mathfrak{p} = (x_1)$ as $A$-ideals and $B = A_\mathfrak{p}$. Then $B/\mathfrak{m} B = 0$. Let $b = x_1$. As $B/\mathfrak{m} B$ vanishes, it has no zero-divisors, so $b$ is not a zero-divisor there.

But $B/b B \neq 0$ is obviously not $A$-flat, as it does not preserve the injectiveness of $A \overset{\cdot b}{\hookrightarrow} A$.

Of course, one could object that implicitly all $B \otimes_A k(\mathfrak{m})$ should be taken nonzero, but in this case $B$ would be faithfully flat over $A$ so as to strongly narrow the scope of the proposition.

A comparison with Kurke, Pfister, Roczen, Henselsche Ringe und algebraische Geometrie led me to suppose, that the proposition should be worded slightly different, namely:

"$b$ is not a zero-divisor in $B \otimes_A k(A \cap \mathfrak{n})$ for all $\mathfrak{n} \subseteq B$, maximal"

(see, 1.4.5. Korollar, there)

Is the remark above correct?


1 Answer 1


Have a look at Milne's errata and notes page: http://www.jmilne.org/math/Books/add/ECPUP.pdf


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