Lemma 15.31.5 (0CEQ)—The Stacks project

Lemma 15.31.5. Let $A' \to B'$ be a ring map. Let $I \subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$ . Let $f'_1, \ldots , f'_ r \in B'$ . Assume

$A' \to B'$ is flat and of finite presentation,
$I$ is locally nilpotent,
the images $f_1, \ldots , f_ r \in B$ form a quasi-regular sequence,
$B/(f_1, \ldots , f_ r)$ is flat over $A$ .

Then $B'/(f'_1, \ldots , f'_ r)$ is flat over $A'$ .

Proof. Set $C' = B'/(f'_1, \ldots , f'_ r)$ . We have to show $A' \to C'$ is flat. Let $\mathfrak r' \subset C'$ be a prime ideal lying over $\mathfrak p' \subset A'$ . We let $\mathfrak q' \subset B'$ be the inverse image of $\mathfrak r'$ . By Algebra, Lemma 10.39.18 it suffices to show that $A'_{\mathfrak p'} \to C'_{\mathfrak q'}$ is flat. Algebra, Lemma 10.128.6 tells us it suffices to show that $f'_1, \ldots , f'_ r$ map to a regular sequence in

$B'_{\mathfrak q'}/\mathfrak p'B'_{\mathfrak q'} = B_\mathfrak q/\mathfrak p B_\mathfrak q = (B \otimes _ A \kappa (\mathfrak p))_\mathfrak q$

with obvious notation. What we know is that $f_1, \ldots , f_ r$ is a quasi-regular sequence in $B$ and that $B/(f_1, \ldots , f_ r)$ is flat over $A$ . By Lemma 15.31.4 the images $\overline{f}_1, \ldots , \overline{f}_ r$ of $f'_1, \ldots , f'_ r$ in $B \otimes _ A \kappa (\mathfrak p)$ form a quasi-regular sequence. Since $(B \otimes _ A \kappa (\mathfrak p))_\mathfrak q$ is a Noetherian local ring, we conclude by Lemma 15.30.7. $\square$

The Stacks project

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