The Stacks project

Lemma 38.30.5. Let $A \to B$ be an ├ętale ring map. Let $a \in A$ be a nonzerodivisor. Let $J \subset B$ be a finite type ideal with $V(J) \subset V(aB)$. For every $\mathfrak q \subset B$ there exists a finite type ideal $I \subset A$ with $V(I) \subset V(a)$ and $g \in B$, $g \not\in \mathfrak q$ such that $IB_ g = JJ'$ for some finite type ideal $J' \subset B_ g$.

Proof. We may replace $B$ by a principal localization at an element $g \in B$, $g \not\in \mathfrak q$. Thus we may assume that $B$ is standard ├ętale, see Algebra, Proposition 10.144.4. Thus we may assume $B$ is a localization of $C = A[x]/(f)$ for some monic $f \in A[x]$ of some degree $d$. Say $B = C_ h$ for some $h \in C$. Choose elements $h_1, \ldots , h_ n \in C$ which generate $J$ over $B$. The condition $V(J) \subset V(aB)$ signifies that $a^ m = \sum b_ i h_ i$ in $B$ for some large $m$. Set $h_{n + 1} = a^ m$. As in Lemma 38.30.4 we take $I = \text{Fit}_0(C/(h_1, \ldots , h_{r + 1}))$. Since the module $C/(h_1, \ldots , h_{r + 1})$ is annihilated by $a^ m$ we see that $a^{dm} \in I$ which implies that $V(I) \subset V(a)$. $\square$


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