The Stacks project

Lemma 58.19.2. In Situation 58.19.1. Assume one of the following holds

  1. $\dim (A/\mathfrak p) \geq 2$ for every minimal prime $\mathfrak p \subset A$ with $f \not\in \mathfrak p$, or

  2. every connected component of $U$ meets $U_0$.

Then

\[ \textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0},\quad V \longmapsto V_0 = V \times _ U U_0 \]

is a faithful functor.

Proof. Case (2) is immediate from Lemma 58.17.5. Assumption (1) implies every irreducible component of $U$ meets $U_0$, see Algebra, Lemma 10.60.13. Hence (1) follows from (2). $\square$

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