Lemma 88.25.6. Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let T \subset |X| be a closed subset. Let s, t : R \to U be two morphisms of algebraic spaces over X. Assume
R, U are locally of finite type over X,
the base change of s and t to X \setminus T is an étale equivalence relation, and
the formal completion (t_{/T}, s_{/T}) : R_{/T} \to U_{/T} \times _{X_{/T}} U_{/T} is an equivalence relation too (see proof for notation).
Then (t, s) : R \to U \times _ X U is an étale equivalence relation.
Proof.
The notation using the subscript {}_{/T} in the statement refers to the construction which to a morphism f : X' \to X of algebraic spaces associates the morphism f_{/T} : X'_{/f^{-1}T} \to X_{/T} of formal algebraic spaces, see Section 88.23. The morphisms s, t : R \to U are étale over X \setminus T by assumption. Since the formal completions of the maps s, t : R \to U are étale, we see that s and t are étale for example by More on Morphisms, Lemma 37.12.3. Applying Lemma 88.25.3 to the morphisms \text{id} : R \times _{U \times _ X U} R \to R \times _{U \times _ X U} R and \Delta : R \to R \times _{U \times _ X U} R we conclude that (t, s) is a monomorphism. Applying it again to (t \circ \text{pr}_0, s \circ \text{pr}_1) : R \times _{s, U, t} R \to U \times _ X U and (t, s) : R \to U \times _ X U we find that “transitivity” holds. We omit the proof of the other two axioms of an equivalence relation.
\square
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