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The Stacks project

Lemma 39.20.5. Let \tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} . Let S be a scheme. Let j : R \to U \times _ S U be a pre-equivalence relation over S and g : U' \to U a morphism of schemes over S. Let j' : R' \to U' \times _ S U' be the restriction of j to U'. Assume U, U', R, S are objects of a \tau -site \mathit{Sch}_\tau . The map of quotient sheaves

U'/R' \longrightarrow U/R

is injective. If g defines a surjection h_{U'} \to h_ U of sheaves in the \tau -topology (for example if \{ g : U' \to U\} is a \tau -covering), then U'/R' \to U/R is an isomorphism.

Proof. Suppose \xi , \xi ' \in (U'/R')(T) are sections which map to the same section of U/R. Then we can find a \tau -covering \mathcal{T} = \{ T_ i \to T\} of T such that \xi |_{T_ i}, \xi '|_{T_ i} are given by a_ i, a_ i' \in U'(T_ i). By Lemma 39.20.4 and the axioms of a site we may after refining \mathcal{T} assume there exist morphisms r_ i : T_ i \to R such that g \circ a_ i = s \circ r_ i, g \circ a_ i' = t \circ r_ i. Since by construction R' = R \times _{U \times _ S U} (U' \times _ S U') we see that (r_ i, (a_ i, a_ i')) \in R'(T_ i) and this shows that a_ i and a_ i' define the same section of U'/R' over T_ i. By the sheaf condition this implies \xi = \xi '.

If h_{U'} \to h_ U is a surjection of sheaves, then of course U'/R' \to U/R is surjective also. If \{ g : U' \to U\} is a \tau -covering, then the map of sheaves h_{U'} \to h_ U is surjective, see Sites, Lemma 7.12.4. Hence U'/R' \to U/R is surjective also in this case. \square


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