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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