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The Stacks project

Lemma 39.20.4. 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. Assume U, R, S are objects of a \tau -site \mathit{Sch}_\tau . For T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau ) and a, b \in U(T) the following are equivalent:

  1. a and b map to the same element of (U/R)(T), and

  2. there exists a \tau -covering \{ f_ i : T_ i \to T\} of T and morphisms r_ i : T_ i \to R such that a \circ f_ i = s \circ r_ i and b \circ f_ i = t \circ r_ i.

In other words, in this case the map of \tau -sheaves

h_ R \longrightarrow h_ U \times _{U/R} h_ U

is surjective.

Proof. Omitted. Hint: The reason this works is that the presheaf (39.20.0.1) in this case is really given by T \mapsto U(T)/j(R(T)) as j(R(T)) \subset U(T) \times U(T) is an equivalence relation, see Definition 39.3.1. \square


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