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Definition 39.3.1. Let $S$ be a scheme. Let $U$ be a scheme over $S$.

  1. A pre-relation on $U$ over $S$ is any morphism of schemes $j : R \to U \times _ S U$. In this case we set $t = \text{pr}_0 \circ j$ and $s = \text{pr}_1 \circ j$, so that $j = (t, s)$.

  2. A relation on $U$ over $S$ is a monomorphism of schemes $j : R \to U \times _ S U$.

  3. A pre-equivalence relation is a pre-relation $j : R \to U \times _ S U$ such that the image of $j : R(T) \to U(T) \times U(T)$ is an equivalence relation for all $T/S$.

  4. We say a morphism $R \to U \times _ S U$ of schemes is an equivalence relation on $U$ over $S$ if and only if for every scheme $T$ over $S$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$.

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