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 $U$.

Comment #6527 by Jeroen Hekking on

Is there a typo in (4)? I guess an equivalence relation is a pre-equivalence relation which is a relation. The latter seems to be missing, or I'm missing something obvious.

Comment #6528 by on

Sorry, but what would be the typo? The condition as currently formulated in (4) is equivalent to (2) + (3) as currently formulated. Right?

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