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Tag 022P

Chapter 38: Groupoid Schemes > Section 38.3: Equivalence relations

Definition 38.3.1. 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 $U$.

    The code snippet corresponding to this tag is a part of the file groupoids.tex and is located in lines 85–105 (see updates for more information).

    \begin{definition}
    \label{definition-equivalence-relation}
    Let $S$ be a scheme. Let $U$ be a scheme over $S$.
    \begin{enumerate}
    \item A {\it 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 $j = (t, s)$.
    \item A {\it relation} on $U$ over $S$ is a monomorphism
    of schemes $j : R \to U \times_S U$.
    \item A {\it 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 $T/S$.
    \item We say a morphism $R \to U \times_S U$ of schemes is
    an {\it 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 $U$.
    \end{enumerate}
    \end{definition}

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