The Stacks project

Lemma 83.3.3. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times _ B U$ be a pre-relation of algebraic spaces over $B$. Let $U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Let $X' \to X$ be any morphism of algebraic spaces.

  1. Setting $U' = X' \times _ X U$, $R' = X' \times _ X R$ we obtain a pre-relation $j' : R' \to U' \times _ B U'$.

  2. If $j$ is a relation, then $j'$ is a relation.

  3. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.

  4. If $j$ is an equivalence relation, then $j'$ is an equivalence relation.

  5. If $j$ comes from a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$, then

    1. $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $X$, and

    2. $j'$ comes from the base change $(U', R', s', t', c')$ of this groupoid to $X'$, see Groupoids in Spaces, Lemma 78.11.6.

  6. If $j$ comes from the action of a group algebraic space $G/B$ on $U$ as in Groupoids in Spaces, Lemma 78.15.1 then $j'$ comes from the induced action of $G$ on $U'$.

Proof. Omitted. Hint: Functorial point of view combined with the picture:

\[ \xymatrix{ R' = X' \times _ X R \ar[dd] \ar[rr] \ar[rd] & & X' \times _ X U = U' \ar '[d][dd] \ar[rd] \\ & R \ar[dd] \ar[rr] & & U \ar[dd] \\ U' = X' \times _ X U \ar '[r][rr] \ar[rd] & & X' \ar[rd] \\ & U \ar[rr] & & X } \]
$\square$

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