The Stacks project

Definition 39.19.1. Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$.

  1. A subset $W \subset U$ is set-theoretically $R$-invariant if $t(s^{-1}(W)) \subset W$.

  2. An open $W \subset U$ is $R$-invariant if $t(s^{-1}(W)) \subset W$.

  3. A closed subscheme $Z \subset U$ is called $R$-invariant if $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see Schemes, Definition 26.17.7.

  4. A monomorphism of schemes $T \to U$ is $R$-invariant if $T \times _{U, t} R = R \times _{s, U} T$ as schemes over $R$.

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View Definition 39.19.1 as pdf