The Stacks project

77.18 Invariant subspaces

In this section we discuss briefly the notion of an invariant subspace.

Definition 77.18.1. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over the base $B$.

  1. We say an open subspace $W \subset U$ is $R$-invariant if $t(s^{-1}(W)) \subset W$.

  2. A locally closed subspace $Z \subset U$ is called $R$-invariant if $t^{-1}(Z) = s^{-1}(Z)$ as locally closed subspaces of $R$.

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

For an open subspace $W \subset U$ the $R$-invariance is also equivalent to requiring that $s^{-1}(W) = t^{-1}(W)$. If $W \subset U$ is $R$-invariant then the restriction of $R$ to $W$ is just $R_ W = s^{-1}(W) = t^{-1}(W)$. Similarly, if $Z \subset U$ is an $R$-invariant locally closed subspace, then the restriction of $R$ to $Z$ is just $R_ Z = s^{-1}(Z) = t^{-1}(Z)$.

Lemma 77.18.2. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.

  1. If $s$ and $t$ are open, then for every open $W \subset U$ the open $s(t^{-1}(W))$ is $R$-invariant.

  2. If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subspaces.

Proof. Assume $s$ and $t$ open and $W \subset U$ open. Since $s$ is open we see that $W' = s(t^{-1}(W))$ is an open subspace of $U$. Now it is quite easy to using the functorial point of view that this is an $R$-invariant open subset of $U$, but we are going to argue this directly by some diagrams, since we think it is instructive. Note that $t^{-1}(W')$ is the image of the morphism

\[ A := t^{-1}(W) \times _{s|_{t^{-1}(W)}, U, t} R \xrightarrow {\text{pr}_1} R \]

and that $s^{-1}(W')$ is the image of the morphism

\[ B := R \times _{s, U, s|_{t^{-1}(W)}} t^{-1}(W) \xrightarrow {\text{pr}_0} R. \]

The algebraic spaces $A$, $B$ on the left of the arrows above are open subspaces of $R \times _{s, U, t} R$ and $R \times _{s, U, s} R$ respectively. By Lemma 77.11.4 the diagram

\[ \xymatrix{ R \times _{s, U, t} R \ar[rd]_{\text{pr}_1} \ar[rr]_{(\text{pr}_1, c)} & & R \times _{s, U, s} R \ar[ld]^{\text{pr}_0} \\ & R & } \]

is commutative, and the horizontal arrow is an isomorphism. Moreover, it is clear that $(\text{pr}_1, c)(A) = B$. Hence we conclude $s^{-1}(W') = t^{-1}(W')$, and $W'$ is $R$-invariant. This proves (1).

Assume now that $s$, $t$ are both open and quasi-compact. Then, if $W \subset U$ is a quasi-compact open, then also $W' = s(t^{-1}(W))$ is a quasi-compact open, and invariant by the discussion above. Letting $W$ range over images of affines ├ętale over $U$ we see (2). $\square$

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