Lemma 78.25.1. Notation and assumption as in Lemma 78.21.1. The morphism of quotient stacks

is fully faithful if and only if $R$ is the restriction of $R'$ via the morphism $f : U \to U'$.

Lemma 78.25.1. Notation and assumption as in Lemma 78.21.1. The morphism of quotient stacks

\[ [f] : [U/R] \longrightarrow [U'/R'] \]

is fully faithful if and only if $R$ is the restriction of $R'$ via the morphism $f : U \to U'$.

**Proof.**
Let $x, y$ be objects of $[U/R]$ over a scheme $T/S$. Let $x', y'$ be the images of $x, y$ in the category $[U'/R']_ T$. The functor $[f]$ is fully faithful if and only if the map of sheaves

\[ \mathit{Isom}(x, y) \longrightarrow \mathit{Isom}(x', y') \]

is an isomorphism for every $T, x, y$. We may test this locally on $T$ (in the fppf topology). Hence, by Lemma 78.24.1 we may assume that $x, y$ come from $a, b \in U(T)$. In that case we see that $x', y'$ correspond to $f \circ a, f \circ b$. By Lemma 78.22.1 the displayed map of sheaves in this case becomes

\[ T \times _{(a, b), U \times _ B U} R \longrightarrow T \times _{f \circ a, f \circ b, U' \times _ B U'} R'. \]

This is an isomorphism if $R$ is the restriction, because in that case $R = (U \times _ B U) \times _{U' \times _ B U'} R'$, see Lemma 78.17.3 and its proof. Conversely, if the last displayed map is an isomorphism for all $T, a, b$, then it follows that $R = (U \times _ B U) \times _{U' \times _ B U'} R'$, i.e., $R$ is the restriction of $R'$. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (3)

Comment #1813 by OS on

Comment #3216 by William Chen on

Comment #3318 by Johan on