Lemma 81.12.5. Let $(R \to R', f)$ be a glueing pair, see above. The functor (81.12.0.1) is fully faithful on the full subcategory of algebraic spaces $Y/X$ which are (a) glueable for $(R \to R', f)$ and (b) have affine diagonal $Y \to Y \times _ X Y$.
Proof. Let $Y, Z$ be two algebraic spaces over $X$ which are both glueable for $(R \to R', f)$ and assume the diagonal of $Z$ is affine. Let $a : U \times _ X Y \to U \times _ X Z$ over $U$ and $b : X' \times _ X Y \to X' \times _ X Z$ over $X'$ be two morphisms of algebraic spaces which induce the same morphism $c : U' \times _ X Y \to U' \times _ X Z$ over $U'$. We want to construct a morphism $f : Y \to Z$ over $X$ which produces the morphisms $a$, $b$ on base change to $U$, $X'$. By the faithfulness of Lemma 81.12.4, it suffices to construct the morphism $f$ étale locally on $Y$ (details omitted). Thus we may and do assume $Y$ is affine.
Let $y \in |Y|$ be a point. If $y$ maps into the open $U \subset X$, then $U \times _ X Y$ is an open of $Y$ on which the morphism $f$ is defined (we can just take $a$). Thus we may assume $y$ maps into the closed subset $V(f)$ of $X$. Since $R/fR = R'/fR'$ there is a unique point $y' \in |X' \times _ X Y|$ mapping to $y$. Denote $z' = b(y') \in |X' \times _ X Z|$ and $z \in |Z|$ the images of $y'$. Choose an étale neighbourhood $(W, w) \to (Z, z)$ with $W$ affine. Observe that
\[ (U \times _ X W) \times _{U \times _ X Z, a} (U \times _ X Y),\quad (U' \times _ X W) \times _{U' \times _ X Z, c} (U' \times _ X Y), \]
and
\[ (X' \times _ X W) \times _{X' \times _ X Z, b} (X' \times _ X Y) \]
form an object of $\textit{Spaces}(U \leftarrow U' \to X')$ with affine parts (this is where we use that $Z$ has affine diagonal). Hence by Lemma 81.12.2 there exists a unique affine scheme $V$ glueable for $(R \to R', f)$ such that
\[ (U \times _ X V, U' \times _ X V, X' \times _ X V) \]
is the triple displayed above. By fully faithfulness for the affine case (Lemma 81.12.2) we get a unique morphisms $V \to W$ and $V \to Y$ agreeing with the first and second projection morphisms over $U$ and $X'$ in the construction above. By Lemma 81.12.3 the morphism $V \to Y$ is étale. To finish the proof, it suffices to show that there is a point $v \in |V|$ mapping to $y$ (because then $f$ is defined on an étale neighbourhood of $y$, namely $V$). There is a unique point $w' \in |X' \times _ X W|$ mapping to $w$. By uniqueness $w'$ is mapped to $z'$ under the map $|X' \times _ X W| \to |X' \times _ X Z|$. Then we consider the cartesian diagram
\[ \xymatrix{ X' \times _ X V \ar[r] \ar[d] & X' \times _ X W \ar[d] \\ X' \times _ X Y \ar[r] & X' \times _ X Z } \]
to see that there is a point $v' \in |X' \times _ X V|$ mapping to $y'$ and $w'$, see Properties of Spaces, Lemma 66.4.3. Of course the image $v$ of $v'$ in $|V|$ maps to $y$ and the proof is complete. $\square$
Post a comment
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)