Lemma 81.12.6. Let $(R \to R', f)$ be a glueing pair, see above. Any object $(V, V', Y')$ of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ quasi-affine is isomorphic to the image under the functor (81.12.0.1) of a separated algebraic space $Y$ over $X$.
Proof. Choose $n'$, $T' \to Y'$ and $n_1$, $T_1 \to V$ as in Properties, Lemma 28.18.6. Picture
\[ \xymatrix{ & & T_1 \times _ V V' \times _ Y T' \ar[ld] \ar[rd] \\ T_1 \ar[d] & T_1 \times _ V V' \ar[l] \ar[dr] & & V' \times _{Y'} T' \ar[r] \ar[dl] & T' \ar[d] \\ V & & V' \ar[rr] \ar[ll] & & Y' } \]
Observe that $T_1 \times _ V V'$ and $V' \times _{Y'} T'$ are affine (namely the morphisms $V' \to V$ and $V' \to Y'$ are affine as base changes of the affine morphisms $U' \to U$ and $U' \to X'$). By construction we see that
\[ \mathbf{A}^{n'}_{T_1 \times _ V V'} \cong T_1 \times _ V V' \times _{Y'} T' \cong \mathbf{A}^{n_1}_{V' \times _{Y'} T'} \]
In other words, the affine schemes $\mathbf{A}^{n'}_{T_1}$ and $\mathbf{A}^{n_1}_{T'}$ are part of a triple making an affine object of $\textit{Spaces}(U \leftarrow U' \to X')$. By Lemma 81.12.2 there exists a morphism of affine schemes $T \to X$ and isomorphisms $U \times _ X T \cong \mathbf{A}^{n'}_{T_1}$ and $X' \times _ X T \cong \mathbf{A}^{n_1}_{T'}$ compatible with the isomorphisms displayed above. These isomorphisms produce morphisms
\[ U \times _ X T \longrightarrow V \quad \text{and}\quad X' \times _ X T \longrightarrow Y' \]
satisfying the property of Properties, Lemma 28.18.6 with $n = n' + n_1$ and moreover define a morphism from the triple $(U \times _ X T, U' \times _ X T, X' \times _ X T)$ to our triple $(V, V', Y')$ in the category $\textit{Spaces}(U \leftarrow U' \to X')$.
By Lemma 81.12.2 there is an affine scheme $W$ whose image in $\textit{Spaces}(U \leftarrow U' \to X')$ is isomorphic to the triple
\[ ((U \times _ X T) \times _ V (U \times _ X T), (U' \times _ X T) \times _{V'} (U' \times _ X T), (X' \times _ X T) \times _{Y'} (X' \times _ X T)) \]
By fully faithfulness of this construction, we obtain two maps $p_0, p_1 : W \to T$ whose base changes to $U, U', X'$ are the projection morphisms. By Lemma 81.12.3 the morphisms $p_0, p_1$ are flat and of finite presentation and the morphism $(p_0, p_1) : W \to T \times _ X T$ is a closed immersion. In fact, $W \to T \times _ X T$ is an equivalence relation: by the lemmas used above we may check symmetry, reflexivity, and transitivity after base change to $U$ and $X'$, where these are obvious (details omitted). Thus the quotient sheaf
\[ Y = T/W \]
is an algebraic space for example by Bootstrap, Theorem 80.10.1. Since it is clear that $Y/X$ is sent to the triple $(V, V', Y')$. The base change of the diagonal $\Delta : Y \to Y \times _ X Y$ by the quasi-compact surjective flat morphism $T \times _ X T \to Y \times _ X Y$ is the closed immersion $W \to T \times _ X T$. Thus $\Delta $ is a closed immersion by Descent on Spaces, Lemma 74.11.17. Thus the algebraic space $Y$ is separated 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 (4)
Comment #4313 by comment_bot on
Comment #4472 by Johan on
Comment #4956 by Laurent Moret-Bailly on
Comment #4959 by comment_bot on