Lemma 38.32.3. Let $S$ be a quasi-compact and quasi-separated scheme. The collection of morphisms $(u, \overline{u}) : (X', \overline{X}') \to (X, \overline{X})$ such that $u$ is an isomorphism forms a right multiplicative system (Categories, Definition 4.27.1) of arrows in the category of compactifications.
Proof. Axiom RMS1 is trivial to verify. Let us check RMS2 holds. Suppose given a diagram
\[ \xymatrix{ & (X', \overline{X}') \ar[d]_{(u, \overline{u})} \\ (Y, \overline{Y}) \ar[r]^{(f, \overline{f})} & (X, \overline{X}) } \]
with $u : X' \to X$ an isomorphism. Then we let $Y' = Y \times _ X X'$ with the projection map $v : Y' \to Y$ (an isomorphism). We also set $\overline{Y}' = \overline{Y} \times _{\overline{X}} \overline{X}'$ with the projection map $\overline{v} : \overline{Y}' \to \overline{Y}$ It is clear that $Y' \to \overline{Y}'$ is an open immersion. The diagram
\[ \xymatrix{ (Y', \overline{Y}') \ar[r]_{(g, \overline{g})} \ar[d]_{(v, \overline{v})} & (X', \overline{X}') \ar[d]_{(u, \overline{u})} \\ (Y, \overline{Y}) \ar[r]^{(f, \overline{f})} & (X, \overline{X}) } \]
shows that axiom RMS2 holds.
Let us check RMS3 holds. Suppose given a pair of morphisms $(f, \overline{f}), (g, \overline{g}) : (X, \overline{X}) \to (Y, \overline{Y})$ of compactifications and a morphism $(v, \overline{v}) : (Y, \overline{Y}) \to (Y', \overline{Y}')$ such that $v$ is an isomorphism and such that $(v, \overline{v}) \circ (f, \overline{f}) = (v, \overline{v}) \circ (g, \overline{g})$. Then $f = g$. Hence if we let $\overline{X}' \subset \overline{X}$ be the equalizer of $\overline{f}$ and $\overline{g}$, then $(u, \overline{u}) : (X, \overline{X}') \to (X, \overline{X})$ will be a morphism of the category of compactifications such that $(f, \overline{f}) \circ (u, \overline{u}) = (g, \overline{g}) \circ (u, \overline{u})$ as desired. $\square$
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Your email address will not be published.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)
There are also: