The Stacks project

Remark 76.17.7. Another special case of Lemmas 76.17.1, 76.17.2, 76.17.4, and 76.17.5 is where $B$ itself is a thickening $Z \subset Z' = B$ and $Y = Z \times _{Z'} Y'$. Picture

\[ \xymatrix{ (X \subset X') \ar@{..>}[rr]_{(a, ?)} \ar[rd]_{(g, g')} & & (Y \subset Y') \ar[ld]^{(h, h')} \\ & (Z \subset Z') } \]

In this case the map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \times _{Z'} Y'$), hence the pullback $g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma 76.17.4 is just given by the rule

\[ U' \mapsto \{ a' : U' \to Y'\text{ over }Z'\text{ with } a'|_ U = a|_ U\} \]

and we act on this by the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(a^*\Omega _{Y/Z}, \mathcal{C}_{X/X'})$.


Comments (0)


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Your email address will not be published.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).

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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CK7. Beware of the difference between the letter 'O' and the digit '0'.