The Stacks project

Lemma 91.7.6. Let $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ and $g : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ be morphisms of ringed spaces. Let $\mathcal{F}$ be a $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a $\mathcal{O}_ Y$-module. Let $c : \mathcal{F} \to \mathcal{G}$ be a $g$-map. Finally, consider

  1. $0 \to \mathcal{F} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0$ an extension of $f^{-1}\mathcal{O}_ S$-algebras corresponding to $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/S}, \mathcal{F})$, and

  2. $0 \to \mathcal{G} \to \mathcal{O}_{Y'} \to \mathcal{O}_ Y \to 0$ an extension of $g^{-1}f^{-1}\mathcal{O}_ S$-algebras corresponding to $\zeta \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(\mathop{N\! L}\nolimits _{Y/S}, \mathcal{G})$.

See Lemma 91.7.4. Then there is an $S$-morphism $g' : Y' \to X'$ compatible with $g$ and $c$ if and only if $\xi $ and $\zeta $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(Lg^*\mathop{N\! L}\nolimits _{X/S}, \mathcal{G})$.

Proof. The stament makes sense as we have the maps

\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/S}, \mathcal{F}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(Lg^*\mathop{N\! L}\nolimits _{X/S}, Lg^*\mathcal{F}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(Lg^*\mathop{N\! L}\nolimits _{X/S}, \mathcal{G}) \]

using the map $Lg^*\mathcal{F} \to g^*\mathcal{F} \xrightarrow {c} \mathcal{G}$ and

\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(\mathop{N\! L}\nolimits _{Y/S}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}(Lg^*\mathop{N\! L}\nolimits _{X/S}, \mathcal{G}) \]

using the map $Lg^*\mathop{N\! L}\nolimits _{X/S} \to \mathop{N\! L}\nolimits _{Y/S}$. The statement of the lemma can be deduced from Lemma 91.7.1 applied to the diagram

\[ \xymatrix{ & 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{Y'} \ar[r] & \mathcal{O}_ Y \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & \mathcal{O}_ S \ar[u] \ar[r]|\hole & \mathcal{O}_ S \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[ruu] \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_ X \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_ S \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_ S \ar[ruu]|\hole \ar[u] \ar[r] & 0 } \]

and a compatibility between the constructions in the proofs of Lemmas 91.7.4 and 91.7.1 whose statement and proof we omit. (See remark below for a direct argument.) $\square$

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 0GQ1. Beware of the difference between the letter 'O' and the digit '0'.