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.) $\square$


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