Lemma 91.13.6. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be morphisms of ringed topoi. Let $\mathcal{F}$ be a $\mathcal{O}_\mathcal {C}$-module. Let $\mathcal{G}$ be a $\mathcal{O}_\mathcal {D}$-module. Let $c : g^*\mathcal{F} \to \mathcal{G}$ be a $\mathcal{O}_\mathcal {D}$-linear map. Finally, consider
$0 \to \mathcal{F} \to \mathcal{O}_{\mathcal{C}'} \to \mathcal{O}_\mathcal {C} \to 0$ an extension of $f^{-1}\mathcal{O}_\mathcal {B}$-algebras corresponding to $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {C}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{F})$, and
$0 \to \mathcal{G} \to \mathcal{O}_{\mathcal{D}'} \to \mathcal{O}_\mathcal {D} \to 0$ an extension of $g^{-1}f^{-1}\mathcal{O}_\mathcal {B}$-algebras corresponding to $\zeta \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}, \mathcal{G})$.
See Lemma 91.13.4. Then there is a morphism
\[ g' : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_{\mathcal{D}'}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_{\mathcal{C}'}) \]
of ringed topoi over $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ compatible with $g$ and $c$ if and only if $\xi $ and $\zeta $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{G})$.
Proof.
The stament makes sense as we have the maps
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {C}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{F}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, Lg^*\mathcal{F}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}} (Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{G}) \]
using the map $Lg^*\mathcal{F} \to g^*\mathcal{F} \xrightarrow {c} \mathcal{G}$ and
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ Y}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{G}) \]
using the map $Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}} \to \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}$. The statement of the lemma can be deduced from Lemma 91.13.1 applied to the diagram
\[ \xymatrix{ & 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal {D} \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & g^{-1}f^{-1}\mathcal{O}_\mathcal {B} \ar[u] \ar[r]|\hole & g^{-1}f^{-1}\mathcal{O}_\mathcal {B} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[ruu] \ar[r] & \mathcal{O}_{\mathcal{C}'} \ar[r] & \mathcal{O}_\mathcal {C} \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal {B} \ar[ruu]|\hole \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal {B} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } \]
and a compatibility between the constructions in the proofs of Lemmas 91.13.4 and 91.13.1 whose statement and proof we omit. (See remark below for a direct argument.)
$\square$
Comments (0)