Remark 91.13.7 (0GQ8)—The Stacks project

Remark 91.13.7. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$, $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$, $\mathcal{F}$, $\mathcal{G}$, $c : g^*\mathcal{F} \to \mathcal{G}$, $0 \to \mathcal{F} \to \mathcal{O}_{\mathcal{C}'} \to \mathcal{O}_\mathcal {C} \to 0$, $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {C}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{F})$, $0 \to \mathcal{G} \to \mathcal{O}_{\mathcal{D}'} \to \mathcal{O}_\mathcal {D} \to 0$, and $\zeta \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}, \mathcal{G})$ be as in Lemma 91.13.6. Using pushout along $c : g^{-1}\mathcal{F} \to \mathcal{G}$ we can construct an extension

\[ \xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}'_1 \ar[r] & g^{-1}\mathcal{O}_\mathcal {C} \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{F} \ar[u]^ c \ar[r] & g^{-1}\mathcal{O}_{\mathcal{C}'} \ar[u] \ar[r] & g^{-1}\mathcal{O}_\mathcal {C} \ar@{=}[u] \ar[r] & 0 } \]

Using pullback along $g^\sharp : g^{-1}\mathcal{O}_\mathcal {C} \to \mathcal{O}_\mathcal {D}$ we can construct an extension

\[ \xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal {D} \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G} \ar@{=}[u] \ar[r] & \mathcal{O}'_2 \ar[u] \ar[r] & g^{-1}\mathcal{O}_\mathcal {C} \ar[u] \ar[r] & 0 } \]

A diagram chase tells us that there exists a morphism $g' : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_{\mathcal{D}'}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_{\mathcal{C}'})$ over $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ compatible with $g$ and $c$ if and only if $\mathcal{O}'_1$ is isomorphic to $\mathcal{O}'_2$ as $g^{-1}f^{-1}\mathcal{O}_\mathcal {B}$-algebra extensions of $g^{-1}\mathcal{O}_\mathcal {C}$ by $\mathcal{G}$. By Lemma 91.13.4 these extensions are classified by the LHS of

\[ \mathop{\mathrm{Ext}}\nolimits ^1_{g^{-1}\mathcal{O}_\mathcal {C}}( \mathop{N\! L}\nolimits _{g^{-1}\mathcal{O}_\mathcal {C}/g^{-1}f^{-1}\mathcal{O}_\mathcal {B}}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {D}}( Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}}, \mathcal{G}) \]

Here the equality comes from tensor-hom adjunction and the equalities

\[ \mathop{N\! L}\nolimits _{g^{-1}\mathcal{O}_\mathcal {C}/g^{-1}f^{-1}\mathcal{O}_\mathcal {B}} = g^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}} \quad \text{and}\quad Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}} = g^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}} \otimes _{g^{-1}\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ Y \]

For the first of these see Modules on Sites, Lemma 18.35.3; the second follows from the definition of derived pullback. Thus, in order to see that Lemma 91.13.6 is true, it suffices to show that $\mathcal{O}'_1$ corresponds to the image of $\xi $ and that $\mathcal{O}'_2$ correspond to the image of $\zeta $. The correspondence between $\xi $ and $\mathcal{O}'_1$ is immediate from the construction of the class $\xi $ in Remark 91.13.5. For the correspondence between $\zeta $ and $\mathcal{O}'_2$, we first choose a commutative diagram

\[ \xymatrix{ \mathcal{E} \ar[d]_{\beta '} \ar[rd]^\beta \\ \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal {D} } \]

such that $g^{-1}f^{-1}\mathcal{O}_\mathcal {B}[\mathcal{E}] \to \mathcal{O}_\mathcal {D}$ is surjective with kernel $\mathcal{K}$. Next choose a commutative diagram

\[ \xymatrix{ \mathcal{E} \ar[d]_{\beta '} & \mathcal{E}' \ar[l]^\varphi \ar[d]_{\alpha '} \ar[rd]^\alpha \\ \mathcal{O}_{\mathcal{D}'} & \mathcal{O}'_2 \ar[l] \ar[r] & g^{-1}\mathcal{O}_\mathcal {C} } \]

such that $g^{-1}f^{-1}\mathcal{O}_\mathcal {B}[\mathcal{E}'] \to g^{-1}\mathcal{O}_\mathcal {C}$ is surjective with kernel $\mathcal{J}$. (For example just take $\mathcal{E}' = \mathcal{E} \amalg \mathcal{O}'_2$ as a sheaf of sets.) The map $\varphi $ induces a map of complexes $\mathop{N\! L}\nolimits (\alpha ) \to \mathop{N\! L}\nolimits (\beta )$ (notation as in Modules, Section 17.31) and in particular $\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2$. Then $\mathop{N\! L}\nolimits (\alpha ) \cong \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}$ and $\mathop{N\! L}\nolimits (\beta ) \cong \mathop{N\! L}\nolimits _{g^{-1}\mathcal{O}_\mathcal {C}/g^{-1}f^{-1}\mathcal{O}_\mathcal {B}}$ and the map of complexes $\mathop{N\! L}\nolimits (\alpha ) \to \mathop{N\! L}\nolimits (\beta )$ represents the map $Lg^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {B}} \to \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {D}/\mathcal{O}_\mathcal {B}}$ used in the statement of Lemma 91.13.6 (see first part of its proof). Now $\zeta $ corresponds to the class of the map $\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}$ induced by $\beta '$, see Remark 91.13.5. Similarly, the extension $\mathcal{O}'_2$ corresponds to the map $\mathcal{J}/\mathcal{J}^2 \to \mathcal{G}$ induced by $\alpha '$. The commutative diagram above shows that this map is the composition of the map $\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}$ induced by $\beta '$ with the map $\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2$. This proves the compatibility we were looking for.

The Stacks project

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