The Stacks project

Remark 91.10.11. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let

\[ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_3), \mathcal{O}'_3) \]

be a complex first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, see Remark 91.10.10. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2, 3$ be pairs consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume given a short exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0 \]

such that

\[ \vcenter { \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathcal{I}_3 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } } \]

are commutative. Finally, assume given an extension

\[ 0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0 \]

as in (91.10.0.1) with $\mathcal{K} = \mathcal{K}_2$ of $\mathcal{O}'_2$-modules with $c_{\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark 91.10.7 to obtain an extension $\mathcal{F}'_1$ of $\mathcal{O}'_1$-modules (we'll describe $\mathcal{F}'_1$ in this special case below). By Remark 91.10.6 using the canonical splitting $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of Remark 91.10.10 we obtain $\xi _{\mathcal{F}'_1} \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_1)$. Finally, we have the obstruction

\[ o(\mathcal{F}, \mathcal{K}_3, c_3) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_3) \]

see Lemma 91.10.4. In this situation we claim that the canonical map

\[ \partial : \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_3) \]

coming from the short exact sequence $0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$ sends $\xi _{\mathcal{F}'_1}$ to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$.

To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$ where $\mathcal{K}$ is an injective $\mathcal{O}$-module. We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$. Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$. Picture:

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an $\mathcal{O}'_3$-module via $\mathcal{O}'_3 \to \mathcal{O}'_2$. Then $c_{\mathcal{E}'_3} = j \circ c_3$. The proof of Lemma 91.10.4 constructs $o(\mathcal{F}, \mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0 \]

On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$ hence the class $\xi _{\mathcal{F}'_1}$ is the class of the extension

\[ 0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0 \]

seen as a sequence of $\mathcal{O}$-modules using $\pi ^\sharp $ where $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

which is $\mathcal{O}$-linear (with the $\mathcal{O}$-module structures given above).

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