Remark 91.4.10 (08LF)—The Stacks project

Remark 91.4.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module. Let

$(X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3})$

be a complex first order thickenings of $(X, \mathcal{O}_ X)$ , see Remark 91.4.9. Let $(\mathcal{K}_ i, c_ i)$ , $i = 1, 2, 3$ be pairs consisting of an $\mathcal{O}_ X$ -module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K}_ i$ . Assume given a short exact sequence of $\mathcal{O}_ X$ -modules

$0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$

such that

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

are commutative. Finally, assume given an extension

$0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0$

as in (91.4.0.1) with $\mathcal{K} = \mathcal{K}_2$ of $\mathcal{O}_{X'_2}$ -modules with $c_{\mathcal{F}'_2} = c_2$ . In this situation we can apply the functoriality of Remark 91.4.7 to obtain an extension $\mathcal{F}'_1$ on $X'_1$ (we'll describe $\mathcal{F}'_1$ in this special case below). By Remark 91.4.6 using the canonical splitting $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_ X)$ of Remark 91.4.9 we obtain $\xi _{\mathcal{F}'_1} \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}_1)$ . Finally, we have the obstruction

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

see Lemma 91.4.4. In this situation we claim that the canonical map

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

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 $\mathcal{O}_ X$ -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 $\mathcal{O}_{X'_3}$ -module via $\mathcal{O}_{X'_3} \to \mathcal{O}_{X'_2}$ . Then $c_{\mathcal{E}'_3} = j \circ c_3$ . The proof of Lemma 91.4.4 constructs $o(\mathcal{F}, \mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\mathcal{O}_ X$ -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 $\mathcal{O}_ X$ -modules using $\pi ^\sharp$ where $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_ X)$ 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 $\mathcal{O}_ X$ -linear (with the $\mathcal{O}_ X$ -module structures given above).

The Stacks project

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