Remark 91.4.8 (08LD)—The Stacks project

Remark 91.4.8. Let $(X, \mathcal{O}_ X)$ , $(X, \mathcal{O}_ X) \to (X'_ i, \mathcal{O}_{X'_ i})$ , $\mathcal{I}_ i$ , and $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be as in Remark 91.4.7. Assume that we are given trivializations $\pi _ i : X'_ i \to X$ such that $\pi _1 = h \circ \pi _2$ . In other words, assume $h$ is a morphism of trivialized first order thickening of $(X, \mathcal{O}_ X)$ . Let $(\mathcal{K}_ i, c_ i)$ , $i = 1, 2$ be a pair 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 furthermore given a map of $\mathcal{O}_ X$ -modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

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

is commutative. In this situation the construction of Remark 91.4.6 induces a commutative diagram

$\xymatrix{ \{ \mathcal{F}'_2\text{ as in (08L4) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{ \mathcal{F}'_1\text{ as in (08L4) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}_1) }$

where the vertical map on the right is given by functoriality of $\mathop{\mathrm{Ext}}\nolimits$ and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left is the one from Remark 91.4.7.

The Stacks project

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