The Stacks project

Remark 91.10.8. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $\mathcal{I}_ i$, and $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark 91.10.7. Assume that we are given trivializations $\pi _ i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ such that $\pi _1 = h \circ \pi _2$. In other words, assume $h$ is a morphism of trivialized first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair 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 furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

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

is commutative. In this situation the construction of Remark 91.10.6 induces a commutative diagram

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

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 91.10.7.

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