The Stacks project

Remark 91.10.7. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_ i$. Let $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be a morphism of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. Picture

\[ \xymatrix{ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[ld] \ar[rd] & \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \ar[rr]^ h & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2) } \]

Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. 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. Then there is a canonical functoriality

\[ \left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (08MB) with } \\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (08MB) with } \\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\} \]

Namely, thinking of all sheaves $\mathcal{O}$, $\mathcal{O}'_ i$, $\mathcal{F}$, $\mathcal{K}_ i$, etc as sheaves on $\mathcal{C}$, we set given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

We omit the construction of the $\mathcal{O}'_1$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).

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