The Stacks project

Remark 91.4.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(X, \mathcal{O}_ X) \to (X'_ i, \mathcal{O}_{X'_ i})$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_ i$. Let $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be a morphism of first order thickenings of $(X, \mathcal{O}_ X)$. Picture

\[ \xymatrix{ & (X, \mathcal{O}_ X) \ar[ld] \ar[rd] & \\ (X'_1, \mathcal{O}_{X'_1}) \ar[rr]^ h & & (X'_2, \mathcal{O}_{X'_2}) } \]

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

\[ \left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (08L4) with } \\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (08L4) with } \\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\} \]

Namely, thinking of all sheaves $\mathcal{O}_ X$, $\mathcal{O}_{X'_ i}$, $\mathcal{F}$, $\mathcal{K}_ i$, etc as sheaves on $X$, 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 $\mathcal{O}_{X'_1}$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).

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