Lemma 91.4.1 (08L5)—The Stacks project

Lemma 91.4.1. Let $i : (X, \mathcal{O}_ X) \to (X', \mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given extensions

$0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad \text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0$

as in (91.4.0.1) and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{K} \to \mathcal{L}$ .

If there exists an $\mathcal{O}_{X'}$ -module map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$ , then the diagram

$\xymatrix{ \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi } & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} }$

is commutative.
The set of $\mathcal{O}_{X'}$ -module maps $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$ is, if nonempty, a principal homogeneous space under $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{L})$ .

Proof. Part (1) is immediate from the description of the maps. For (2), if $\varphi '$ and $\varphi ''$ are two maps $\mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$ and $\psi$ , then $\varphi ' - \varphi ''$ factors as

$\mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}'$

The map in the middle comes from a unique element of $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{L})$ by Modules, Lemma 17.13.4. Conversely, given an element $\alpha$ of this group we can add the composition (as displayed above with $\alpha$ in the middle) to $\varphi '$ . Some details omitted. $\square$

The Stacks project

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