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 $\psi : \mathcal{K} \to \mathcal{L}$.

  1. 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.

  2. 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. $\square$

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View Lemma 91.4.1 as pdf