Remark 99.7.9 (0CZU): Obstructions for quotients

Remark 99.7.9 (Obstructions for quotients). In Situation 99.7.1 assume that $\mathcal{F}$ is flat over $B$ . Let $T \subset T'$ be an first order thickening of schemes over $B$ with ideal sheaf $\mathcal{J}$ . Then $X_ T \subset X_{T'}$ is a first order thickening of algebraic spaces whose ideal sheaf $\mathcal{I}$ is a quotient of $f_ T^*\mathcal{J}$ . We will think of sheaves on $X_{T'}$ , resp. $T'$ as sheaves on $X_ T$ , resp. $T$ using the fundamental equivalence described in More on Morphisms of Spaces, Section 76.9. Let

$0 \to \mathcal{K} \to \mathcal{F}_ T \to \mathcal{Q} \to 0$

define an element $x$ of $Q_{\mathcal{F}/X/B}(T)$ . Since $\mathcal{F}_{T'}$ is flat over $T'$ we have a short exact sequence

$0 \to f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T \xrightarrow {i} \mathcal{F}_{T'} \xrightarrow {\pi } \mathcal{F}_ T \to 0$

and we have $f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T = \mathcal{I} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T$ , see Deformation Theory, Lemma 91.11.2. Let us use the abbreviation $f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{G} = \mathcal{G} \otimes _{\mathcal{O}_ T} \mathcal{J}$ for an $\mathcal{O}_{X_ T}$ -module $\mathcal{G}$ . Since $\mathcal{Q}$ is flat over $T$ , we obtain a short exact sequence

$0 \to \mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{F}_ T \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \to 0$

Combining the above we obtain an canonical extension

$0 \to \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \pi ^{-1}(\mathcal{K})/i(\mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J}) \to \mathcal{K} \to 0$

of $\mathcal{O}_{X_ T}$ -modules. This defines a canonical class

$o_ x(T') \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X_ T}}(\mathcal{K}, \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J})$

If $o_ x(T')$ is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a $\mathcal{O}_{X_{T'}}$ -submodule $\mathcal{K}' \subset \pi ^{-1}(\mathcal{K})$ sitting in a short exact sequence $0 \to \mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{K}' \to \mathcal{K} \to 0$ . Then it follows from the lemma reference above that $\mathcal{Q}' = \mathcal{F}_{T'}/\mathcal{K}'$ is a lift of $x$ to an element of $Q_{\mathcal{F}/X/B}(T')$ . Conversely, the reader sees that the existence of a lift implies that $o_ x(T')$ is zero. Moreover, if $x \in Q_{\mathcal{F}/X/B}^{fp}(T)$ , then automatically $x' \in Q_{\mathcal{F}/X/B}^{fp}(T')$ by Deformation Theory, Lemma 91.11.3. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).

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