The Stacks project

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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