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Remark 99.7.10 (Deformations of quotients). In Situation 99.7.1 assume that $\mathcal{F}$ is flat over $B$. We continue the discussion of Remark 99.7.9. Assume $o_ x(T') = 0$. Then we claim that the set of lifts $x' \in Q_{\mathcal{F}/X/B}(T')$ is a principal homogeneous space under the group

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ T}}(\mathcal{K}, \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J}) \]

Namely, given any $\mathcal{F}_{T'} \to \mathcal{Q}'$ flat over $T'$ lifting the quotient $\mathcal{Q}$ we obtain a commutative diagram with exact rows and columns

\[ \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K} \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{F}_ T \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{Q} \otimes \mathcal{J} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] \ar[d] & \mathcal{F}_{T'} \ar[r] \ar[d] & \mathcal{Q}' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[d] \ar[r] & \mathcal{F}_ T \ar[d] \ar[r] & \mathcal{Q} \ar[d] \ar[r] & 0 \\ & 0 & 0 & 0 } \]

(to see this use the observations made in the previous remark). Given a map $\varphi : \mathcal{K} \to \mathcal{Q} \otimes \mathcal{J}$ we can consider the subsheaf $\mathcal{K}'_\varphi \subset \mathcal{F}_{T'}$ consisting of those local sections $s$ whose image in $\mathcal{F}_ T$ is a local section $k$ of $\mathcal{K}$ and whose image in $\mathcal{Q}'$ is the local section $\varphi (k)$ of $\mathcal{Q} \otimes \mathcal{J}$. Then set $\mathcal{Q}'_\varphi = \mathcal{F}_{T'}/\mathcal{K}'_\varphi $. Conversely, any second lift of $x$ corresponds to one of the qotients constructed in this manner. 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).

Comments (2)

Comment #3781 by Aron Heleodoro on

In the Remark 0CZV. "to one the qotients constructed" should be "to one of the quotients constructed".

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