The Stacks project

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be as in the statement of the lemma. Choose $i_0 \in I$ and replace $I$ by $\{ i \in I \mid i \geq i_0\} $. We may set $B = S = T_{i_0}$ and we may replace $X$ by $X_{T_0}$ and $\mathcal{F}$ by the pullback to $X_{T_0}$. Then $X_ T = \mathop{\mathrm{lim}}\nolimits X_{T_ i}$, see Limits of Spaces, Lemma 70.4.1. Let $\mathcal{F}_ T \to \mathcal{Q}$ be an element of $\text{Q}^{fp}_{\mathcal{F}/X/B}(T)$. By Limits of Spaces, Lemma 70.7.2 there exists an $i$ and a map $\mathcal{F}_{T_ i} \to \mathcal{Q}_ i$ of $\mathcal{O}_{X_{T_ i}}$-modules of finite presentation whose pullback to $X_ T$ is the given quotient map.

We still have to check that, after possibly increasing $i$, the map $\mathcal{F}_{T_ i} \to \mathcal{Q}_ i$ is surjective and $\mathcal{Q}_ i$ is flat over $T_ i$. To do this, choose an affine scheme $U$ and a surjective étale morphism $U \to X$ (see Properties of Spaces, Lemma 66.6.3). We may check surjectivity and flatness over $T_ i$ after pulling back to the étale cover $U_{T_ i} \to X_{T_ i}$ (by definition). This reduces us to the case where $X = \mathop{\mathrm{Spec}}(B_0)$ is an affine scheme of finite presentation over $B = S = T_0 = \mathop{\mathrm{Spec}}(A_0)$. Writing $T_ i = \mathop{\mathrm{Spec}}(A_ i)$, then $T = \mathop{\mathrm{Spec}}(A)$ with $A = \mathop{\mathrm{colim}}\nolimits A_ i$ we have reached the following algebra problem. Let $M_ i \to N_ i$ be a map of finitely presented $B_0 \otimes _{A_0} A_ i$-modules such that $M_ i \otimes _{A_ i} A \to N_ i \otimes _{A_ i} A$ is surjective and $N_ i \otimes _{A_ i} A$ is flat over $A$. Show that for some $i' \geq i$ $M_ i \otimes _{A_ i} A_{i'} \to N_ i \otimes _{A_ i} A_{i'}$ is surjective and $N_ i \otimes _{A_ i} A_{i'}$ is flat over $A$. The first follows from Algebra, Lemma 10.127.5 and the second from Algebra, Lemma 10.168.1. $\square$