The Stacks project

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $\mathcal{F}_ i = \mathcal{F}_{T_ i}$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$ (Topologies on Spaces, Lemma 73.9.3) and that $X_{T_ i \times _ T T_{i'}} = X_ i \times _{X_ T} X_{i'}$. Suppose that $\mathcal{F}_ i \to \mathcal{Q}_ i$ is a collection of elements of $\text{Q}_{\mathcal{F}/X/B}(T_ i)$ such that $\mathcal{Q}_ i$ and $\mathcal{Q}_{i'}$ restrict to the same element of $\text{Q}_{\mathcal{F}/X/B}(T_ i \times _ T T_{i'})$. By Remark 99.7.3 we obtain a surjective map of quasi-coherent $\mathcal{O}_{X_ T}$-modules $\mathcal{F}_ T \to \mathcal{Q}$ whose restriction to $X_ i$ recovers the given quotients. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{Q}$ is flat over $T$. Finally, in the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_ i$ are of finite presentation, then Descent on Spaces, Lemma 74.6.2 guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. $\square$

Proof. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $t, s : R \to U$.

Let $v : T \to \text{Q}_{\mathcal{F}/X/B}$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma 65.9.1), we obtain a morphism $h : T \to B$ such that $h_ U = h \circ p$. By Descent on Spaces, Proposition 74.4.1 the quotient $\mathcal{Q}_ U$ descends to a quotient $\mathcal{F}_ T \to \mathcal{Q}$ over $X_ T$. Since $U \to T$ is surjective and flat, it follows from Morphisms of Spaces, Lemma 67.31.5 that $\mathcal{Q}$ is flat over $T$.

Conversely, let $(h, \mathcal{F}_ T \to \mathcal{Q})$ be a pair over $T$. Then we get a natural transformation $v : T \to \text{Q}_{\mathcal{F}/X/B}$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, \mathcal{F}_{T'} \to a^*\mathcal{Q})$. We omit the verification that the construction of this and the previous paragraph are mutually inverse.

In the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$ we add: given a morphism $h : T \to B$, a quasi-coherent sheaf on $X_ T$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module if and only if the pullback to $X_ U$ is of finite presentation as an $\mathcal{O}_{X_ U}$-module. This follows from the fact that $X_ U \to X_ T$ is surjective and étale and Descent on Spaces, Lemma 74.6.2. $\square$

Proof. Let $\mathcal{F}_{i, T} \to \mathcal{Q}_ i$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. By Remark 99.7.3 we obtain a surjection $\mathcal{F}_ T \to \mathcal{Q}$ of quasi-coherent $\mathcal{O}_{X_ T}$-modules whose restriction to $X_{i, T}$ recovers $\mathcal{F}_ i \to \mathcal{Q}_ i$. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{Q}$ is flat over $T$ as desired. In the case of the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_ i$ is of finite presentation, then $\mathcal{Q}$ is of finite presentation too by Descent on Spaces, Lemma 74.6.2. $\square$

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. $\square$

Proof. Let us construct an inverse map. Namely, suppose we have $\mathcal{F}_ Y \to \mathcal{A}$, $\mathcal{F}_{Z'} \to \mathcal{B}'$, and an isomorphism $\mathcal{A}|_{X_ Z} \to \mathcal{B}'|_{X_ Z}$ compatible with the given surjections. Then we apply Pushouts of Spaces, Lemma 81.6.6 to get a quasi-coherent module $\mathcal{A}'$ on $X_{Y'}$ flat over $Y'$. Since this sheaf is constructed as a fibre product (see proof of cited lemma) there is a canonical map $\mathcal{F}_{Y'} \to \mathcal{A}'$. That this map is surjective can be seen because it factors as

and the first arrow is surjective by More on Algebra, Lemma 15.6.5 and the second by More on Algebra, Lemma 15.6.6.

In the case of $Q^{fp}_{\mathcal{F}/X/B}$ all we have to show is that the construction above produces a finitely presented module. This is explained in More on Algebra, Remark 15.7.8 in the commutative algebra setting. The current case of modules over algebraic spaces follows from this by étale localization. $\square$