The Stacks project

Proof. (This lemma is the analogue of Lemma 32.8.7 for modules.) For $i' \geq i$ denote $X_{i'} = S_{i'} \times _{S_ i} X_ i$, $\mathcal{F}_{i'} = (X_{i'} \to X_ i)^*\mathcal{F}_ i$ and similarly for $Y_{i'}$. Denote $\varphi _{i'}$ the base change of $\varphi _ i$ to $S_{i'}$. Also set $X = S \times _{S_ i} X_ i$, $Y =S \times _{S_ i} X_ i$, $\mathcal{F} = (X \to X_ i)^*\mathcal{F}_ i$ and $\varphi $ the base change of $\varphi _ i$ to $S$. Let $Y_ i = \bigcup _{j = 1, \ldots , m} V_{j, i}$ be a finite affine open covering such that each $V_{j, i}$ maps into some affine open of $S_ i$. For each $j = 1, \ldots m$ let $\varphi _ i^{-1}(V_{j, i}) = \bigcup _{k = 1, \ldots , m(j)} U_{k, j, i}$ be a finite affine open covering. For $i' \geq i$ we denote $V_{j, i'}$ the inverse image of $V_{j, i}$ in $Y_{i'}$ and $U_{k, j, i'}$ the inverse image of $U_{k, j, i}$ in $X_{i'}$. Similarly we have $U_{k, j} \subset X$ and $V_ j \subset Y$. Then $U_{k, j} = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} U_{k, j, i'}$ and $V_ j = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} V_ j$ (see Lemma 32.2.2). Since $X_{i'} = \bigcup _{k, j} U_{k, j, i'}$ is a finite open covering it suffices to prove the lemma for each of the morphisms $U_{k, j, i} \to V_{j, i}$ and the sheaf $\mathcal{F}_ i|_{U_{k, j, i}}$. Hence we see that the lemma reduces to the case that $X_ i$ and $Y_ i$ are affine and map into an affine open of $S_ i$, i.e., we may also assume that $S$ is affine.

In the affine case we reduce to the following algebra result. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$. For some $i \in I$ suppose given a map $A_ i \to B_ i$ of finitely presented $R_ i$-algebras. Let $N_ i$ be a finitely presented $B_ i$-module. Then, if $R \otimes _{R_ i} N_ i$ is flat over $R \otimes _{R_ i} A_ i$, then for some $i' \geq i$ the module $R_{i'} \otimes _{R_ i} N_ i$ is flat over $R_{i'} \otimes _{R_ i} A$. This is exactly the result proved in Algebra, Lemma 10.168.1 part (3). $\square$