The Stacks project

Lemma 70.3.8. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that

  1. $F$ is a sheaf, and

  2. there exists an fppf covering $\{ U_ j \to S\} _{j \in J}$ such that $F|_{(\mathit{Sch}/U_ j)_{fppf}}$ is limit preserving.

Then $F$ is limit preserving.

Proof. Let $T$ be an affine scheme over $S$. Let $I$ be a directed set, and let $T_ i$ be an inverse system of affine schemes over $S$ such that $T = \mathop{\mathrm{lim}}\nolimits T_ i$. We have to show that the canonical map $\mathop{\mathrm{colim}}\nolimits F(T_ i) \to F(T)$ is bijective.

Choose some $0 \in I$ and choose a standard fppf covering $\{ V_{0, k} \to T_{0}\} _{k = 1, \ldots , m}$ which refines the pullback $\{ U_ j \times _ S T_0 \to T_0\} $ of the given fppf covering of $S$. For each $i \geq 0$ we set $V_{i, k} = T_ i \times _{T_0} V_{0, k}$, and we set $V_ k = T \times _{T_0} V_{0, k}$. Note that $V_ k = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} V_{i, k}$, see Limits, Lemma 32.2.3.

Suppose that $x, x' \in \mathop{\mathrm{colim}}\nolimits F(T_ i)$ map to the same element of $F(T)$. Say $x, x'$ are given by elements $x_ i, x'_ i \in F(T_ i)$ for some $i \in I$ (we may choose the same $i$ for both as $I$ is directed). By assumption (2) and the fact that $x_ i, x'_ i$ map to the same element of $F(T)$ this implies that

\[ x_ i|_{V_{i', k}} = x'_ i|_{V_{i', k}} \]

for some suitably large $i' \in I$. We can choose the same $i'$ for each $k$ as $k \in \{ 1, \ldots , m\} $ ranges over a finite set. Since $\{ V_{i', k} \to T_{i'}\} $ is an fppf covering and $F$ is a sheaf this implies that $x_ i|_{T_{i'}} = x'_ i|_{T_{i'}}$ as desired. This proves that the map $\mathop{\mathrm{colim}}\nolimits F(T_ i) \to F(T)$ is injective.

To show surjectivity we argue in a similar fashion. Let $x \in F(T)$. By assumption (2) for each $k$ we can choose a $i$ such that $x|_{V_ k}$ comes from an element $x_{i, k} \in F(V_{i, k})$. As before we may choose a single $i$ which works for all $k$. By the injectivity proved above we see that

\[ x_{i, k}|_{V_{i', k} \times _{T_{i'}} V_{i', l}} = x_{i, l}|_{V_{i', k} \times _{T_{i'}} V_{i', l}} \]

for some large enough $i'$. Hence by the sheaf condition of $F$ the elements $x_{i, k}|_{V_{i', k}}$ glue to an element $x_{i'} \in F(T_{i'})$ as desired. $\square$

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