The Stacks project

Lemma 69.3.7. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. If $F$ is limit preserving then its sheafification $F^\# $ is limit preserving.

Proof. Assume $F$ is limit preserving. It suffices to show that $F^+$ is limit preserving, since $F^\# = (F^+)^+$, see Sites, Theorem 7.10.10. Let $T$ be an affine scheme over $S$, and let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be written as the directed limit of an inverse system of affine $S$ schemes. Recall that $F^+(T)$ is the colimit of $\check H^0(\mathcal{V}, F)$ where the limit is over all coverings of $T$ in $(\mathit{Sch}/S)_{fppf}$. Any fppf covering of an affine scheme can be refined by a standard fppf covering, see Topologies, Lemma 34.7.4. Hence we can write

\[ F^+(T) = \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F). \]

Any $\mathcal{V} = \{ T_ k \to T\} _{k = 1, \ldots , n}$ in the colimit may be written as $V_ i \times _{T_ i} T$ for some $i$ and some standard fppf covering $\mathcal{V}_ i = \{ T_{i, k} \to T_ i\} _{k = 1, \ldots , n}$ of $T_ i$. Denote $\mathcal{V}_{i'} = \{ T_{i', k} \to T_{i'}\} _{k = 1, \ldots , n}$ the base change for $i' \geq i$. Then we see that

\begin{align*} \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \check H^0(\mathcal{V}_ i, F) & = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \text{Equalizer}( \xymatrix{ \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_{i', k} \times _{T_{i'}} T_{i', l}) } \\ & = \text{Equalizer}( \xymatrix{ \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{\mathrm{colim}}\nolimits _{k' \geq k} \prod F(T_{i', k} \times _{T_{i'}} T_{i', l}) } \\ & = \text{Equalizer}( \xymatrix{ \prod F(T_ k) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_ k \times _ T T_ l) } \\ & = \check H^0(\mathcal{V}, F) \end{align*}

Here the second equality holds because filtered colimits are exact. The third equality holds because $F$ is limit preserving and because $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} = T_ k$ and $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} \times _{T_{i'}} T_{i', l} = T_ k \times _ T T_ l$ by Limits, Lemma 32.2.3. If we use this for all coverings at the same time we obtain

\begin{align*} F^+(T) & = \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F) \\ & = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}_ i\text{ standard covering }T_ i} \check H^0(T \times _{T_ i}\mathcal{V}_ i, F) \\ & = \mathop{\mathrm{colim}}\nolimits _{i \in I} F^+(T_ i) \end{align*}

The switch of the order of the colimits is allowed by Categories, Lemma 4.14.10. $\square$

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