The Stacks project

Lemma 61.10.3. Let $S$ be a scheme. Let $F$ be a contravariant functor defined on the category of all schemes over $S$. If

  1. $F$ satisfies the sheaf property for the h topology, and

  2. $F$ is limit preserving (Limits, Remark 32.6.2),

then $F$ satisfies the sheaf property for the V topology.

Proof. We will prove this by verifying (1) and (2') of Topologies, Lemma 34.10.12. The sheaf property for Zariski coverings follows from the fact that $F$ has the sheaf property for all h coverings. Finally, suppose that $X \to Y$ is a morphism of affine schemes over $S$ such that $\{ X \to Y\} $ is a V covering. By Lemma 61.10.2 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\} $ is an h covering for each $i$. We obtain

\begin{align*} & \text{Equalizer}( \xymatrix{ F(X) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X \times _ Y X) } ) \\ & = \text{Equalizer}( \xymatrix{ \mathop{\mathrm{colim}}\nolimits F(X_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{\mathrm{colim}}\nolimits F(X_ i \times _ Y X_ i) } ) \\ & = \mathop{\mathrm{colim}}\nolimits \text{Equalizer}( \xymatrix{ F(X_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & F(X_ i \times _ Y X_ i) } ) \\ & = \mathop{\mathrm{colim}}\nolimits F(Y) = F(Y) \end{align*}

which is what we wanted to show. The first equality because $F$ is limit preserving and $X = \mathop{\mathrm{lim}}\nolimits X_ i$ and $X \times _ Y X = \mathop{\mathrm{lim}}\nolimits X_ i \times _ Y X_ i$. The second equality because filtered colimits are exact. The third equality because $F$ satisfies the sheaf property for h coverings. $\square$


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