The Stacks project

Remark 96.6.3. In the situation of Lemma 96.6.2 the morphism of ringed topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ is flat as is clear from the equality $f^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_\mathcal {Y}$. This is a bit counter intuitive, for example because a closed immersion of algebraic stacks is typically not flat (as a morphism of algebraic stacks). However, exactly the same thing happens when taking a closed immersion $i : X \to Y$ of schemes: in this case the associated morphism of big $\tau$-sites $i : (\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau$ also is flat.