The Stacks project

Remark 96.19.4. Let us “compute” the value of the relative Čech complex on an object $x$ of $\mathcal{X}$. Say $p(x) = U$. Consider the $2$-fibre product diagram (which serves to introduce the notation $g : \mathcal{V} \to \mathcal{Y}$)

Note that the morphism $\mathcal{V}_ n \to \mathcal{U}_ n$ of the proof of Lemma 96.18.1 induces an equivalence $\mathcal{V}_ n = (\mathit{Sch}/U)_{fppf} \times _{x, \mathcal{X}} \mathcal{U}_ n$. Hence we see from (96.5.0.1) that

In words: The value of the relative Čech complex on an object $x$ of $\mathcal{X}$ is the Čech complex of the base change of $f$ to $\mathcal{X}/x \cong (\mathit{Sch}/U)_{fppf}$. This implies for example that Lemma 96.18.2 implies Lemma 96.19.3 and more generally that results on the (usual) Čech complex imply results for the relative Čech complex.