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 $g : \mathcal{V} \to \mathcal{Y}$)

\[ \xymatrix{ \mathcal{V} \ar@{=}[r] \ar[d]_ g & (\mathit{Sch}/U)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \ar[r] \ar[d] & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar@{=}[r] & (\mathit{Sch}/U)_{fppf} \ar[r]^-x & \mathcal{X} } \]

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

\[ \Gamma (x, \mathcal{K}^\bullet (f, \mathcal{F})) = \check{\mathcal{C}}^\bullet (\mathcal{V} \to \mathcal{Y}, x^{-1}\mathcal{F}) \]

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.

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