The Stacks project

Lemma 85.28.4. Let $f : X \to S$ be a morphism of schemes which has a section1. Let $(X/S)_\bullet $ be the simplicial scheme associated to $X \to S$, see Definition 85.27.3. Then pullback defines an equivalence between the category of quasi-coherent $\mathcal{O}_ S$-modules and the category of quasi-coherent modules on $((X/S)_\bullet )_{Zar}$.

Proof. Let $\sigma : S \to X$ be a section of $f$. Let $(\mathcal{F}, \alpha )$ be a pair as in Lemma 85.12.5. Set $\mathcal{G} = \sigma ^*\mathcal{F}$. Consider the diagram

\[ \xymatrix{ X \ar[r]_-{(\sigma \circ f, 1)} \ar[d]_ f & X \times _ S X \ar[d]^{\text{pr}_0} \ar[r]_-{\text{pr}_1} & X \\ S \ar[r]^\sigma & X } \]

Note that $\text{pr}_0 = d^1_1$ and $\text{pr}_1 = d^1_0$. Hence we see that $(\sigma \circ f, 1)^*\alpha $ defines an isomorphism

\[ f^*\mathcal{G} = (\sigma \circ f, 1)^*\text{pr}_0^*\mathcal{F} \longrightarrow (\sigma \circ f, 1)^*\text{pr}_1^*\mathcal{F} = \mathcal{F} \]

We omit the verification that this isomorphism is compatible with $\alpha $ and the canonical isomorphism $\text{pr}_0^*f^*\mathcal{G} \to \text{pr}_1^*f^*\mathcal{G}$. $\square$

[1] In fact, it would be enough to assume that $f$ has fpqc locally on $S$ a section, since we have descent of quasi-coherent modules by Descent, Section 35.5.

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