The Stacks project

83.28 Quasi-coherent modules on simplicial schemes

Lemma 83.28.1. Let $f : V \to U$ be a morphism of simplicial schemes. Given a quasi-coherent module $\mathcal{F}$ on $U_{Zar}$ the pullback $f^*\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$.

Proof. Recall that $\mathcal{F}$ is cartesian with $\mathcal{F}_ n$ quasi-coherent, see Lemma 83.12.10. By Lemma 83.2.4 we see that $(f^*\mathcal{F})_ n = f_ n^*\mathcal{F}_ n$ (some details omitted). Hence $(f^*\mathcal{F})_ n$ is quasi-coherent. The same fact and the cartesian property for $\mathcal{F}$ imply the cartesian property for $f^*\mathcal{F}$. Thus $\mathcal{F}$ is quasi-coherent by Lemma 83.12.10 again. $\square$

Lemma 83.28.2. Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^ n_ j : U_ n \to U_{n - 1}$ are flat and the morphisms $V_ n \to U_ n$ are quasi-compact and quasi-separated. For a quasi-coherent module $\mathcal{G}$ on $V_{Zar}$ the pushforward $f_*\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$.

Proof. If $\mathcal{F} = f_* \mathcal{G}$, then $\mathcal{F}_ n = f_{n , *}\mathcal{G}_ n$ by Lemma 83.2.4. The maps $\mathcal{F}(\varphi )$ are defined using the base change maps, see Cohomology, Section 20.17. The sheaves $\mathcal{F}_ n$ are quasi-coherent by Schemes, Lemma 26.24.1 and the fact that $\mathcal{G}_ n$ is quasi-coherent by Lemma 83.12.10. The base change maps along the degeneracies $d^ n_ j$ are isomorphisms by Cohomology of Schemes, Lemma 30.5.2 and the fact that $\mathcal{G}$ is cartesian by Lemma 83.12.10. Hence $\mathcal{F}$ is cartesian by Lemma 83.12.2. Thus $\mathcal{F}$ is quasi-coherent by Lemma 83.12.10. $\square$

Lemma 83.28.3. Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^ n_ j : U_ n \to U_{n - 1}$ are flat and the morphisms $V_ n \to U_ n$ are quasi-compact and quasi-separated. Then $f^*$ and $f_*$ form an adjoint pair of functors between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$.

Proof. We have seen in Lemmas 83.28.1 and 83.28.2 that the statement makes sense. The adjointness property follows immediately from the fact that each $f_ n^*$ is adjoint to $f_{n, *}$. $\square$

Lemma 83.28.4. 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 83.27.3. 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 83.12.5. 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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