35.11 Quasi-coherent modules and affines
Let $S$ be a scheme1. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Recall that $(\textit{Aff}/S)_\tau $ is the full subcategory of $(\mathit{Sch}/S)_\tau $ whose objects are affine turned into a site by declaring the coverings to be the standard $\tau $-coverings. By Topologies, Lemmas 34.3.10, 34.4.11, 34.5.9, 34.6.9, and 34.7.11 we have an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau )$ whose pullback functor is given by restriction. Recalling that $\mathcal{O}$ denotes the structure sheaf on $(\mathit{Sch}/S)_\tau $, let us temporarily and pedantically denote $\mathcal{O}_{\textit{Aff}}$ the restriction of $\mathcal{O}$ to $(\textit{Aff}/S)_\tau $. Then we obtain an equivalence
35.11.0.1
\begin{equation} \label{descent-equation-alternative-ringed} (\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_\tau ), \mathcal{O}_{\textit{Aff}}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau ), \mathcal{O}) \end{equation}
of ringed topoi. Having said this we can compare quasi-coherent modules as well.
Lemma 35.11.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_{\textit{Aff}}$-modules on $(\textit{Aff}/S)_{fppf}$. The following are equivalent
for every morphism $U \to U'$ of $(\textit{Aff}/S)_{fppf}$ the map $\mathcal{F}(U') \otimes _{\mathcal{O}(U')} \mathcal{O}(U) \to \mathcal{F}(U)$ is an isomorphism,
$\mathcal{F}$ is a sheaf on $(\textit{Aff}/S)_{Zar}$ and a quasi-coherent module on the ringed site $((\textit{Aff}/S)_{Zar}, \mathcal{O}_{\textit{Aff}})$ in the sense of Modules on Sites, Definition 18.23.1,
same as in (3) for the étale topology,
same as in (3) for the smooth topology,
same as in (3) for the syntomic topology,
same as in (3) for the fppf topology,
$\mathcal{F}$ corresponds to a quasi-coherent module on $(\mathit{Sch}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, or $(\mathit{Sch}/S)_{fppf}$ via the equivalence (35.11.0.1),
$\mathcal{F}$ comes from a unique quasi-coherent $\mathcal{O}_ S$-module $\mathcal{G}$ by the procedure described in Section 35.8.
Proof.
Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 18.23.2) we see that the equivalence (35.11.0.1) induces an equivalence between categories of quasi-coherent modules. Proposition 35.8.9 says the topology we use to study quasi-coherent modules on $\mathit{Sch}/S$ does not matter and it also tells us that (8) is the same as (7). Hence we see that (2) – (8) are all equivalent.
Assume the equivalent conditions (2) – (8) hold and let $\mathcal{G}$ be as in (8). Let $h : U \to U' \to S$ be a morphism of $\textit{Aff}/S$. Denote $f : U \to S$ and $f' : U' \to S$ the structure morphisms, so that $f = f' \circ h$. We have $\mathcal{F}(U') = \Gamma (U', (f')^*\mathcal{G})$ and $\mathcal{F}(U) = \Gamma (U, f^*\mathcal{G}) = \Gamma (U, h^*(f')^*\mathcal{G})$. Hence (1) holds by Schemes, Lemma 26.7.3.
Assume (1) holds. To finish the proof it suffices to prove (2). Let $U$ be an object of $(\textit{Aff}/S)_{Zar}$. Say $U = \mathop{\mathrm{Spec}}(R)$. A standard open covering $U = U_1 \cup \ldots \cup U_ n$ is given by $U_ i = D(f_ i)$ for some elements $f_1, \ldots , f_ n \in R$ generating the unit ideal of $R$. By property (1) we see that
\[ \mathcal{F}(U_ i) = \mathcal{F}(U) \otimes _ R R_{f_ i} = \mathcal{F}(U)_{f_ i} \]
and
\[ \mathcal{F}(U_ i \cap U_ j) = \mathcal{F}(U) \otimes _ R R_{f_ if_ j} = \mathcal{F}(U)_{f_ if_ j} \]
Thus we conclude from Algebra, Lemma 10.24.1 that $\mathcal{F}$ is a sheaf on $(\textit{Aff}/S)_{Zar}$. Choose a presentation
\[ \bigoplus \nolimits _{k \in K} R \longrightarrow \bigoplus \nolimits _{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0 \]
by free $R$-modules. By property (1) and the right exactness of tensor product we see that for every morphism $U' \to U$ in $(\textit{Aff}/S)_{Zar}$ we obtain a presentation
\[ \bigoplus \nolimits _{k \in K} \mathcal{O}_{Aff}(U') \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}_{Aff}(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0 \]
In other words, we see that the restriction of $\mathcal{F}$ to the localized category $(\textit{Aff}/S)_{Zar}/U$ has a presentation
\[ \bigoplus \nolimits _{k \in K} \mathcal{O}_{Aff}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}_{Aff}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow \mathcal{F}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow 0 \]
With apologies for the horrible notation, this finishes the proof.
$\square$
We continue the discussion started in the introduction to this section. Let $\tau \in \{ Zariski, {\acute{e}tale}\} $. Recall that $S_{affine, \tau }$ is the full subcategory of $S_\tau $ whose objects are affine turned into a site by declaring the coverings to be the standard $\tau $ coverings. See Topologies, Definitions 34.3.7 and 34.4.8. By Topologies, Lemmas 34.3.11, resp. 34.4.12 we have an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits (S_{affine, \tau }) \to \mathop{\mathit{Sh}}\nolimits (S_\tau )$, whose pullback functor is given by restriction. Recalling that $\mathcal{O}$ denotes the structure sheaf on $S_\tau $ let us temporarily and pedantically denote $\mathcal{O}_{affine}$ the restriction of $\mathcal{O}$ to $S_{affine, \tau }$. Then we obtain an equivalence
35.11.1.1
\begin{equation} \label{descent-equation-alternative-small-ringed} (\mathop{\mathit{Sh}}\nolimits (S_{affine, \tau }), \mathcal{O}_{affine}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (S_\tau ), \mathcal{O}) \end{equation}
of ringed topoi. Having said this we can compare quasi-coherent modules as well.
Lemma 35.11.2. Let $S$ be a scheme. Let $\tau \in \{ Zariski, {\acute{e}tale}\} $. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_{affine}$-modules on $S_{affine, \tau }$. The following are equivalent
for every morphism $U \to U'$ of $S_{affine, \tau }$ the map $\mathcal{F}(U') \otimes _{\mathcal{O}(U')} \mathcal{O}(U) \to \mathcal{F}(U)$ is an isomorphism,
$\mathcal{F}$ is a sheaf on $S_{affine, \tau }$ and a quasi-coherent module on the ringed site $(S_{affine, \tau }, \mathcal{O}_{affine})$ in the sense of Modules on Sites, Definition 18.23.1,
$\mathcal{F}$ corresponds to a quasi-coherent module on $S_\tau $ via the equivalence (35.11.1.1),
$\mathcal{F}$ comes from a unique quasi-coherent $\mathcal{O}_ S$-module $\mathcal{G}$ by the procedure described in Section 35.8.
Proof.
Let us prove this in the case of the étale topology.
Assume (1) holds. To show that $\mathcal{F}$ is a sheaf, let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be a covering of $S_{affine, {\acute{e}tale}}$. The sheaf condition for $\mathcal{F}$ and $\mathcal{U}$, by our assumption on $\mathcal{F}$. reduces to showing that
\[ 0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U) \otimes _{\mathcal{O}(U)} \mathcal{O}(U_ i) \to \prod \mathcal{F}(U) \otimes _{\mathcal{O}(U)} \mathcal{O}(U_ i \times _ U U_ j) \]
is exact. This is true because $\mathcal{O}(U) \to \prod \mathcal{O}(U_ i)$ is faithfully flat (by Lemma 35.9.1 and the fact that coverings in $S_{affine, {\acute{e}tale}}$ are standard étale coverings) and we may apply Lemma 35.3.6. Next, we show that $\mathcal{F}$ is quasi-coherent on $S_{affine, {\acute{e}tale}}$. Namely, for $U$ in $S_{affine, {\acute{e}tale}}$, set $R = \mathcal{O}(U)$ and choose a presentation
\[ \bigoplus \nolimits _{k \in K} R \longrightarrow \bigoplus \nolimits _{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0 \]
by free $R$-modules. By property (1) and the right exactness of tensor product we see that for every morphism $U' \to U$ in $S_{affine, {\acute{e}tale}}$ we obtain a presentation
\[ \bigoplus \nolimits _{k \in K} \mathcal{O}(U') \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0 \]
In other words, we see that the restriction of $\mathcal{F}$ to the localized category $S_{affine, etale}/U$ has a presentation
\[ \bigoplus \nolimits _{k \in K} \mathcal{O}_{affine}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}_{affine}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow \mathcal{F}|_{S_{affine, {\acute{e}tale}}/U} \longrightarrow 0 \]
as required to show that $\mathcal{F}$ is quasi-coherent. With apologies for the horrible notation, this finishes the proof that (1) implies (2).
Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 18.23.2) we see that the equivalence (35.11.1.1) induces an equivalence between categories of quasi-coherent modules. Thus we have the equivalence of (2) and (3).
The equivalence of (3) and (4) follows from Proposition 35.8.9.
Let us assume (4) and prove (1). Namely, let $\mathcal{G}$ be as in (4). Let $h : U \to U' \to S$ be a morphism of $S_{affine, {\acute{e}tale}}$. Denote $f : U \to S$ and $f' : U' \to S$ the structure morphisms, so that $f = f' \circ h$. We have $\mathcal{F}(U') = \Gamma (U', (f')^*\mathcal{G})$ and $\mathcal{F}(U) = \Gamma (U, f^*\mathcal{G}) = \Gamma (U, h^*(f')^*\mathcal{G})$. Hence (1) holds by Schemes, Lemma 26.7.3.
We omit the proof in the case of the Zariski topology.
$\square$
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