**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.
$\square$

## Comments (0)