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### 31.7. Quasi-coherent sheaves and topologies

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. Consider the functor $$\tag{31.7.0.1} (\textit{Sch}/S)^{opp} \longrightarrow \textit{Ab}, \quad (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}).$$

Lemma 31.7.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $\tau \in \{Zariski, fpqc, fppf, {\acute{e}tale}, smooth, syntomic\}$. The functor defined in (31.7.0.1) satisfies the sheaf condition with respect to any $\tau$-covering $\{T_i \to T\}_{i \in I}$ of any scheme $T$ over $S$.

Proof. For $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$ a $\tau$-covering is also a fpqc-covering, see the results in Topologies, Lemmas 30.4.2, 30.5.2, 30.6.2, 30.7.2, and 30.8.6. Hence it suffices to prove the theorem for a fpqc covering. Assume that $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering where $f : T \to S$ is given. Suppose that we have a family of sections $s_i \in \Gamma(T_i , f_i^*f^*\mathcal{F})$ such that $s_i|_{T_i \times_T T_j} = s_j|_{T_i \times_T T_j}$. We have to find the correspond section $s \in \Gamma(T, f^*\mathcal{F})$. We can reinterpret the $s_i$ as a family of maps $\varphi_i : f_i^*\mathcal{O}_T = \mathcal{O}_{T_i} \to f_i^*f^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_T$ and $f^*\mathcal{F}$ on $T$. Hence by Proposition 31.5.2 we see that we may (uniquely) descend these to a map $\mathcal{O}_T \to f^*\mathcal{F}$ which gives us our section $s$. $\square$

We may in particular make the following definition.

Definition 31.7.2. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$. Let $S$ be a scheme. Let $\textit{Sch}_\tau$ be a big site containing $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.

1. The {\it structure sheaf of the big site $(\textit{Sch}/S)_\tau$} is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_S$.
2. If $\tau = {\acute{e}tale}$ the structure sheaf of the small site $S_{\acute{e}tale}$ is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_S$.
3. The {\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$} on the big site $(\textit{Sch}/S)_\tau$ is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).
4. Let $\tau = {\acute{e}tale}$ (resp. $\tau = Zariski$). The {\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$} on the small site $S_{\acute{e}tale}$ (resp. $S_{Zar}$) is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).

Note how we use the same notation $\mathcal{F}^a$ in each case. No confusion can really arise from this as by definition the rule that defines the sheaf $\mathcal{F}^a$ is independent of the site we choose to look at.

Remark 31.7.3. In Topologies, Lemma 30.3.11 we have seen that the small Zariski site of a scheme $S$ is equivalent to $S$ as a topological space in the sense that the category of sheaves are naturally equivalent. Now that $S_{Zar}$ is also endowed with a structure sheaf $\mathcal{O}$ we see that sheaves of modules on the ringed site $(S_{Zar}, \mathcal{O})$ agree with sheaves of modules on the ringed space $(S, \mathcal{O}_S)$.

Remark 31.7.4. Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section 30.9 becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\textit{Sch}/T)_\tau \to (\textit{Sch}/S)_{\tau'}$, or $f_{sites} : (\textit{Sch}/S)_\tau \to S_{\tau'}$ is given by the continuous functor $S'/S \mapsto T \times_S S'/S$. Hence, given $S'/S$ we let $$f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(S \times_S S'/T)$$ be the usual map $\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times_S S')$. Similarly, the morphism $i_f : \mathop{\textit{Sh}}\nolimits(T_\tau) \to \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_\tau)$ for $\tau \in \{Zar, {\acute{e}tale}\}$, see Topologies, Lemmas 30.3.12 and 30.4.12, becomes a morphism of ringed topoi because $i_f^{-1}\mathcal{O} = \mathcal{O}$. Here are some special cases:

1. The morphism of big sites $f_{big} : (\textit{Sch}/X)_{fppf} \to (\textit{Sch}/Y)_{fppf}$, becomes a morphism of ringed sites $$(f_{big}, f_{big}^\sharp) : ((\textit{Sch}/X)_{fppf}, \mathcal{O}_X) \longrightarrow ((\textit{Sch}/Y)_{fppf}, \mathcal{O}_Y)$$ as in Modules on Sites, Definition 17.6.1. Similarly for the big syntomic, smooth, étale and Zariski sites.
2. The morphism of small sites $f_{small} : X_{\acute{e}tale} \to Y_{\acute{e}tale}$ becomes a morphism of ringed sites $$(f_{small}, f_{small}^\sharp) : (X_{\acute{e}tale}, \mathcal{O}_X) \longrightarrow (Y_{\acute{e}tale}, \mathcal{O}_Y)$$ as in Modules on Sites, Definition 17.6.1. Similarly for the small Zariski site.

Let $S$ be a scheme. It is clear that given an $\mathcal{O}$-module on (say) $(\textit{Sch}/S)_{Zar}$ the pullback to (say) $(\textit{Sch}/S)_{fppf}$ is just the fppf-sheafification. To see what happens when comparing big and small sites we have the following.

Lemma 31.7.5. Let $S$ be a scheme. Denote $$\begin{matrix} \text{id}_{\tau, Zar} & : & (\textit{Sch}/S)_\tau \to S_{Zar}, & \tau \in \{Zar, {\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{\tau, {\acute{e}tale}} & : & (\textit{Sch}/S)_\tau \to S_{\acute{e}tale}, & \tau \in \{{\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{small, {\acute{e}tale}, Zar} & : & S_{\acute{e}tale} \to S_{Zar}, \end{matrix}$$ the morphisms of ringed sites of Remark 31.7.4. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_S$-modules which we view a sheaf of $\mathcal{O}$-modules on $S_{Zar}$. Then

1. $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of the Zariski sheaf $$(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})$$ on $(\textit{Sch}/S)_\tau$, and
2. $(\text{id}_{small, {\acute{e}tale}, Zar})^*\mathcal{F}$ is the étale sheafification of the Zariski sheaf $$(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})$$ on $S_{\acute{e}tale}$.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on $S_{\acute{e}tale}$. Then
1. $(\text{id}_{\tau, {\acute{e}tale}})^*\mathcal{G}$ is the $\tau$-sheafification of the étale sheaf $$(f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{G})$$ where $f_{small} : T_{\acute{e}tale} \to S_{\acute{e}tale}$ is the morphism of ringed small étale sites of Remark 31.7.4.

Proof. Proof of (1). We first note that the result is true when $\tau = Zar$ because in that case we have the morphism of topoi $i_f : \mathop{\textit{Sh}}\nolimits(T_{Zar}) \to \mathop{\textit{Sh}}\nolimits(\textit{Sch}/S)_{Zar})$ such that $\text{id}_{\tau, Zar} \circ i_f = f_{small}$ as morphisms $T_{Zar} \to S_{Zar}$, see Topologies, Lemmas 30.3.12 and 30.3.16. Since pullback is transitive (see Modules on Sites, Lemma 17.13.3) we see that $i_f^*(\text{id}_{\tau, Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}$ as desired. Hence, by the remark preceding this lemma we see that $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of the presheaf $T \mapsto \Gamma(T, f^*\mathcal{F})$.

The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas 30.4.12 and 30.4.16. We omit the proof of (2). $\square$

Remark 31.7.6. Remark 31.7.4 and Lemma 31.7.5 have the following applications:

1. Let $S$ be a scheme. The construction $\mathcal{F} \mapsto \mathcal{F}^a$ is the pullback under the morphism of ringed sites $\text{id}_{\tau, Zar} : ((\textit{Sch}/S)_\tau, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$ or the morphism $\text{id}_{small, {\acute{e}tale}, Zar} : (S_{\acute{e}tale}, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$.
2. Let $f : X \to Y$ be a morphism of schemes. For any of the morphisms $f_{sites}$ of ringed sites of Remark 31.7.4 we have $$(f^*\mathcal{F})^a = f_{sites}^*\mathcal{F}^a.$$ This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma 17.13.3.

Lemma 31.7.7. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$.

1. The sheaf $\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on $(\textit{Sch}/S)_\tau$, as defined in Modules on Sites, Definition 17.23.1.
2. If $\tau = {\acute{e}tale}$ (resp. $\tau = Zariski$), then the sheaf $\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on $S_{\acute{e}tale}$ (resp. $S_{Zar}$) as defined in Modules on Sites, Definition 17.23.1.

Proof. Let $\{S_i \to S\}$ be a Zariski covering such that we have exact sequences $$\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{S_i} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{S_i} \longrightarrow \mathcal{F} \longrightarrow 0$$ for some index sets $K_i$ and $J_i$. This is possible by the definition of a quasi-coherent sheaf on a ringed space (See Modules, Definition 16.10.1).

Proof of (1). Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$. It is clear that $\mathcal{F}^a|_{(\textit{Sch}/S_i)_\tau}$ also sits in an exact sequence $$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau} \longrightarrow \mathcal{F}^a|_{(\textit{Sch}/S_i)_\tau} \longrightarrow 0$$ Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites, Lemma 17.23.3.

Proof of (2). Let $\tau = {\acute{e}tale}$. It is clear that $\mathcal{F}^a|_{(S_i)_{\acute{e}tale}}$ also sits in an exact sequence $$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(S_i)_{\acute{e}tale}} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(S_i)_{\acute{e}tale}} \longrightarrow \mathcal{F}^a|_{(S_i)_{\acute{e}tale}} \longrightarrow 0$$ Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites, Lemma 17.23.3. The case $\tau = Zariski$ is similar (actually, it is really tautological since the corresponding ringed topoi agree). $\square$

Lemma 31.7.8. Let $S$ be a scheme. Let

1. $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$\ and $\mathcal{C} = (\textit{Sch}/S)_\tau$, or
2. let $\tau = {\acute{e}tale}$ and $\mathcal{C} = S_{\acute{e}tale}$, or
3. let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$.
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ be affine. Let $\{U_i \to U\}_{i = 1, \ldots, n}$ be a standard affine $\tau$-covering in $\mathcal{C}$. Then
1. $\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}$ is a $\tau$-covering of $U$,
2. $\mathcal{U}$ is a refinement of $\mathcal{V}$, and
3. the induced map on Cech complexes (Cohomology on Sites, Equation (20.9.2.1)) $$\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$$ is an isomorphism of complexes.

Proof. This follows because $$\textstyle(\coprod_{i_0 = 1, \ldots, n} U_{i_0}) \times_U \ldots \times_U (\coprod_{i_p = 1, \ldots, n} U_{i_p}) = \coprod_{i_0, \ldots, i_p \in \{1, \ldots, n\}} U_{i_0} \times_U \ldots \times_U U_{i_p}$$ and the fact that $\mathcal{F}(\coprod_a V_a) = \prod_a \mathcal{F}(V_a)$ since disjoint unions are $\tau$-coverings. $\square$

Lemma 31.7.9. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau$, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in Lemma 31.7.8. Then there is an isomorphism of complexes $$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a) \cong s((A/R)_\bullet \otimes_R M)$$ (see Section 31.3) where $R = \Gamma(U, \mathcal{O}_U)$, $M = \Gamma(U, \mathcal{F}^a)$ and $R \to A$ is a faithfully flat ring map. In particular $$\check{H}^p(\mathcal{U}, \mathcal{F}^a) = 0$$ for all $p \geq 1$.

Proof. By Lemma 31.7.8 we see that $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)$ is isomorphic to $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}^a)$ where $\mathcal{V} = \{V \to U\}$ with $V = \coprod_{i = 1, \ldots n} U_i$ affine also. Set $A = \Gamma(V, \mathcal{O}_V)$. Since $\{V \to U\}$ is a $\tau$-covering we see that $R \to A$ is faithfully flat. On the other hand, by definition of $\mathcal{F}^a$ we have that the degree $p$ term $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{F}^a)$ is $$\Gamma(V \times_U \ldots \times_U V, \mathcal{F}^a) = \Gamma(\mathop{\rm Spec}(A \otimes_R \ldots \otimes_R A), \mathcal{F}^a) = A \otimes_R \ldots \otimes_R A \otimes_R M$$ We omit the verification that the maps of the chech complex agree with the maps in the complex $s((A/R)_\bullet \otimes_R M)$. The vanishing of cohomology is Lemma 31.3.6. $\square$

Proposition 31.7.10. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$.

1. There is a canonical isomorphism $$H^q(S, \mathcal{F}) = H^q((\textit{Sch}/S)_\tau, \mathcal{F}^a).$$
2. There are canonical isomorphisms $$H^q(S, \mathcal{F}) = H^q(S_{Zar}, \mathcal{F}^a) = H^q(S_{\acute{e}tale}, \mathcal{F}^a).$$

Proof. The result for $q = 0$ is clear from the definition of $\mathcal{F}^a$. Let $\mathcal{C} = (\textit{Sch}/S)_\tau$, or $\mathcal{C} = S_{\acute{e}tale}$, or $\mathcal{C} = S_{Zar}$.

We are going to apply Cohomology on Sites, Lemma 20.11.8 with $\mathcal{F} = \mathcal{F}^a$, $\mathcal{B} \subset \mathop{\rm Ob}\nolimits(\mathcal{C})$ the set of affine schemes in $\mathcal{C}$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ the set of standard affine $\tau$-coverings. Assumption (3) of the lemma is satisfied by Lemma 31.7.9. Hence we conclude that $H^p(U, \mathcal{F}^a) = 0$ for every affine object $U$ of $\mathcal{C}$.

Next, let $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$ be any separated object. Denote $f : U \to S$ the structure morphism. Let $U = \bigcup U_i$ be an affine open covering. We may also think of this as a $\tau$-covering $\mathcal{U} = \{U_i \to U\}$ of $U$ in $\mathcal{C}$. Note that $U_{i_0} \times_U \ldots \times_U U_{i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ is affine as we assumed $U$ separated. By Cohomology on Sites, Lemma 20.11.6 and the result above we see that $$H^p(U, \mathcal{F}^a) = \check{H}^p(\mathcal{U}, \mathcal{F}^a) = H^p(U, f^*\mathcal{F})$$ the last equality by Cohomology of Schemes, Lemma 26.2.5. In particular, if $S$ is separated we can take $U = S$ and $f = \text{id}_S$ and the proposition is proved. We suggest the reader skip the rest of the proof (or rewrite it to give a clearer exposition).

Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ on $S$. Choose an injective resolution $\mathcal{F}^a \to \mathcal{J}^\bullet$ on $\mathcal{C}$. Denote $\mathcal{J}^n|_S$ the restriction of $\mathcal{J}^n$ to opens of $S$; this is a sheaf on the topological space $S$ as open coverings are $\tau$-coverings. We get a complex $$0 \to \mathcal{F} \to \mathcal{J}^0|_S \to \mathcal{J}^1|_S \to \ldots$$ which is exact since its sections over any affine open $U \subset S$ is exact (by the vanshing of $H^p(U, \mathcal{F}^a)$, $p > 0$ seen above). Hence by Derived Categories, Lemma 12.17.6 there exists map of complexes $\mathcal{J}^\bullet|_S \to \mathcal{I}^\bullet$ which in particular induces a map $$R\Gamma(\mathcal{C}, \mathcal{F}^a) = \Gamma(S, \mathcal{J}^\bullet) \longrightarrow \Gamma(S, \mathcal{I}^\bullet) = R\Gamma(S, \mathcal{F}).$$ Taking cohomology gives the map $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ which we have to prove is an isomorphism. Let $\mathcal{U} : S = \bigcup U_i$ be an affine open covering which we may think of as a $\tau$-covering also. By the above we get a map of double complexes $$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}|_S) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).$$ This map induces a map of spectral sequences $${}^\tau\! E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a)) \longrightarrow E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$$ The first spectral sequence converges to $H^{p + q}(\mathcal{C}, \mathcal{F})$ and the second to $H^{p + q}(S, \mathcal{F})$. On the other hand, we have seen that the induced maps ${}^\tau\! E_2^{p, q} \to E_2^{p, q}$ are bijections (as all the intersections are separated being opens in affines). Whence also the maps $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ are isomorphisms, and we win. $\square$

Proposition 31.7.11. Let $S$ be a scheme. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$.

1. The functor $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence of categories $$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}((\textit{Sch}/S)_\tau, \mathcal{O})$$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$.
2. Let $\tau = {\acute{e}tale}$, or $\tau = Zariski$. The functor $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence of categories $$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}(S_\tau, \mathcal{O})$$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the small $\tau$ site of $S$.

Proof. We have seen in Lemma 31.7.7 that the functor is well defined. It is straightforward to show that the functor is fully faithful (we omit the verification). To finish the proof we will show that a quasi-cohernet $\mathcal{O}$-module on $(\textit{Sch}/S)_\tau$ gives rise to a descent datum for quasi-coherent sheaves relative to a $\tau$-covering of $S$. Having produced this descent datum we will appeal to Proposition 31.5.2 to get the corresponding quasi-coherent sheaf on $S$.

Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$. By Modules on Sites, Definition 17.23.1 there exists a $\tau$-covering $\{S_i \to S\}_{i \in I}$ of $S$ such that each of the restrictions $\mathcal{G}|_{(\textit{Sch}/S_i)_\tau}$ has a global presentation $$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\textit{Sch}/S_i)_\tau} \longrightarrow 0$$ for some index sets $J_i$ and $K_i$. We claim that this implies that $\mathcal{G}|_{(\textit{Sch}/S_i)_\tau}$ is $\mathcal{F}_i^a$ for some quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$. Namely, this is clear for the direct sums $\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau}$ and $\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\textit{Sch}/S_i)_\tau}$. Hence we see that $\mathcal{G}|_{(\textit{Sch}/S_i)_\tau}$ is a cokernel of a map $\varphi : \mathcal{K}_i^a \to \mathcal{L}_i^a$ for some quasi-coherent sheaves $\mathcal{K}_i$, $\mathcal{L}_i$ on $S_i$. By the fully faithfulness of $( )^a$ we see that $\varphi = \phi^a$ for some map of quasi-coherent sheaves $\phi : \mathcal{K}_i \to \mathcal{L}_i$ on $S_i$. Then it is clear that $\mathcal{G}|_{(\textit{Sch}/S_i)_\tau} \cong \text{Coker}(\phi)^a$ as claimed.

Since $\mathcal{G}$ lives on all of the category $(\textit{Sch}/S_i)_\tau$ we see that $$(\text{pr}_0^*\mathcal{F}_i)^a \cong \mathcal{G}|_{(\textit{Sch}/(S_i \times_S S_j))_\tau} \cong (\text{pr}_1^*\mathcal{F})^a$$ as $\mathcal{O}$-modules on $(\textit{Sch}/(S_i \times_S S_j))_\tau$. Hence, using fully faithfulness again we get canonical isomorphisms $$\phi_{ij} : \text{pr}_0^*\mathcal{F}_i \longrightarrow \text{pr}_1^*\mathcal{F}_j$$ of quasi-coherent modules over $S_i \times_S S_j$. We omit the verification that these satisfy the cocycle condition. Since they do we see by effectivity of descent for quasi-coherent sheaves and the covering $\{S_i \to S\}$ (Proposition 31.5.2) that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ with $\mathcal{F}|_{S_i} \cong \mathcal{F}_i$ compatible with the given descent data. In other words we are given $\mathcal{O}$-module isomorphisms $$\phi_i : \mathcal{F}^a|_{(\textit{Sch}/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\textit{Sch}/S_i)_\tau}$$ which agree over $S_i \times_S S_j$. Hence, since $\mathop{\mathcal{H}\!{\it om}}\nolimits_\mathcal{O}(\mathcal{F}^a, \mathcal{G})$ is a sheaf (Modules on Sites, Lemma 17.27.1), we conclude that there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^a \to \mathcal{G}$ recovering the isomorphisms $\phi_i$ above. Hence this is an isomorphism and we win.

The case of the sites $S_{\acute{e}tale}$ and $S_{Zar}$ is proved in the exact same manner. $\square$

Lemma 31.7.12. Let $S$ be a scheme. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$. Let $\mathcal{P}$ be one of the properties of modules (The list is: free, finite free, generated by global sections, generated by $r$ global sections, generated by finitely many global sections, having a global presentation, having a global finite presentation, locally free, finite locally free, locally generated by sections, locally generated by $r$ sections, finite type, of finite presentation, coherent, or flat.) defined in Modules on Sites, Definitions 17.17.1, 17.23.1, and 17.28.1. The equivalences of categories $$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}((\textit{Sch}/S)_\tau, \mathcal{O}) \quad\text{and}\quad \textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}(S_\tau, \mathcal{O})$$ defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in Proposition 31.7.11 have the property $$\mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module}$$ except (possibly) when $\mathcal{P}$ is ''locally free'' or ''coherent''. If $\mathcal{P}=$''coherent'' the equivalence holds for $\textit{QCoh}(\mathcal{O}_S) \to \textit{QCoh}(S_\tau, \mathcal{O})$ when $S$ is locally Noetherian and $\tau$ is Zariski or étale.

Proof. This is immediate for the global properties, i.e., those defined in Modules on Sites, Definition 17.17.1. For the local properties we can use Modules on Sites, Lemma 17.23.3 to translate ''$\mathcal{F}^a$ has $\mathcal{P}$'' into a property on the members of a covering of $X$. Hence the result follows from Lemmas 31.6.1, 31.6.2, 31.6.3, 31.6.4, and 31.6.5. Being coherent for a quasi-coherent module is the same as being of finite type over a locally Noetherian scheme (see Cohomology of Schemes, Lemma 26.10.1) hence this reduces to the case of finite type modules (details omitted). $\square$

Lemma 31.7.13. Let $S$ be a scheme. Let $\tau \in \{Zariski, fppf, {\acute{e}tale}, smooth, syntomic\}$. The functors $$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{Mod}((\textit{Sch}/S)_\tau, \mathcal{O}) \quad\text{and}\quad \textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{Mod}(S_\tau, \mathcal{O})$$ defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in Proposition 31.7.11 are

1. fully faithful,
2. compatible with direct sums,
3. compatible with colimits,
4. right exact,
5. exact as a functor $\textit{QCoh}(\mathcal{O}_S) \to \textit{Mod}(S_\tau, \mathcal{O})$,
6. not exact as a functor $\textit{QCoh}(\mathcal{O}_S) \to \textit{Mod}((\textit{Sch}/S)_\tau, \mathcal{O})$ in general,
7. given two quasi-coherent $\mathcal{O}_S$-modules $\mathcal{F}$, $\mathcal{G}$ we have $(\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G})^a = \mathcal{F}^a \otimes_\mathcal{O} \mathcal{G}^a$,
8. given two quasi-coherent $\mathcal{O}_S$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation we have $(\mathop{\mathcal{H}\!{\it om}}\nolimits_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a = \mathop{\mathcal{H}\!{\it om}}\nolimits_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)$, and
9. given a short exact sequence $0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$ of $\mathcal{O}$-modules then $\mathcal{E}$ is quasi-coherent (Warning: This is misleading. See part (6).), i.e., $\mathcal{E}$ is in the essential image of the functor.

Proof. Part (1) we saw in Proposition 31.7.11.

We have seen in Schemes, Section 22.24 that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent sheaf. Moreover, in Remark 31.7.6 we saw that $\mathcal{F} \mapsto \mathcal{F}^a$ is the pullback functor for a morphism of ringed sites, hence commutes with all colimits, see Modules on Sites, Lemma 17.14.3. Thus (3) and its special case (3) hold.

This also shows that the functor is right exact (i.e., commutes with finite colimits), hence (4).

The functor $\textit{QCoh}(\mathcal{O}_S) \to \textit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$, $\mathcal{F} \mapsto \mathcal{F}^a$ is left exact because an étale morphism is flat, see Morphisms, Lemma 25.37.12. This proves (5).

To see (6), suppose that $S = \mathop{\rm Spec}(\mathbf{Z})$. Then $2 : \mathcal{O}_S \to \mathcal{O}_S$ is injective but the associated map of $\mathcal{O}$-modules on $(\textit{Sch}/S)_\tau$ isn't injective because $2 : \mathbf{F}_2 \to \mathbf{F}_2$ isn't injective and $\mathop{\rm Spec}(\mathbf{F}_2)$ is an object of $(\textit{Sch}/S)_\tau$.

We omit the proofs of (7) and (8).

Let $0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$ be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{F}_1$ and $\mathcal{F}_2$ quasi-coherent on $S$. Consider the restriction $$0 \to \mathcal{F}_1 \to \mathcal{E}|_{S_{Zar}} \to \mathcal{F}_2$$ to $S_{Zar}$. By Proposition 31.7.10 we see that on any affine $U \subset S$ we have $H^1(U, \mathcal{F}_1^a) = H^1(U, \mathcal{F}_1) = 0$. Hence the sequence above is also exact on the right. By Schemes, Section 22.24 we conclude that $\mathcal{F} = \mathcal{E}|_{S_{Zar}}$ is quasi-coherent. Thus we obtain a commutative diagram $$\xymatrix{ & \mathcal{F}_1^a \ar[r] \ar[d] & \mathcal{F}^a \ar[r] \ar[d] & \mathcal{F}_2^a \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1^a \ar[r] & \mathcal{E} \ar[r] & \mathcal{F}_2^a \ar[r] & 0 }$$ To finish the proof it suffices to show that the top row is also right exact. To do this, denote once more $U = \mathop{\rm Spec}(A) \subset S$ an affine open of $S$. We have seen above that $0 \to \mathcal{F}_1(U) \to \mathcal{E}(U) \to \mathcal{F}_2(U) \to 0$ is exact. For any affine scheme $V/U$, $V = \mathop{\rm Spec}(B)$ the map $\mathcal{F}_1^a(V) \to \mathcal{E}(V)$ is injective. We have $\mathcal{F}_1^a(V) = \mathcal{F}_1(U) \otimes_A B$ by definition. The injection $\mathcal{F}_1^a(V) \to \mathcal{E}(V)$ factors as $$\mathcal{F}_1(U) \otimes_A B \to \mathcal{E}(U) \otimes_A B \to \mathcal{E}(U)$$ Considering $A$-algebras $B$ of the form $B = A \oplus M$ we see that $\mathcal{F}_1(U) \to \mathcal{E}(U)$ is universally injective (see Algebra, Definition 9.79.1). Since $\mathcal{E}(U) = \mathcal{F}(U)$ we conclude that $\mathcal{F}_1 \to \mathcal{F}$ remains injective after any base change, or equivalently that $\mathcal{F}_1^a \to \mathcal{F}^a$ is injective. $\square$

Proposition 31.7.14. Let $f : T \to S$ be a morphism of schemes.

1. The equivalences of categories of Proposition 31.7.11 are compatible with pullback. More precisely, we have $f^*(\mathcal{G}^a) = (f^*\mathcal{G})^a$ for any quasi-coherent sheaf $\mathcal{G}$ on $S$.
2. The equivalences of categories of Proposition 31.7.11 part (1) are not compatible with pushforward in general.
3. If $f$ is quasi-compact and quasi-separated, and $\tau \in \{Zariski, {\acute{e}tale}\}$ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram $$\xymatrix{ \textit{QCoh}(\mathcal{O}_T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^a} & & \textit{QCoh}(\mathcal{O}_S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^a} \\ \textit{QCoh}(T_\tau, \mathcal{O}) \ar[rr]^{f_{small, *}} & & \textit{QCoh}(S_\tau, \mathcal{O}) }$$ is commutative, i.e., $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$.

Proof. Part (1) follows from the discussion in Remark 31.7.6. Part (2) is just a warning, and can be explained in the following way: First the statement cannot be made precise since $f_*$ does not transform quasi-coherent sheaves into quasi-coherent sheaves in general. Even if this is the case for $f$ (and any base change of $f$), then the compatibility over the big sites would mean that formation of $f_*\mathcal{F}$ commutes with any base change, which does not hold in general. An explicit example is the quasi-compact open immersion $j : X = \mathbf{A}^2_k \setminus \{0\} \to \mathbf{A}^2_k = Y$ where $k$ is a field. We have $j_*\mathcal{O}_X = \mathcal{O}_Y$ but after base change to $\mathop{\rm Spec}(k)$ by the $0$ map we see that the pushfoward is zero.

Let us prove (3) in case $\tau = {\acute{e}tale}$. Note that $f$, and any base change of $f$, transforms quasi-coherent sheaves into quasi-coherent sheaves, see Schemes, Lemma 22.24.1. The equality $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$ means that for any étale morphism $g : U \to S$ we have $\Gamma(U, g^*f_*\mathcal{F}) = \Gamma(U \times_S T, (g')^*\mathcal{F})$ where $g' : U \times_S T \to T$ is the projection. This is true by Cohomology of Schemes, Lemma 26.5.2. $\square$

Lemma 31.7.15. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the étale or Zariski topology, there are canonical isomorphisms $R^if_{small, *}(\mathcal{F}^a) = (R^if_*\mathcal{F})^a$.

Proof. We prove this for the étale topology; we omit the proof in the case of the Zariski topology. By Cohomology of Schemes, Lemma 26.4.4 the sheaves $R^if_*\mathcal{F}$ are quasi-coherent so that the assertion makes sense. The sheaf $R^if_{small, *}\mathcal{F}^a$ is the sheaf associated to the presheaf $$U \longmapsto H^i(U \times_S T, \mathcal{F}^a)$$ where $g : U \to S$ is an object of $S_{\acute{e}tale}$, see Cohomology on Sites, Lemma 20.8.4. By our conventions the right hand side is the étale cohomology of the restriction of $\mathcal{F}^a$ to the localization $T_{\acute{e}tale}/U \times_S T$ which equals $(U \times_S T)_{\acute{e}tale}$. By Proposition 31.7.10 this is presheaf the same as the presheaf $$U \longmapsto H^i(U \times_S T, (g')^*\mathcal{F}),$$ where $g' : U \times_S T \to T$ is the projection. If $U$ is affine then this is the same as $H^0(U, R^if'_*(g')^*\mathcal{F})$, see Cohomology of Schemes, Lemma 26.4.5. By Cohomology of Schemes, Lemma 26.5.2 this is equal to $H^0(U, g^*R^if_*\mathcal{F})$ which is the value of $(R^if_*\mathcal{F})^a$ on $U$. Thus the values of the sheaves of modules $R^if_{small, *}(\mathcal{F}^a)$ and $(R^if_*\mathcal{F})^a$ on every affine object of $S_{\acute{e}tale}$ are canonically isomorphic which implies they are canonically isomorphic. $\square$

The results in this section say there is virtually no difference between quasi-coherent sheaves on $S$ and quasi-coherent sheaves on any of the sites associated to $S$ in the chapter on topologies. Hence one often sees statements on quasi-coherent sheaves formulated in either language, without restatements in the other.

\section{Quasi-coherent sheaves and topologies}
\label{section-quasi-coherent-sheaves}

\noindent
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Consider the functor
$$\label{equation-quasi-coherent-preseheaf} (\Sch/S)^{opp} \longrightarrow \textit{Ab}, \quad (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}).$$

\begin{lemma}
\label{lemma-sheaf-condition-holds}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Let $\tau \in \{Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
The functor defined in (\ref{equation-quasi-coherent-preseheaf})
satisfies the sheaf condition with respect to any $\tau$-covering
$\{T_i \to T\}_{i \in I}$ of any scheme $T$ over $S$.
\end{lemma}

\begin{proof}
For $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$ a $\tau$-covering
is also a fpqc-covering, see the results in
Topologies, Lemmas
\ref{topologies-lemma-zariski-etale},
\ref{topologies-lemma-zariski-etale-smooth},
\ref{topologies-lemma-zariski-etale-smooth-syntomic},
\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, and
\ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.
Hence it suffices to prove the theorem
for a fpqc covering. Assume that $\{f_i : T_i \to T\}_{i \in I}$
is an fpqc covering where $f : T \to S$ is given. Suppose that
we have a family of sections $s_i \in \Gamma(T_i , f_i^*f^*\mathcal{F})$
such that $s_i|_{T_i \times_T T_j} = s_j|_{T_i \times_T T_j}$.
We have to find the correspond section $s \in \Gamma(T, f^*\mathcal{F})$.
We can reinterpret the $s_i$ as a family of maps
$\varphi_i : f_i^*\mathcal{O}_T = \mathcal{O}_{T_i} \to f_i^*f^*\mathcal{F}$
compatible with the canonical descent data associated to the
quasi-coherent sheaves $\mathcal{O}_T$ and $f^*\mathcal{F}$ on $T$.
Hence by Proposition \ref{proposition-fpqc-descent-quasi-coherent}
we see that we may (uniquely) descend
these to a map $\mathcal{O}_T \to f^*\mathcal{F}$ which gives
us our section $s$.
\end{proof}

\noindent
We may in particular make the following definition.

\begin{definition}
\label{definition-structure-sheaf}
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
Let $S$ be a scheme.
Let $\Sch_\tau$ be a big site containing $S$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
\begin{enumerate}
\item The {\it structure sheaf of the big site $(\Sch/S)_\tau$}
is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$ which is
denoted $\mathcal{O}$ or $\mathcal{O}_S$.
\item If $\tau = \etale$ the structure sheaf of the small site
$S_\etale$ is the sheaf of rings $T/S \mapsto \Gamma(T, \mathcal{O}_T)$
which is denoted $\mathcal{O}$ or $\mathcal{O}_S$.
\item The {\it sheaf of $\mathcal{O}$-modules associated to
$\mathcal{F}$} on the big site $(\Sch/S)_\tau$
is the sheaf of $\mathcal{O}$-modules
$(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$
which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).
\item Let $\tau = \etale$ (resp.\ $\tau = Zariski$). The
{\it sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$}
on the small site $S_\etale$ (resp.\ $S_{Zar}$) is the sheaf of
$\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})$
which is denoted $\mathcal{F}^a$ (and often simply $\mathcal{F}$).
\end{enumerate}
\end{definition}

\noindent
Note how we use the same notation $\mathcal{F}^a$ in each case.
No confusion can really arise from this as by definition the rule
that defines the sheaf $\mathcal{F}^a$ is independent of the site
we choose to look at.

\begin{remark}
\label{remark-Zarsiki-site-space}
In Topologies, Lemma \ref{topologies-lemma-Zariski-usual}
we have seen that the small Zariski site of a scheme $S$ is
equivalent to $S$ as a topological space in the sense that the
category of sheaves are naturally equivalent. Now that $S_{Zar}$
is also endowed with a structure sheaf $\mathcal{O}$ we see
that sheaves of modules on the ringed site $(S_{Zar}, \mathcal{O})$
agree with sheaves of modules on the ringed space $(S, \mathcal{O}_S)$.
\end{remark}

\begin{remark}
\label{remark-change-topologies-ringed}
Let $f : T \to S$ be a morphism of schemes.
Each of the morphisms of sites $f_{sites}$ listed in
Topologies, Section \ref{topologies-section-change-topologies}
becomes a morphism of ringed sites. Namely, each of these morphisms of sites
$f_{sites} : (\Sch/T)_\tau \to (\Sch/S)_{\tau'}$, or
$f_{sites} : (\Sch/S)_\tau \to S_{\tau'}$ is given by the continuous
functor $S'/S \mapsto T \times_S S'/S$. Hence, given $S'/S$ we let
$$f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(S \times_S S'/T)$$
be the usual map
$\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times_S S')$.
Similarly, the morphism
$i_f : \Sh(T_\tau) \to \Sh((\Sch/S)_\tau)$
for $\tau \in \{Zar, \etale\}$, see
Topologies, Lemmas \ref{topologies-lemma-put-in-T} and
\ref{topologies-lemma-put-in-T-etale},
becomes a morphism of ringed topoi because $i_f^{-1}\mathcal{O} = \mathcal{O}$.
Here are some special cases:
\begin{enumerate}
\item The morphism of big sites
$f_{big} : (\Sch/X)_{fppf} \to (\Sch/Y)_{fppf}$,
becomes a morphism of ringed sites
$$(f_{big}, f_{big}^\sharp) : ((\Sch/X)_{fppf}, \mathcal{O}_X) \longrightarrow ((\Sch/Y)_{fppf}, \mathcal{O}_Y)$$
as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}.
Similarly for the big syntomic, smooth, \'etale and Zariski sites.
\item The morphism of small sites
$f_{small} : X_\etale \to Y_\etale$
becomes a morphism of ringed sites
$$(f_{small}, f_{small}^\sharp) : (X_\etale, \mathcal{O}_X) \longrightarrow (Y_\etale, \mathcal{O}_Y)$$
as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}.
Similarly for the small Zariski site.
\end{enumerate}
\end{remark}

\noindent
Let $S$ be a scheme. It is clear that given an $\mathcal{O}$-module on (say)
$(\Sch/S)_{Zar}$ the pullback to (say) $(\Sch/S)_{fppf}$
is just the fppf-sheafification. To see what happens when comparing
big and small sites we have the following.

\begin{lemma}
\label{lemma-compare-sites}
Let $S$ be a scheme. Denote
$$\begin{matrix} \text{id}_{\tau, Zar} & : & (\Sch/S)_\tau \to S_{Zar}, & \tau \in \{Zar, \etale, smooth, syntomic, fppf\} \\ \text{id}_{\tau, \etale} & : & (\Sch/S)_\tau \to S_\etale, & \tau \in \{\etale, smooth, syntomic, fppf\} \\ \text{id}_{small, \etale, Zar} & : & S_\etale \to S_{Zar}, \end{matrix}$$
the morphisms of ringed sites of
Remark \ref{remark-change-topologies-ringed}.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_S$-modules
which we view a sheaf of $\mathcal{O}$-modules on $S_{Zar}$. Then
\begin{enumerate}
\item $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification
of the Zariski sheaf
$$(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})$$
on $(\Sch/S)_\tau$, and
\item $(\text{id}_{small, \etale, Zar})^*\mathcal{F}$ is the
\'etale sheafification of the Zariski sheaf
$$(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})$$
on $S_\etale$.
\end{enumerate}
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules
on $S_\etale$. Then
\begin{enumerate}
\item[(3)] $(\text{id}_{\tau, \etale})^*\mathcal{G}$ is the
$\tau$-sheafification of the \'etale sheaf
$$(f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{G})$$
where $f_{small} : T_\etale \to S_\etale$
is the morphism of ringed small \'etale sites of
Remark \ref{remark-change-topologies-ringed}.
\end{enumerate}
\end{lemma}

\begin{proof}
Proof of (1). We first note that the result is true when $\tau = Zar$
because in that case we have the morphism of topoi
$i_f : \Sh(T_{Zar}) \to \Sh(\Sch/S)_{Zar})$
such that $\text{id}_{\tau, Zar} \circ i_f = f_{small}$ as morphisms
$T_{Zar} \to S_{Zar}$, see
Topologies, Lemmas \ref{topologies-lemma-put-in-T} and
\ref{topologies-lemma-morphism-big-small}.
Since pullback is transitive (see
Modules on Sites,
Lemma \ref{sites-modules-lemma-push-pull-composition-modules})
we see that
$i_f^*(\text{id}_{\tau, Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}$
as desired. Hence, by the remark preceding this lemma we see that
$(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of
the presheaf $T \mapsto \Gamma(T, f^*\mathcal{F})$.

\medskip\noindent
The proof of (3) is exactly the same as the proof of (1), except that it
uses
Topologies, Lemmas \ref{topologies-lemma-put-in-T-etale} and
\ref{topologies-lemma-morphism-big-small-etale}.
We omit the proof of (2).
\end{proof}

\begin{remark}
\label{remark-change-topologies-ringed-sites}
Remark \ref{remark-change-topologies-ringed}
and
Lemma \ref{lemma-compare-sites}
have the following applications:
\begin{enumerate}
\item Let $S$ be a scheme.
The construction $\mathcal{F} \mapsto \mathcal{F}^a$ is
the pullback under the morphism of ringed sites
$\text{id}_{\tau, Zar} : ((\Sch/S)_\tau, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$
or the morphism
$\text{id}_{small, \etale, Zar} : (S_\etale, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$.
\item Let $f : X \to Y$ be a morphism of schemes.
For any of the morphisms $f_{sites}$ of ringed sites of
Remark \ref{remark-change-topologies-ringed}
we have
$$(f^*\mathcal{F})^a = f_{sites}^*\mathcal{F}^a.$$
This follows from (1) and the fact that pullbacks are compatible with
compositions of morphisms of ringed sites, see
Modules on Sites,
Lemma \ref{sites-modules-lemma-push-pull-composition-modules}.
\end{enumerate}
\end{remark}

\begin{lemma}
\label{lemma-quasi-coherent-gives-quasi-coherent}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item The sheaf $\mathcal{F}^a$ is a quasi-coherent
$\mathcal{O}$-module on $(\Sch/S)_\tau$, as defined in
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
\item If $\tau = \etale$ (resp.\ $\tau = Zariski$), then the sheaf
$\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on
$S_\etale$ (resp.\ $S_{Zar}$) as defined in
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\{S_i \to S\}$ be a Zariski covering such that we have exact sequences
$$\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{S_i} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{S_i} \longrightarrow \mathcal{F} \longrightarrow 0$$
for some index sets $K_i$ and $J_i$. This is possible by the definition
of a quasi-coherent sheaf on a ringed space
(See Modules, Definition \ref{modules-definition-quasi-coherent}).

\medskip\noindent
Proof of (1). Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
It is clear that $\mathcal{F}^a|_{(\Sch/S_i)_\tau}$ also
sits in an exact sequence
$$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow 0$$
Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites,
Lemma \ref{sites-modules-lemma-local-final-object}.

\medskip\noindent
Proof of (2). Let $\tau = \etale$.
It is clear that $\mathcal{F}^a|_{(S_i)_\etale}$ also sits
in an exact sequence
$$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \mathcal{F}^a|_{(S_i)_\etale} \longrightarrow 0$$
Hence $\mathcal{F}^a$ is quasi-coherent by Modules on Sites,
Lemma \ref{sites-modules-lemma-local-final-object}.
The case $\tau = Zariski$ is similar (actually, it is really
tautological since the corresponding ringed topoi agree).
\end{proof}

\begin{lemma}
\label{lemma-standard-covering-Cech}
Let $S$ be a scheme. Let
\begin{enumerate}
\item[(a)] $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$\
and $\mathcal{C} = (\Sch/S)_\tau$, or
\item[(b)] let $\tau = \etale$ and $\mathcal{C} = S_\etale$, or
\item[(c)] let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$.
\end{enumerate}
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$.
Let $U \in \Ob(\mathcal{C})$ be affine.
Let $\{U_i \to U\}_{i = 1, \ldots, n}$ be a standard affine
$\tau$-covering in $\mathcal{C}$. Then
\begin{enumerate}
\item $\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}$ is a
$\tau$-covering of $U$,
\item $\mathcal{U}$ is a refinement of $\mathcal{V}$, and
\item the induced map on Cech complexes
(Cohomology on Sites,
Equation (\ref{sites-cohomology-equation-map-cech-complexes}))
$$\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$$
is an isomorphism of complexes.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows because
$$\textstyle(\coprod_{i_0 = 1, \ldots, n} U_{i_0}) \times_U \ldots \times_U (\coprod_{i_p = 1, \ldots, n} U_{i_p}) = \coprod_{i_0, \ldots, i_p \in \{1, \ldots, n\}} U_{i_0} \times_U \ldots \times_U U_{i_p}$$
and the fact that $\mathcal{F}(\coprod_a V_a) = \prod_a \mathcal{F}(V_a)$
since disjoint unions are $\tau$-coverings.
\end{proof}

\begin{lemma}
\label{lemma-standard-covering-Cech-quasi-coherent}
Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$.
Let $\tau$, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in
Lemma \ref{lemma-standard-covering-Cech}. Then there is an isomorphism
of complexes
$$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a) \cong s((A/R)_\bullet \otimes_R M)$$
(see Section \ref{section-descent-modules})
where $R = \Gamma(U, \mathcal{O}_U)$, $M = \Gamma(U, \mathcal{F}^a)$
and $R \to A$ is a faithfully flat ring map. In particular
$$\check{H}^p(\mathcal{U}, \mathcal{F}^a) = 0$$
for all $p \geq 1$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-standard-covering-Cech} we see that
$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)$
is isomorphic to $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}^a)$
where $\mathcal{V} = \{V \to U\}$ with $V = \coprod_{i = 1, \ldots n} U_i$
affine also. Set $A = \Gamma(V, \mathcal{O}_V)$. Since $\{V \to U\}$
is a $\tau$-covering we see that $R \to A$ is faithfully flat.
On the other hand, by definition of $\mathcal{F}^a$ we have
that the degree $p$ term $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{F}^a)$
is
$$\Gamma(V \times_U \ldots \times_U V, \mathcal{F}^a) = \Gamma(\Spec(A \otimes_R \ldots \otimes_R A), \mathcal{F}^a) = A \otimes_R \ldots \otimes_R A \otimes_R M$$
We omit the verification that the maps of the chech complex agree with
the maps in the complex $s((A/R)_\bullet \otimes_R M)$. The vanishing
of cohomology is Lemma \ref{lemma-ff-exact}.
\end{proof}

\begin{proposition}
\label{proposition-same-cohomology-quasi-coherent}
Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item There is a canonical isomorphism
$$H^q(S, \mathcal{F}) = H^q((\Sch/S)_\tau, \mathcal{F}^a).$$
\item There are canonical isomorphisms
$$H^q(S, \mathcal{F}) = H^q(S_{Zar}, \mathcal{F}^a) = H^q(S_\etale, \mathcal{F}^a).$$
\end{enumerate}
\end{proposition}

\begin{proof}
The result for $q = 0$ is clear from the definition of $\mathcal{F}^a$.
Let $\mathcal{C} = (\Sch/S)_\tau$, or $\mathcal{C} = S_\etale$,
or $\mathcal{C} = S_{Zar}$.

\medskip\noindent
We are going to apply
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}
with $\mathcal{F} = \mathcal{F}^a$,
$\mathcal{B} \subset \Ob(\mathcal{C})$ the set of affine schemes
in $\mathcal{C}$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ the
set of standard affine $\tau$-coverings. Assumption (3) of
the lemma is satisfied by
Lemma \ref{lemma-standard-covering-Cech-quasi-coherent}.
Hence we conclude that $H^p(U, \mathcal{F}^a) = 0$ for every
affine object $U$ of $\mathcal{C}$.

\medskip\noindent
Next, let $U \in \Ob(\mathcal{C})$ be any separated object.
Denote $f : U \to S$ the structure morphism.
Let $U = \bigcup U_i$ be an affine open covering.
We may also think of this as a $\tau$-covering
$\mathcal{U} = \{U_i \to U\}$ of $U$ in $\mathcal{C}$.
Note that
$U_{i_0} \times_U \ldots \times_U U_{i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ is affine as we assumed $U$ separated.
By
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cech-spectral-sequence-application}
and the result above we see that
$$H^p(U, \mathcal{F}^a) = \check{H}^p(\mathcal{U}, \mathcal{F}^a) = H^p(U, f^*\mathcal{F})$$
the last equality by
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cech-cohomology-quasi-coherent}.
In particular, if $S$ is separated we can take $U = S$ and
$f = \text{id}_S$ and the proposition is proved.
We suggest the reader skip the rest of the proof (or rewrite it
to give a clearer exposition).

\medskip\noindent
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ on $S$.
Choose an injective resolution $\mathcal{F}^a \to \mathcal{J}^\bullet$
on $\mathcal{C}$. Denote $\mathcal{J}^n|_S$ the restriction of $\mathcal{J}^n$
to opens of $S$; this is a sheaf on the topological space $S$ as open
coverings are $\tau$-coverings. We get a complex
$$0 \to \mathcal{F} \to \mathcal{J}^0|_S \to \mathcal{J}^1|_S \to \ldots$$
which is exact since its sections over any affine open $U \subset S$
is exact (by the vanshing of $H^p(U, \mathcal{F}^a)$, $p > 0$ seen
above). Hence by
Derived Categories, Lemma \ref{derived-lemma-morphisms-lift}
there exists map of complexes
$\mathcal{J}^\bullet|_S \to \mathcal{I}^\bullet$ which in particular
induces a map
$$R\Gamma(\mathcal{C}, \mathcal{F}^a) = \Gamma(S, \mathcal{J}^\bullet) \longrightarrow \Gamma(S, \mathcal{I}^\bullet) = R\Gamma(S, \mathcal{F}).$$
Taking cohomology gives the map
$H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ which
we have to prove is an isomorphism.
Let $\mathcal{U} : S = \bigcup U_i$ be an affine open covering
which we may think of as a $\tau$-covering also.
By the above we get a map of double complexes
$$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}|_S) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).$$
This map induces a map of spectral sequences
$${}^\tau\! E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a)) \longrightarrow E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$$
The first spectral sequence converges to
$H^{p + q}(\mathcal{C}, \mathcal{F})$ and the second to
$H^{p + q}(S, \mathcal{F})$. On the other hand, we have seen
that the induced maps ${}^\tau\! E_2^{p, q} \to E_2^{p, q}$ are
bijections (as all the intersections are separated being opens in affines).
Whence also the maps $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$
are isomorphisms, and we win.
\end{proof}

\begin{proposition}
\label{proposition-equivalence-quasi-coherent}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
\begin{enumerate}
\item The functor $\mathcal{F} \mapsto \mathcal{F}^a$
defines an equivalence of categories
$$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}((\Sch/S)_\tau, \mathcal{O})$$
between the category of quasi-coherent sheaves on $S$ and the category
of quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$.
\item Let $\tau = \etale$, or $\tau = Zariski$.
The functor $\mathcal{F} \mapsto \mathcal{F}^a$
defines an equivalence of categories
$$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}(S_\tau, \mathcal{O})$$
between the category of quasi-coherent sheaves on $S$ and the category
of quasi-coherent $\mathcal{O}$-modules on the small $\tau$ site of $S$.
\end{enumerate}
\end{proposition}

\begin{proof}
We have seen in Lemma \ref{lemma-quasi-coherent-gives-quasi-coherent}
that the functor is well defined.
It is straightforward to show that the functor is fully faithful (we omit
the verification). To finish the proof we will show that a
quasi-cohernet $\mathcal{O}$-module on $(\Sch/S)_\tau$ gives
rise to a descent datum for quasi-coherent sheaves relative to a
$\tau$-covering of $S$. Having produced this descent datum we will appeal
to Proposition \ref{proposition-fpqc-descent-quasi-coherent} to get the
corresponding quasi-coherent sheaf on $S$.

\medskip\noindent
Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on
the big $\tau$ site of $S$. By
Modules on Sites, Definition \ref{sites-modules-definition-site-local}
there exists a $\tau$-covering $\{S_i \to S\}_{i \in I}$ of $S$
such that each of the restrictions
$\mathcal{G}|_{(\Sch/S_i)_\tau}$ has a global presentation
$$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau} \longrightarrow 0$$
for some index sets $J_i$ and $K_i$. We claim that this implies
that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is $\mathcal{F}_i^a$
for some quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$. Namely,
this is clear for the direct sums
$\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$
and
$\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$.
Hence we see that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is a
cokernel of a map $\varphi : \mathcal{K}_i^a \to \mathcal{L}_i^a$
for some quasi-coherent sheaves $\mathcal{K}_i$, $\mathcal{L}_i$
on $S_i$. By the fully faithfulness of $(\ )^a$ we see that
$\varphi = \phi^a$ for some map of quasi-coherent sheaves
$\phi : \mathcal{K}_i \to \mathcal{L}_i$ on $S_i$. Then it is
clear that
$\mathcal{G}|_{(\Sch/S_i)_\tau} \cong \text{Coker}(\phi)^a$
as claimed.

\medskip\noindent
Since $\mathcal{G}$ lives on all of the category
$(\Sch/S_i)_\tau$ we see that
$$(\text{pr}_0^*\mathcal{F}_i)^a \cong \mathcal{G}|_{(\Sch/(S_i \times_S S_j))_\tau} \cong (\text{pr}_1^*\mathcal{F})^a$$
as $\mathcal{O}$-modules on $(\Sch/(S_i \times_S S_j))_\tau$.
Hence, using fully faithfulness again we get canonical isomorphisms
$$\phi_{ij} : \text{pr}_0^*\mathcal{F}_i \longrightarrow \text{pr}_1^*\mathcal{F}_j$$
of quasi-coherent modules over $S_i \times_S S_j$. We omit the verification
that these satisfy the cocycle condition. Since they do we see by
effectivity of descent for quasi-coherent sheaves and the covering
$\{S_i \to S\}$ (Proposition \ref{proposition-fpqc-descent-quasi-coherent})
that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$
with $\mathcal{F}|_{S_i} \cong \mathcal{F}_i$ compatible
with the given descent data. In other words we are given
$\mathcal{O}$-module isomorphisms
$$\phi_i : \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau}$$
which agree over $S_i \times_S S_j$. Hence, since
$\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G})$ is
a sheaf (Modules on Sites, Lemma \ref{sites-modules-lemma-internal-hom}),
we conclude that
there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^a \to \mathcal{G}$
recovering the isomorphisms $\phi_i$ above. Hence this is an isomorphism
and we win.

\medskip\noindent
The case of the sites $S_\etale$ and $S_{Zar}$ is proved in the
exact same manner.
\end{proof}

\begin{lemma}
\label{lemma-equivalence-quasi-coherent-properties}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
Let $\mathcal{P}$ be one of the properties of modules\footnote{The list is:
free, finite free, generated by global sections,
generated by $r$ global sections, generated by finitely many global sections,
having a global presentation, having a global finite presentation,
locally free, finite locally free, locally generated by sections,
locally generated by $r$ sections, finite type, of finite presentation,
coherent, or flat.} defined in
Modules on Sites, Definitions \ref{sites-modules-definition-global},
\ref{sites-modules-definition-site-local}, and
\ref{sites-modules-definition-flat}.
The equivalences of categories
$$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{QCoh}(S_\tau, \mathcal{O})$$
defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in
Proposition \ref{proposition-equivalence-quasi-coherent}
have the property
$$\mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module}$$
except (possibly) when $\mathcal{P}$ is locally free'' or coherent''.
If $\mathcal{P}=$coherent'' the equivalence
holds for $\textit{QCoh}(\mathcal{O}_S) \to \textit{QCoh}(S_\tau, \mathcal{O})$
when $S$ is locally Noetherian and $\tau$ is Zariski or \'etale.
\end{lemma}

\begin{proof}
This is immediate for the global properties, i.e., those defined in
Modules on Sites, Definition \ref{sites-modules-definition-global}.
For the local properties we can use
Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}
to translate $\mathcal{F}^a$ has $\mathcal{P}$'' into a property
on the members of a covering of $X$. Hence the result follows from
Lemmas \ref{lemma-finite-type-descends},
\ref{lemma-finite-presentation-descends},
\ref{lemma-locally-generated-by-r-sections-descends},
\ref{lemma-flat-descends}, and
\ref{lemma-finite-locally-free-descends}.
Being coherent for a quasi-coherent module is the same as being
of finite type over a locally Noetherian scheme (see
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian})
hence this reduces
to the case of finite type modules (details omitted).
\end{proof}

\begin{lemma}
\label{lemma-equivalence-quasi-coherent-limits}
Let $S$ be a scheme.
Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$.
The functors
$$\textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{Mod}((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \textit{QCoh}(\mathcal{O}_S) \longrightarrow \textit{Mod}(S_\tau, \mathcal{O})$$
defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in
Proposition \ref{proposition-equivalence-quasi-coherent}
are
\begin{enumerate}
\item fully faithful,
\item compatible with direct sums,
\item compatible with colimits,
\item right exact,
\item exact as a functor
$\textit{QCoh}(\mathcal{O}_S) \to \textit{Mod}(S_\tau, \mathcal{O})$,
\item {\bf not} exact as a functor
$\textit{QCoh}(\mathcal{O}_S) \to \textit{Mod}((\Sch/S)_\tau, \mathcal{O})$
in general,
\item given two quasi-coherent $\mathcal{O}_S$-modules
$\mathcal{F}$, $\mathcal{G}$ we have
$(\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G})^a = \mathcal{F}^a \otimes_\mathcal{O} \mathcal{G}^a$,
\item given two quasi-coherent $\mathcal{O}_S$-modules
$\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$
is of finite presentation we have
$(\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a = \SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)$, and
\item given a short exact sequence
$0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$
of $\mathcal{O}$-modules then $\mathcal{E}$ is
quasi-coherent\footnote{Warning: This is misleading. See part (6).}, i.e.,
$\mathcal{E}$ is in the essential image of the functor.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) we saw in
Proposition \ref{proposition-equivalence-quasi-coherent}.

\medskip\noindent
We have seen in
Schemes, Section \ref{schemes-section-quasi-coherent}
that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent
sheaf. Moreover, in
Remark \ref{remark-change-topologies-ringed-sites}
we saw that $\mathcal{F} \mapsto \mathcal{F}^a$ is the pullback functor
for a morphism of ringed sites, hence commutes with all colimits, see
Modules on Sites, Lemma
\ref{sites-modules-lemma-exactness-pushforward-pullback}.
Thus (3) and its special case (3) hold.

\medskip\noindent
This also shows that the functor is right exact (i.e., commutes with
finite colimits), hence (4).

\medskip\noindent
The functor $\textit{QCoh}(\mathcal{O}_S) \to \textit{QCoh}(S_\etale, \mathcal{O})$,
$\mathcal{F} \mapsto \mathcal{F}^a$
is left exact because an \'etale morphism is flat, see
Morphisms, Lemma \ref{morphisms-lemma-etale-flat}.
This proves (5).

\medskip\noindent
To see (6), suppose that $S = \Spec(\mathbf{Z})$.
Then $2 : \mathcal{O}_S \to \mathcal{O}_S$ is injective but the associated
map of $\mathcal{O}$-modules on $(\Sch/S)_\tau$ isn't
injective because $2 : \mathbf{F}_2 \to \mathbf{F}_2$ isn't injective
and $\Spec(\mathbf{F}_2)$ is an object of $(\Sch/S)_\tau$.

\medskip\noindent
We omit the proofs of (7) and (8).

\medskip\noindent
Let $0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$
be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{F}_1$
and $\mathcal{F}_2$ quasi-coherent on $S$. Consider the restriction
$$0 \to \mathcal{F}_1 \to \mathcal{E}|_{S_{Zar}} \to \mathcal{F}_2$$
to $S_{Zar}$. By
Proposition \ref{proposition-same-cohomology-quasi-coherent}
we see that on any affine $U \subset S$ we have
$H^1(U, \mathcal{F}_1^a) = H^1(U, \mathcal{F}_1) = 0$.
Hence the sequence above is also exact on the right. By
Schemes, Section \ref{schemes-section-quasi-coherent}
we conclude that $\mathcal{F} = \mathcal{E}|_{S_{Zar}}$ is
quasi-coherent. Thus we obtain a commutative diagram
$$\xymatrix{ & \mathcal{F}_1^a \ar[r] \ar[d] & \mathcal{F}^a \ar[r] \ar[d] & \mathcal{F}_2^a \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1^a \ar[r] & \mathcal{E} \ar[r] & \mathcal{F}_2^a \ar[r] & 0 }$$
To finish the proof it suffices to show that the top row is also
right exact. To do this, denote once more $U = \Spec(A) \subset S$
an affine open of $S$. We have seen above that
$0 \to \mathcal{F}_1(U) \to \mathcal{E}(U) \to \mathcal{F}_2(U) \to 0$
is exact. For any affine scheme $V/U$,
$V = \Spec(B)$ the map $\mathcal{F}_1^a(V) \to \mathcal{E}(V)$
is injective. We have $\mathcal{F}_1^a(V) = \mathcal{F}_1(U) \otimes_A B$
by definition. The injection
$\mathcal{F}_1^a(V) \to \mathcal{E}(V)$ factors as
$$\mathcal{F}_1(U) \otimes_A B \to \mathcal{E}(U) \otimes_A B \to \mathcal{E}(U)$$
Considering $A$-algebras $B$ of the form $B = A \oplus M$
we see that $\mathcal{F}_1(U) \to \mathcal{E}(U)$ is
universally injective (see
Algebra, Definition \ref{algebra-definition-universally-injective}).
Since $\mathcal{E}(U) = \mathcal{F}(U)$ we conclude that
$\mathcal{F}_1 \to \mathcal{F}$ remains injective after any base change,
or equivalently that $\mathcal{F}_1^a \to \mathcal{F}^a$ is injective.
\end{proof}

\begin{proposition}
\label{proposition-equivalence-quasi-coherent-functorial}
Let $f : T \to S$ be a morphism of schemes.
\begin{enumerate}
\item The equivalences of categories of
Proposition \ref{proposition-equivalence-quasi-coherent}
are compatible with pullback.
More precisely, we have $f^*(\mathcal{G}^a) = (f^*\mathcal{G})^a$
for any quasi-coherent sheaf $\mathcal{G}$ on $S$.
\item The equivalences of categories of
Proposition \ref{proposition-equivalence-quasi-coherent} part (1)
are {\bf not} compatible with pushforward in general.
\item If $f$ is quasi-compact and quasi-separated, and
$\tau \in \{Zariski, \etale\}$ then $f_*$ and $f_{small, *}$
preserve quasi-coherent sheaves and the diagram
$$\xymatrix{ \textit{QCoh}(\mathcal{O}_T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^a} & & \textit{QCoh}(\mathcal{O}_S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^a} \\ \textit{QCoh}(T_\tau, \mathcal{O}) \ar[rr]^{f_{small, *}} & & \textit{QCoh}(S_\tau, \mathcal{O}) }$$
is commutative, i.e., $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$.
\end{enumerate}
\end{proposition}

\begin{proof}
Part (1) follows from the discussion in
Remark \ref{remark-change-topologies-ringed-sites}.
Part (2) is just a warning, and can be explained in the following way:
First the statement cannot be made precise since $f_*$ does not
transform quasi-coherent sheaves into quasi-coherent sheaves in general.
Even if this is the case for $f$ (and any base change of $f$), then the
compatibility over the big sites would mean that formation of $f_*\mathcal{F}$
commutes with any base change, which does not hold in general.
An explicit example is the quasi-compact open immersion
$j : X = \mathbf{A}^2_k \setminus \{0\} \to \mathbf{A}^2_k = Y$
where $k$ is a field. We have $j_*\mathcal{O}_X = \mathcal{O}_Y$
but after base change to $\Spec(k)$ by the $0$ map
we see that the pushfoward is zero.

\medskip\noindent
Let us prove (3) in case $\tau = \etale$. Note that $f$, and any
base change of $f$, transforms quasi-coherent sheaves
into quasi-coherent sheaves, see
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
The equality $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$
means that for any \'etale morphism $g : U \to S$ we have
$\Gamma(U, g^*f_*\mathcal{F}) = \Gamma(U \times_S T, (g')^*\mathcal{F})$
where $g' : U \times_S T \to T$ is the projection. This is true by
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
\end{proof}

\begin{lemma}
\label{lemma-higher-direct-images-small-etale}
Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \'etale
or Zariski topology, there are canonical isomorphisms
$R^if_{small, *}(\mathcal{F}^a) = (R^if_*\mathcal{F})^a$.
\end{lemma}

\begin{proof}
We prove this for the \'etale topology; we omit the proof in the case
of the Zariski topology. By Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}
the sheaves $R^if_*\mathcal{F}$ are quasi-coherent so that the assertion
makes sense. The sheaf $R^if_{small, *}\mathcal{F}^a$ is the sheaf associated
to the presheaf
$$U \longmapsto H^i(U \times_S T, \mathcal{F}^a)$$
where $g : U \to S$ is an object of $S_\etale$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}.
By our conventions the right hand side is the \'etale
cohomology of the restriction of $\mathcal{F}^a$ to the localization
$T_\etale/U \times_S T$ which equals
$(U \times_S T)_\etale$. By
Proposition \ref{proposition-same-cohomology-quasi-coherent}
this is presheaf the same as the presheaf
$$U \longmapsto H^i(U \times_S T, (g')^*\mathcal{F}),$$
where $g' : U \times_S T \to T$ is the projection. If $U$ is affine
then this is the same as $H^0(U, R^if'_*(g')^*\mathcal{F})$, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}.
By
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}
this is equal to $H^0(U, g^*R^if_*\mathcal{F})$ which is the value
of $(R^if_*\mathcal{F})^a$ on $U$.
Thus the values of the sheaves of modules
$R^if_{small, *}(\mathcal{F}^a)$ and $(R^if_*\mathcal{F})^a$
on every affine object of $S_\etale$ are canonically isomorphic
which implies they are canonically isomorphic.
\end{proof}

\noindent
The results in this section say there is virtually no difference between
quasi-coherent sheaves on $S$ and quasi-coherent sheaves on any of the
sites associated to $S$ in the chapter on topologies. Hence one often
sees statements on quasi-coherent sheaves formulated in either language,
without restatements in the other.


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