The Stacks project

34.8 Quasi-coherent sheaves and topologies

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. Consider the functor

34.8.0.1
\begin{equation} \label{descent-equation-quasi-coherent-presheaf} (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Ab}, \quad (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F}). \end{equation}

Lemma 34.8.1. 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] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. The functor defined in (34.8.0.1) 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, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $ a $\tau $-covering is also a fpqc-covering, see the results in Topologies, Lemmas 33.4.2, 33.5.2, 33.6.2, 33.7.2, and 33.9.6. 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 34.5.2 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 34.8.2. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Let $S$ be a scheme. Let $\mathit{Sch}_\tau $ be a big site containing $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module.

  1. The structure sheaf of the big site $(\mathit{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 sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$ on the big site $(\mathit{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 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 34.8.3. In Topologies, Lemma 33.3.11 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 categories 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 34.8.4. Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section 33.11 becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\mathit{Sch}/T)_\tau \to (\mathit{Sch}/S)_{\tau '}$, or $f_{sites} : (\mathit{Sch}/S)_\tau \to S_{\tau '}$ 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 $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau )$ for $\tau \in \{ Zar, {\acute{e}tale}\} $, see Topologies, Lemmas 33.3.12 and 33.4.12, 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 $f_{big} : (\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/Y)_{fppf}$, becomes a morphism of ringed sites

    \[ (f_{big}, f_{big}^\sharp ) : ((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \longrightarrow ((\mathit{Sch}/Y)_{fppf}, \mathcal{O}_ Y) \]

    as in Modules on Sites, Definition 18.6.1. 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 18.6.1. Similarly for the small Zariski site.

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

Lemma 34.8.5. Let $S$ be a scheme. Denote

\[ \begin{matrix} \text{id}_{\tau , Zar} & : & (\mathit{Sch}/S)_\tau \to S_{Zar}, & \tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{\tau , {\acute{e}tale}} & : & (\mathit{Sch}/S)_\tau \to S_{\acute{e}tale}, & \tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{small, {\acute{e}tale}, Zar} & : & S_{\acute{e}tale}\to S_{Zar}, \end{matrix} \]

the morphisms of ringed sites of Remark 34.8.4. 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 $(\mathit{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 34.8.4.

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 $i_ f : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ such that $\text{id}_{\tau , Zar} \circ i_ f = f_{small}$ as morphisms $T_{Zar} \to S_{Zar}$, see Topologies, Lemmas 33.3.12 and 33.3.16. Since pullback is transitive (see Modules on Sites, Lemma 18.13.3) 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 33.4.12 and 33.4.16. We omit the proof of (2). $\square$

Remark 34.8.6. Remark 34.8.4 and Lemma 34.8.5 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 $\text{id}_{\tau , Zar} : ((\mathit{Sch}/S)_\tau , \mathcal{O}) \to (S_{Zar}, \mathcal{O})$ 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 34.8.4 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 18.13.3.

Lemma 34.8.7. 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] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $.

  1. The sheaf $\mathcal{F}^ a$ is a quasi-coherent $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $, as defined in Modules on Sites, Definition 18.23.1.

  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 18.23.1.

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 17.10.1).

Proof of (1). Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. It is clear that $\mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau }$ also sits in an exact sequence

\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0 \]

Hence $\mathcal{F}^ a$ is quasi-coherent by Modules on Sites, Lemma 18.23.3.

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 18.23.3. The case $\tau = Zariski$ is similar (actually, it is really tautological since the corresponding ringed topoi agree). $\square$

Lemma 34.8.8. Let $S$ be a scheme. Let

  1. $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $ and $\mathcal{C} = (\mathit{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 $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be affine. Let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ 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 Čech complexes (Cohomology on Sites, Equation (21.9.2.1))

    \[ \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

    is an isomorphism of complexes.

Proof. This follows because

\[ (\coprod \nolimits _{i_0 = 1, \ldots , n} U_{i_0}) \times _ U \ldots \times _ U (\coprod \nolimits _{i_ p = 1, \ldots , n} U_{i_ p}) = \coprod \nolimits _{i_0, \ldots , i_ p \in \{ 1, \ldots , n\} } U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \]

and the fact that $\mathcal{F}(\coprod _ a V_ a) = \prod _ a \mathcal{F}(V_ a)$ since disjoint unions are $\tau $-coverings. $\square$

Lemma 34.8.9. 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 34.8.8. 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 34.3) 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 34.8.8 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

\[ \Gamma (V \times _ U \ldots \times _ U V, \mathcal{F}^ a) = \Gamma (\mathop{\mathrm{Spec}}(A \otimes _ R \ldots \otimes _ R A), \mathcal{F}^ a) = A \otimes _ R \ldots \otimes _ R A \otimes _ R M \]

We omit the verification that the maps of the Čech complex agree with the maps in the complex $s((A/R)_\bullet \otimes _ R M)$. The vanishing of cohomology is Lemma 34.3.6. $\square$

slogan

Proposition 34.8.10. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $.

  1. There is a canonical isomorphism

    \[ H^ q(S, \mathcal{F}) = H^ q((\mathit{Sch}/S)_\tau , \mathcal{F}^ a). \]
  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} = (\mathit{Sch}/S)_\tau $, or $\mathcal{C} = S_{\acute{e}tale}$, or $\mathcal{C} = S_{Zar}$.

We are going to apply Cohomology on Sites, Lemma 21.11.9 with $\mathcal{F} = \mathcal{F}^ a$, $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ 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 34.8.9. Hence we conclude that $H^ p(U, \mathcal{F}^ a) = 0$ for every affine object $U$ of $\mathcal{C}$.

Next, let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ 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 21.11.7 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 29.2.6. 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 vanishing of $H^ p(U, \mathcal{F}^ a)$, $p > 0$ seen above). Hence by Derived Categories, Lemma 13.18.6 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 34.8.11. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $.

  1. The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence of categories

    \[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \]

    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

    \[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]

    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 34.8.7 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-coherent $\mathcal{O}$-module on $(\mathit{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 34.5.2 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 18.23.1 there exists a $\tau $-covering $\{ S_ i \to S\} _{i \in I}$ of $S$ such that each of the restrictions $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ has a global presentation

\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0 \]

for some index sets $J_ i$ and $K_ i$. We claim that this implies that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ is $\mathcal{F}_ i^ a$ for some quasi-coherent sheaf $\mathcal{F}_ i$ on $S_ i$. Namely, this is clear for the direct sums $\bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$ and $\bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$. Hence we see that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ 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 $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \cong \mathop{\mathrm{Coker}}(\phi )^ a$ as claimed.

Since $\mathcal{G}$ lives on all of the category $(\mathit{Sch}/S_ i)_\tau $ we see that

\[ (\text{pr}_0^*\mathcal{F}_ i)^ a \cong \mathcal{G}|_{(\mathit{Sch}/(S_ i \times _ S S_ j))_\tau } \cong (\text{pr}_1^*\mathcal{F})^ a \]

as $\mathcal{O}$-modules on $(\mathit{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 34.5.2) 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

\[ \phi _ i : \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \]

which agree over $S_ i \times _ S S_ j$. Hence, since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G})$ is a sheaf (Modules on Sites, Lemma 18.27.1), 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 34.8.12. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Let $\mathcal{P}$ be one of the properties of modules1 defined in Modules on Sites, Definitions 18.17.1, 18.23.1, and 18.28.1. The equivalences of categories

\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{and}\quad \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]

defined by the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ seen in Proposition 34.8.11 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 $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_\tau , \mathcal{O})$ 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 18.17.1. For the local properties we can use Modules on Sites, Lemma 18.23.3 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 34.7.1, 34.7.3, 34.7.4, 34.7.5, and 34.7.6. 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 29.9.1) hence this reduces to the case of finite type modules (details omitted). $\square$

Lemma 34.8.13. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. The functors

\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{and}\quad \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \textit{Mod}(S_\tau , \mathcal{O}) \]

defined by the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ seen in Proposition 34.8.11 are

  1. fully faithful,

  2. compatible with direct sums,

  3. compatible with colimits,

  4. right exact,

  5. exact as a functor $\mathit{QCoh}(\mathcal{O}_ S) \to \textit{Mod}(S_{\acute{e}tale}, \mathcal{O})$,

  6. not exact as a functor $\mathit{QCoh}(\mathcal{O}_ S) \to \textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$ 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 $(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{G}))^ a = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G}^ a)$, 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-coherent2, i.e., $\mathcal{E}$ is in the essential image of the functor.

Proof. Part (1) we saw in Proposition 34.8.11.

We have seen in Schemes, Section 25.24 that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent sheaf. Moreover, in Remark 34.8.6 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 18.14.3. 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 $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$, $\mathcal{F} \mapsto \mathcal{F}^ a$ is left exact because an étale morphism is flat, see Morphisms, Lemma 28.34.12. This proves (5).

To see (6), suppose that $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Then $2 : \mathcal{O}_ S \to \mathcal{O}_ S$ is injective but the associated map of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $ isn't injective because $2 : \mathbf{F}_2 \to \mathbf{F}_2$ isn't injective and $\mathop{\mathrm{Spec}}(\mathbf{F}_2)$ is an object of $(\mathit{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 34.8.10 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 25.24 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 $U = \mathop{\mathrm{Spec}}(A) \subset S$ 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 $V/U$, $V = \mathop{\mathrm{Spec}}(B)$ 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 10.81.1). 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 34.8.14. Let $f : T \to S$ be a morphism of schemes.

  1. The equivalences of categories of Proposition 34.8.11 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 34.8.11 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

    \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^ a} & & \mathit{QCoh}(\mathcal{O}_ S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^ a} \\ \mathit{QCoh}(T_\tau , \mathcal{O}) \ar[rr]^{f_{small, *}} & & \mathit{QCoh}(S_\tau , \mathcal{O}) } \]

    is commutative, i.e., $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$.

Proof. Part (1) follows from the discussion in Remark 34.8.6. 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 $\mathop{\mathrm{Spec}}(k)$ by the $0$ map we see that the pushforward 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 25.24.1. 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 29.5.2. $\square$

Lemma 34.8.15. 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 29.4.5 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 21.8.4. 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 34.8.10 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 29.4.6. By Cohomology of Schemes, Lemma 29.5.2 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.

[1] 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.
[2] Warning: This is misleading. See part (6).

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