59.99 Comparing big and small topoi
Let $S$ be a scheme. In Topologies, Lemma 34.4.14 we have introduced comparison morphisms $\pi _ S : (\mathit{Sch}/S)_{\acute{e}tale}\to S_{\acute{e}tale}$ and $i_ S : \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ with $\pi _ S \circ i_ S = \text{id}$ and $\pi _{S, *} = i_ S^{-1}$. More generally, if $f : T \to S$ is an object of $(\mathit{Sch}/S)_{\acute{e}tale}$, then there is a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ such that $f_{small} = \pi _ S \circ i_ f$, see Topologies, Lemmas 34.4.13 and 34.4.17. In Descent, Remark 35.8.4 we have extended these to a morphism of ringed sites
\[ \pi _ S : ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}) \to (S_{\acute{e}tale}, \mathcal{O}_ S) \]
and morphisms of ringed topoi
\[ i_ S : (\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}), \mathcal{O}_ S) \to (\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}), \mathcal{O}) \]
and
\[ i_ f : (\mathop{\mathit{Sh}}\nolimits (T_{\acute{e}tale}), \mathcal{O}_ T) \to (\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})) \]
Note that the restriction $i_ S^{-1} = \pi _{S, *}$ (see Topologies, Definition 34.4.15) transforms $\mathcal{O}$ into $\mathcal{O}_ S$. Similarly, $i_ f^{-1}$ transforms $\mathcal{O}$ into $\mathcal{O}_ T$. See Descent, Remark 35.8.4. Hence $i_ S^*\mathcal{F} = i_ S^{-1}\mathcal{F}$ and $i_ f^*\mathcal{F} = i_ f^{-1}\mathcal{F}$ for any $\mathcal{O}$-module $\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$. In particular $i_ S^*$ and $i_ f^*$ are exact functors. The functor $i_ S^*$ is often denoted $\mathcal{F} \mapsto \mathcal{F}|_{S_{\acute{e}tale}}$ (and this does not conflict with the notation in Topologies, Definition 34.4.15).
Lemma 59.99.1. Let $S$ be a scheme. Let $T$ be an object of $(\mathit{Sch}/S)_{\acute{e}tale}$.
If $\mathcal{I}$ is injective in $\textit{Ab}((\mathit{Sch}/S)_{\acute{e}tale})$, then
$i_ f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(T_{\acute{e}tale})$,
$\mathcal{I}|_{S_{\acute{e}tale}}$ is injective in $\textit{Ab}(S_{\acute{e}tale})$,
If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}((\mathit{Sch}/S)_{\acute{e}tale})$, then
$i_ f^{-1}\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(T_{\acute{e}tale})$,
$\mathcal{I}^\bullet |_{S_{\acute{e}tale}}$ is a K-injective complex in $\textit{Ab}(S_{\acute{e}tale})$,
The corresponding statements for modules do not hold.
Proof.
Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi _{S, *} = i_ S^{-1}$ is a right adjoint of the exact functor $\pi _ S^{-1}$, see Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9.
Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use that $i_ f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$. This functor is constructed in Topologies, Lemma 34.4.13 for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma 18.16.2. It is shown in Modules on Sites, Lemma 18.16.3 that it is exact. Second proof. We can use that $i_ f = i_ T \circ f_{big}$ as is shown in Topologies, Lemma 34.4.17. Since $f_{big}$ is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas 21.7.1 and 21.20.1. Then we apply the already proved parts (1)(b) and (2)(b) to the functor $i_ T^{-1}$ to conclude.
Let $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$ and consider the map $2 : \mathcal{O}_ S \to \mathcal{O}_ S$. This is an injective map of $\mathcal{O}_ S$-modules on $S_{\acute{e}tale}$. However, the pullback $\pi _ S^*(2) : \mathcal{O} \to \mathcal{O}$ is not injective as we see by evaluating on $\mathop{\mathrm{Spec}}(\mathbf{F}_2)$. Now choose an injection $\alpha : \mathcal{O} \to \mathcal{I}$ into an injective $\mathcal{O}$-module $\mathcal{I}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$. Then consider the diagram
\[ \xymatrix{ \mathcal{O}_ S \ar[d]_2 \ar[rr]_{\alpha |_{S_{\acute{e}tale}}} & & \mathcal{I}|_{S_{\acute{e}tale}} \\ \mathcal{O}_ S \ar@{..>}[rru] } \]
Then the dotted arrow cannot exist in the category of $\mathcal{O}_ S$-modules because it would mean (by adjunction) that the injective map $\alpha $ factors through the noninjective map $\pi _ S^*(2)$ which cannot be the case. Thus $\mathcal{I}|_{S_{\acute{e}tale}}$ is not an injective $\mathcal{O}_ S$-module.
$\square$
Let $f : T \to S$ be a morphism of schemes. The commutative diagram of Topologies, Lemma 34.4.17 (3) leads to a commutative diagram of ringed sites
\[ \xymatrix{ (T_{\acute{e}tale}, \mathcal{O}_ T) \ar[d]_{f_{small}} & ((\mathit{Sch}/T)_{\acute{e}tale}, \mathcal{O}) \ar[d]^{f_{big}} \ar[l]^{\pi _ T} \\ (S_{\acute{e}tale}, \mathcal{O}_ S) & ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}) \ar[l]_{\pi _ S} } \]
as one easily sees by writing out the definitions of $f_{small}^\sharp $, $f_{big}^\sharp $, $\pi _ S^\sharp $, and $\pi _ T^\sharp $. In particular this means that
59.99.1.1
\begin{equation} \label{etale-cohomology-equation-compare-big-small} (f_{big, *}\mathcal{F})|_{S_{\acute{e}tale}} = f_{small, *}(\mathcal{F}|_{T_{\acute{e}tale}}) \end{equation}
for any sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{\acute{e}tale}$ and if $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then (59.99.1.1) is an isomorphism of $\mathcal{O}_ S$-modules on $S_{\acute{e}tale}$.
Lemma 59.99.2. Let $f : T \to S$ be a morphism of schemes.
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale})$ we have $ (Rf_{big, *}K)|_{S_{\acute{e}tale}} = Rf_{small, *}(K|_{T_{\acute{e}tale}}) $ in $D(S_{\acute{e}tale})$.
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale}, \mathcal{O})$ we have $ (Rf_{big, *}K)|_{S_{\acute{e}tale}} = Rf_{small, *}(K|_{T_{\acute{e}tale}}) $ in $D(\textit{Mod}(S_{\acute{e}tale}, \mathcal{O}_ S))$.
More generally, let $g : S' \to S$ be an object of $(\mathit{Sch}/S)_{\acute{e}tale}$. Consider the fibre product
\[ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^ f \\ S' \ar[r]^ g & S } \]
Then
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale})$ we have $i_ g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(S'_{\acute{e}tale})$.
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale}, \mathcal{O})$ we have $i_ g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(S'_{\acute{e}tale}, \mathcal{O}_{S'}))$.
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale})$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\mathit{Sch}/S')_{\acute{e}tale})$.
For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale}, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\textit{Mod}(S'_{\acute{e}tale}, \mathcal{O}_{S'}))$.
Proof.
Part (1) follows from Lemma 59.99.1 and (59.99.1.1) on choosing a K-injective complex of abelian sheaves representing $K$.
Part (3) follows from Lemma 59.99.1 and Topologies, Lemma 34.4.19 on choosing a K-injective complex of abelian sheaves representing $K$.
Part (5) is Cohomology on Sites, Lemma 21.21.1.
Part (6) is Cohomology on Sites, Lemma 21.21.2.
Part (2) can be proved as follows. Above we have seen that $\pi _ S \circ f_{big} = f_{small} \circ \pi _ T$ as morphisms of ringed sites. Hence we obtain $R\pi _{S, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi _{T, *}$ by Cohomology on Sites, Lemma 21.19.2. Since the restriction functors $\pi _{S, *}$ and $\pi _{T, *}$ are exact, we conclude.
Part (4) follows from part (6) and part (2) applied to $f' : T' \to S'$.
$\square$
Let $S$ be a scheme and let $\mathcal{H}$ be an abelian sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. Recall that $H^ n_{\acute{e}tale}(U, \mathcal{H})$ denotes the cohomology of $\mathcal{H}$ over an object $U$ of $(\mathit{Sch}/S)_{\acute{e}tale}$.
Lemma 59.99.3. Let $f : T \to S$ be a morphism of schemes. Then
For $K$ in $D(S_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(S, \pi _ S^{-1}K) = H^ n(S_{\acute{e}tale}, K)$.
For $K$ in $D(S_{\acute{e}tale}, \mathcal{O}_ S)$ we have $H^ n_{\acute{e}tale}(S, L\pi _ S^*K) = H^ n(S_{\acute{e}tale}, K)$.
For $K$ in $D(S_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(T, \pi _ S^{-1}K) = H^ n(T_{\acute{e}tale}, f_{small}^{-1}K)$.
For $K$ in $D(S_{\acute{e}tale}, \mathcal{O}_ S)$ we have $H^ n_{\acute{e}tale}(T, L\pi _ S^*K) = H^ n(T_{\acute{e}tale}, Lf_{small}^*K)$.
For $M$ in $D((\mathit{Sch}/S)_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(T, M) = H^ n(T_{\acute{e}tale}, i_ f^{-1}M)$.
For $M$ in $D((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$ we have $H^ n_{\acute{e}tale}(T, M) = H^ n(T_{\acute{e}tale}, i_ f^*M)$.
Proof.
To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 59.99.1 and work out the definitions. Part (3) follows from this as $i_ f^{-1}\pi _ S^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3).
Part (6) follows from the very general Cohomology on Sites, Lemma 21.37.5. Then part (4) follows because $Lf_{small}^* = i_ f^* \circ L\pi _ S^*$. Part (2) is a special case of (4).
$\square$
Lemma 59.99.4. Let $S$ be a scheme. For $K \in D(S_{\acute{e}tale})$ the map
\[ K \longrightarrow R\pi _{S, *}\pi _ S^{-1}K \]
is an isomorphism.
Proof.
This is true because both $\pi _ S^{-1}$ and $\pi _{S, *} = i_ S^{-1}$ are exact functors and the composition $\pi _{S, *} \circ \pi _ S^{-1}$ is the identity functor.
$\square$
Lemma 59.99.5. Let $f : T \to S$ be a proper morphism of schemes. Then we have
$\pi _ S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ T^{-1}$ as functors $\mathop{\mathit{Sh}}\nolimits (T_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$,
$\pi _ S^{-1}Rf_{small, *}K = Rf_{big, *}\pi _ T^{-1}K$ for $K$ in $D^+(T_{\acute{e}tale})$ whose cohomology sheaves are torsion,
$\pi _ S^{-1}Rf_{small, *}K = Rf_{big, *}\pi _ T^{-1}K$ for $K$ in $D(T_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$, and
$\pi _ S^{-1}Rf_{small, *}K = Rf_{big, *}\pi _ T^{-1}K$ for all $K$ in $D(T_{\acute{e}tale})$ if $f$ is finite.
Proof.
Proof of (1). Let $\mathcal{F}$ be a sheaf on $T_{\acute{e}tale}$. Let $g : S' \to S$ be an object of $(\mathit{Sch}/S)_{\acute{e}tale}$. Consider the fibre product
\[ \xymatrix{ T' \ar[r]_{f'} \ar[d]_{g'} & S' \ar[d]^ g \\ T \ar[r]^ f & S } \]
Then we have
\[ (f_{big, *}\pi _ T^{-1}\mathcal{F})(S') = (\pi _ T^{-1}\mathcal{F})(T') = ((g'_{small})^{-1}\mathcal{F})(T') = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(S') \]
the second equality by Lemma 59.39.2. On the other hand
\[ (\pi _ S^{-1}f_{small, *}\mathcal{F})(S') = (g_{small}^{-1}f_{small, *}\mathcal{F})(S') \]
again by Lemma 59.39.2. Hence by proper base change for sheaves of sets (Lemma 59.91.5) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi _ S^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi _ T^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi _ S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ T^{-1}$.
Proof of (2). There is a canonical base change map $\pi _ S^{-1}Rf_{small, *}K \to Rf_{big, *}\pi _ T^{-1}K$ for any $K$ in $D(T_{\acute{e}tale})$, see Cohomology on Sites, Remark 21.19.3. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_ g : \mathop{\mathit{Sh}}\nolimits (S'_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ is an isomorphism for any object $g : S' \to S$ of $(\mathit{Sch}/S)_{\acute{e}tale}$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is
\begin{align*} g_{small}^{-1}Rf_{small, *}K & = i_ g^{-1}\pi _ S^{-1}Rf_{small, *}K \\ & \to i_ g^{-1}Rf_{big, *}\pi _ T^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi _ T^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*}
where we have used $\pi _ S \circ i_ g = g_{small}$, $\pi _ T \circ i_{g'} = g'_{small}$, and Lemma 59.99.2. This map is an isomorphism by the proper base change theorem (Lemma 59.91.12) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion.
The proof of part (3) is the same as the proof of part (2), except we use Lemma 59.92.3 instead of Lemma 59.91.12.
Proof of (4). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Proposition 59.55.2 for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma 59.99.2 part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1).
$\square$
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