The Stacks project

82.5 Comparing big and small topoi

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In Topologies on Spaces, Lemma 71.4.8 we have introduced comparison morphisms $\pi _ X : (\textit{Spaces}/X)_{\acute{e}tale}\to X_{spaces, {\acute{e}tale}}$ and $i_ X : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale})$ with $\pi _ X \circ i_ X = \text{id}$ as morphisms of topoi and $\pi _{X, *} = i_ X^{-1}$. More generally, if $f : Y \to X$ is an object of $(\textit{Spaces}/X)_{\acute{e}tale}$, then there is a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale})$ such that $f_{small} = \pi _ X \circ i_ f$, see Topologies on Spaces, Lemmas 71.4.7 and 71.4.11. In Topologies on Spaces, Remark 71.4.14 we have extended these to a morphism of ringed sites

\[ \pi _ X : ((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O}) \to (X_{spaces, {\acute{e}tale}}, \mathcal{O}_ X) \]

and morphisms of ringed topoi

\[ i_ X : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \to (\mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}), \mathcal{O}) \]

and

\[ i_ f : (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \to (\mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O})) \]

Note that the restriction $i_ X^{-1} = \pi _{X, *}$ (see Topologies, Definition 34.4.14) transforms $\mathcal{O}$ into $\mathcal{O}_ X$. Similarly, $i_ f^{-1}$ transforms $\mathcal{O}$ into $\mathcal{O}_ Y$. See Topologies on Spaces, Remark 71.4.14. Hence $i_ X^*\mathcal{F} = i_ X^{-1}\mathcal{F}$ and $i_ f^*\mathcal{F} = i_ f^{-1}\mathcal{F}$ for any $\mathcal{O}$-module $\mathcal{F}$ on $(\textit{Spaces}/X)_{\acute{e}tale}$. In particular $i_ X^*$ and $i_ f^*$ are exact functors. The functor $i_ X^*$ is often denoted $\mathcal{F} \mapsto \mathcal{F}|_{X_{\acute{e}tale}}$ (and this does not conflict with the notation in Topologies on Spaces, Definition 71.4.9).

Lemma 82.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. Then $\pi _ X^{-1}\mathcal{F}$ is given by the rule

\[ (\pi _ X^{-1}\mathcal{F})(Y) = \Gamma (Y_{\acute{e}tale}, f_{small}^{-1}\mathcal{F}) \]

for $f : Y \to X$ in $(\textit{Spaces}/X)_{\acute{e}tale}$. Moreover, $\pi _ Y^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.

Proof. Since pullback is transitive and $f_{small} = \pi _ X \circ i_ f$ (see above) we see that $i_ f^{-1} \pi _ X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$. This shows that $\pi _ X^{-1}$ has the description given in the lemma.

To prove that $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma 71.8.7 to show that for a surjective proper morphism $V \to U$ of algebraic spaces over $X$ we have $(\pi _ X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps $(\pi _ X^{-1}\mathcal{F})(V) \to (\pi _ X^{-1}\mathcal{F})(V \times _ U V)$. This we have seen in Lemma 82.4.1.

The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma 71.8.2.

Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be an fpqc covering of algebraic spaces over $X$. Let $s_ i \in (\pi _ X^{-1}\mathcal{F})(U_ i)$ be sections which agree over $U_ i \times _ U U_ j$. We have to prove there exists a unique $s \in (\pi _ X^{-1}\mathcal{F})(U)$ restricting to $s_ i$ over $U_ i$. Case I: $U$ and $U_ i$ are schemes. This case follows from Étale Cohomology, Lemma 58.39.2. Case II: $U$ is a scheme. Here we choose surjective étale morphisms $T_ i \to U_ i$ where $T_ i$ is a scheme. Then $\mathcal{T} = \{ T_ i \to U\} $ is an fpqc covering by schemes and by case I the result holds for $\mathcal{T}$. We omit the verification that this implies the result for $\mathcal{U}$. Case III: general case. Let $W \to U$ be a surjective étale morphism, where $W$ is a scheme. Then $\mathcal{W} = \{ U_ i \times _ U W \to W\} $ is an fpqc covering (by algebraic spaces) of the scheme $W$. By case II the result hold for $\mathcal{W}$. We omit the verification that this implies the result for $\mathcal{U}$. $\square$

Lemma 82.5.2. Let $S$ be a scheme. Let $Y \to X$ be a morphism of $(\textit{Spaces}/S)_{\acute{e}tale}$.

  1. If $\mathcal{I}$ is injective in $\textit{Ab}((\textit{Spaces}/X)_{\acute{e}tale})$, then

    1. $i_ f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(Y_{\acute{e}tale})$,

    2. $\mathcal{I}|_{X_{\acute{e}tale}}$ is injective in $\textit{Ab}(X_{\acute{e}tale})$,

  2. If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}((\textit{Spaces}/X)_{\acute{e}tale})$, then

    1. $i_ f^{-1}\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(Y_{\acute{e}tale})$,

    2. $\mathcal{I}^\bullet |_{X_{\acute{e}tale}}$ is a K-injective complex in $\textit{Ab}(X_{\acute{e}tale})$,

The corresponding statements for modules do not hold.

Proof. Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi _{X, *} = i_ X^{-1}$ is a right adjoint of the exact functor $\pi _ X^{-1}$, 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 34.4.12 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 $i_ f = i_ Y \circ f_{big}$ as is shown in Topologies, Lemma 34.4.16. 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 $i_ Y^{-1}$ to conclude.

To see a counter example for the case of modules we refer to Étale Cohomology, Lemma 58.93.1. $\square$

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The commutative diagram of Topologies on Spaces, Lemma 71.4.11 (3) leads to a commutative diagram of ringed sites

\[ \xymatrix{ (Y_{spaces, {\acute{e}tale}}, \mathcal{O}_ Y) \ar[d]_{f_{spaces, {\acute{e}tale}}} & ((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O}) \ar[d]^{f_{big}} \ar[l]^{\pi _ Y} \\ (X_{spaces, {\acute{e}tale}}, \mathcal{O}_ X) & ((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O}) \ar[l]_{\pi _ X} } \]

as one easily sees by writing out the definitions of $f_{small}^\sharp $, $f_{big}^\sharp $, $\pi _ X^\sharp $, and $\pi _ Y^\sharp $. In particular this means that

82.5.2.1
\begin{equation} \label{spaces-more-cohomology-equation-compare-big-small} (f_{big, *}\mathcal{F})|_{X_{\acute{e}tale}} = f_{small, *}(\mathcal{F}|_{Y_{\acute{e}tale}}) \end{equation}

for any sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_{\acute{e}tale}$ and if $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then (82.5.2.1) is an isomorphism of $\mathcal{O}_ X$-modules on $X_{\acute{e}tale}$.

Lemma 82.5.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

  1. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $ (Rf_{big, *}K)|_{X_{\acute{e}tale}} = Rf_{small, *}(K|_{Y_{\acute{e}tale}}) $ in $D(X_{\acute{e}tale})$.

  2. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $ (Rf_{big, *}K)|_{X_{\acute{e}tale}} = Rf_{small, *}(K|_{Y_{\acute{e}tale}}) $ in $D(\textit{Mod}(X_{\acute{e}tale}, \mathcal{O}_ X))$.

More generally, let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_{\acute{e}tale}$. Consider the fibre product

\[ \xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]

Then

  1. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $i_ g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(X'_{\acute{e}tale})$.

  2. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $i_ g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

  3. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\textit{Spaces}/X')_{\acute{e}tale})$.

  4. For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

Proof. Part (1) follows from Lemma 82.5.2 and (82.5.2.1) on choosing a K-injective complex of abelian sheaves representing $K$.

Part (3) follows from Lemma 82.5.2 and Topologies, Lemma 34.4.18 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 $\pi _ X \circ f_{big} = f_{small} \circ \pi _ Y$ as morphisms of ringed sites. Hence we obtain $R\pi _{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi _{Y, *}$ by Cohomology on Sites, Lemma 21.19.2. Since the restriction functors $\pi _{X, *}$ and $\pi _{Y, *}$ are exact, we conclude.

Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$. $\square$

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{H}$ be an abelian sheaf on $(\textit{Spaces}/X)_{\acute{e}tale}$. Recall that $H^ n_{\acute{e}tale}(U, \mathcal{H})$ denotes the cohomology of $\mathcal{H}$ over an object $U$ of $(\textit{Spaces}/X)_{\acute{e}tale}$.

Lemma 82.5.4. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then

  1. For $K$ in $D(X_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(X, \pi _ X^{-1}K) = H^ n(X_{\acute{e}tale}, K)$.

  2. For $K$ in $D(X_{\acute{e}tale}, \mathcal{O}_ X)$ we have $H^ n_{\acute{e}tale}(X, L\pi _ X^*K) = H^ n(X_{\acute{e}tale}, K)$.

  3. For $K$ in $D(X_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(Y, \pi _ X^{-1}K) = H^ n(Y_{\acute{e}tale}, f_{small}^{-1}K)$.

  4. For $K$ in $D(X_{\acute{e}tale}, \mathcal{O}_ X)$ we have $H^ n_{\acute{e}tale}(Y, L\pi _ X^*K) = H^ n(Y_{\acute{e}tale}, Lf_{small}^*K)$.

  5. For $M$ in $D((\textit{Spaces}/X)_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(Y, M) = H^ n(Y_{\acute{e}tale}, i_ f^{-1}M)$.

  6. For $M$ in $D((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O})$ we have $H^ n_{\acute{e}tale}(Y, M) = H^ n(Y_{\acute{e}tale}, i_ f^*M)$.

Proof. To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 82.5.2 and work out the definitions. Part (3) follows from this as $i_ f^{-1}\pi _ X^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3).

Part (6) follows from the very general Cohomology on Sites, Lemma 21.36.5. Then part (4) follows because $Lf_{small}^* = i_ f^* \circ L\pi _ X^*$. Part (2) is a special case of (4). $\square$

Lemma 82.5.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D(X_{\acute{e}tale})$ the map

\[ K \longrightarrow R\pi _{X, *}\pi _ X^{-1}K \]

is an isomorphism where $\pi _ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as above.

Proof. This is true because both $\pi _ X^{-1}$ and $\pi _{X, *} = i_ X^{-1}$ are exact functors and the composition $\pi _{X, *} \circ \pi _ X^{-1}$ is the identity functor. $\square$

Lemma 82.5.6. Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Then we have

  1. $\pi _ X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ Y^{-1}$ as functors $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale})$,

  2. $\pi _ X^{-1}Rf_{small, *}K = Rf_{big, *}\pi _ Y^{-1}K$ for $K$ in $D^+(Y_{\acute{e}tale})$ whose cohomology sheaves are torsion, and

  3. $\pi _ X^{-1}Rf_{small, *}K = Rf_{big, *}\pi _ Y^{-1}K$ for all $K$ in $D(Y_{\acute{e}tale})$ if $f$ is finite.

Proof. Proof of (1). Let $\mathcal{F}$ be a sheaf on $Y_{\acute{e}tale}$. Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_{\acute{e}tale}$. Consider the fibre product

\[ \xymatrix{ Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^ g \\ Y \ar[r]^ f & X } \]

Then we have

\[ (f_{big, *}\pi _ Y^{-1}\mathcal{F})(X') = (\pi _ Y^{-1}\mathcal{F})(Y') = ((g'_{small})^{-1}\mathcal{F})(Y') = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X') \]

the second equality by Lemma 82.5.1. On the other hand

\[ (\pi _ X^{-1}f_{small, *}\mathcal{F})(X') = (g_{small}^{-1}f_{small, *}\mathcal{F})(X') \]

again by Lemma 82.5.1. Hence by proper base change for sheaves of sets (Lemma 82.4.4) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi _ X^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi _ Y^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi _ X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ Y^{-1}$.

Proof of (2). There is a canonical base change map $\pi _ X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi _ Y^{-1}K$ for any $K$ in $D(Y_{\acute{e}tale})$, 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 $i_ g : \mathop{\mathit{Sh}}\nolimits (X'_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{\acute{e}tale})$ is an isomorphism for any object $g : X' \to X$ of $(\mathit{Sch}/X)_{\acute{e}tale}$. 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 _ X^{-1}Rf_{small, *}K \\ & \to i_ g^{-1}Rf_{big, *}\pi _ Y^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi _ Y^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*}

where we have used $\pi _ X \circ i_ g = g_{small}$, $\pi _ Y \circ i_{g'} = g'_{small}$, and Lemma 82.5.3. This map is an isomorphism by the proper base change theorem (Lemma 82.4.7) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion.

Proof of (3). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Cohomology of Spaces, Lemma 67.4.1 for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma 82.5.3 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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