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61.17 Comparing big and small topoi

This section is the analogue of √Čtale Cohomology, Section 59.99. In the following we will often denote $\mathcal{F} \mapsto \mathcal{F}|_{S_{pro\text{-}\acute{e}tale}}$ the pullback functor $i_ S^{-1}$ corresponding to the morphism of topoi $i_ S : \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ of Lemma 61.12.13.

Lemma 61.17.1. Let $S$ be a scheme. Let $T$ be an object of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$.

  1. If $\mathcal{I}$ is injective in $\textit{Ab}((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$, then

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

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

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

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

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

Proof. Proof of (1)(a) and (2)(a): $i_ f^{-1}$ is a right adjoint of an exact functor $i_{f, !}$. Namely, recall that $i_ f$ corresponds to a cocontinuous functor $u : T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ which is continuous and commutes with fibre products and equalizers, see Lemma 61.12.12 and its proof. Hence we obtain $i_{f, !}$ by Modules on Sites, Lemma 18.16.2. It is shown in Modules on Sites, Lemma 18.16.3 that it is exact. Then we conclude (1)(a) and (2)(a) hold by Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9.

Parts (1)(b) and (2)(b) are special cases of (1)(a) and (2)(a) as $i_ S = i_{\text{id}_ S}$. $\square$

Lemma 61.17.2. Let $f : T \to S$ be a morphism of schemes. For $K$ in $D((\mathit{Sch}/T)_{pro\text{-}\acute{e}tale})$ we have

\[ (Rf_{big, *}K)|_{S_{pro\text{-}\acute{e}tale}} = Rf_{small, *}(K|_{T_{pro\text{-}\acute{e}tale}}) \]

in $D(S_{pro\text{-}\acute{e}tale})$. More generally, let $S' \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ with structure morphism $g : S' \to S$. 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 $D((\mathit{Sch}/T)_{pro\text{-}\acute{e}tale})$ we have

\[ i_ g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K) \]

in $D(S'_{pro\text{-}\acute{e}tale})$ and

\[ g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K) \]

in $D((\mathit{Sch}/S')_{pro\text{-}\acute{e}tale})$.

Proof. The first equality follows from Lemma 61.17.1 and (61.12.16.1) on choosing a K-injective complex of abelian sheaves representing $K$. The second equality follows from Lemma 61.17.1 and Lemma 61.12.18 on choosing a K-injective complex of abelian sheaves representing $K$. The third equality follows similarly from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 and Lemma 61.12.18 on choosing a K-injective complex of abelian sheaves representing $K$. $\square$

Let $S$ be a scheme and let $\mathcal{H}$ be an abelian sheaf on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. Recall that $H^ n_{pro\text{-}\acute{e}tale}(U, \mathcal{H})$ denotes the cohomology of $\mathcal{H}$ over an object $U$ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$.

Lemma 61.17.3. Let $f : T \to S$ be a morphism of schemes. For $K$ in $D(S_{pro\text{-}\acute{e}tale})$ we have

\[ H^ n_{pro\text{-}\acute{e}tale}(S, \pi _ S^{-1}K) = H^ n(S_{pro\text{-}\acute{e}tale}, K) \]

and

\[ H^ n_{pro\text{-}\acute{e}tale}(T, \pi _ S^{-1}K) = H^ n(T_{pro\text{-}\acute{e}tale}, f_{small}^{-1}K). \]

For $M$ in $D((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ we have

\[ H^ n_{pro\text{-}\acute{e}tale}(T, M) = H^ n(T_{pro\text{-}\acute{e}tale}, i_ f^{-1}M). \]

Proof. To prove the last equality represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 61.17.1 and work out the definitions. The second equality follows from this as $i_ f^{-1} \circ \pi _ S^{-1} = f_{small}^{-1}$. The first equality is a special case of the second one. $\square$

Lemma 61.17.4. Let $S$ be a scheme. For $K \in D(S_{pro\text{-}\acute{e}tale})$ 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. $\square$


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