# The Stacks Project

## Tag 075A

Lemma 53.81.2. Let $f : T \to S$ be a morphism of schemes.

1. 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 $D(S_{\acute{e}tale})$.
2. 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

1. (3)    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 $D(S'_{\acute{e}tale})$.
2. (4)    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 $D(\textit{Mod}(S'_{\acute{e}tale}, \mathcal{O}_{S'}))$.
3. (5)    For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale})$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^{-1}K)$ in $D((\mathit{Sch}/S')_{\acute{e}tale})$.
4. (6)    For $K$ in $D((\mathit{Sch}/T)_{\acute{e}tale}, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^*K)$ in $D(\textit{Mod}(S'_{\acute{e}tale}, \mathcal{O}_{S'}))$.

Proof. Part (1) follows from Lemma 53.81.1 and (53.81.1.1) on choosing a K-injective complex of abelian sheaves representing $K$.

Part (3) follows from Lemma 53.81.1 and Topologies, Lemma 33.4.18 on choosing a K-injective complex of abelian sheaves representing $K$.

Part (5) follows from Cohomology on Sites, Lemmas 21.8.1 and 21.21.1 and Topologies, Lemma 33.4.18 on choosing a K-injective complex of abelian sheaves representing $K$.

Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$ are the restriction functors. Hence (6) follows from Cohomology on Sites, Lemmas 21.8.1 and 21.21.1 and Topologies, Lemma 33.4.18 on choosing a K-injective complex of modules representing $K$.

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 21.20.2. Since the restriction functors $\pi_{S, *}$ and $\pi_{T, *}$ are exact, we conclude.

Part (4) follows from part (6) and part (2) applied to $f' : T' \to S'$. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 14154–14192 (see updates for more information).

\begin{lemma}
\label{lemma-compare-higher-direct-image}
Let $f : T \to S$ be a morphism of schemes.
\begin{enumerate}
\item For $K$ in $D((\Sch/T)_\etale)$ we have
$(Rf_{big, *}K)|_{S_\etale} = Rf_{small, *}(K|_{T_\etale})$
in $D(S_\etale)$.
\item For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have
$(Rf_{big, *}K)|_{S_\etale} = Rf_{small, *}(K|_{T_\etale})$
in $D(\textit{Mod}(S_\etale, \mathcal{O}_S))$.
\end{enumerate}
More generally, let $g : S' \to S$ be an object of $(\Sch/S)_\etale$.
Consider the fibre product
$$\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S }$$
Then
\begin{enumerate}
\item[(3)] For $K$ in $D((\Sch/T)_\etale)$ we have
$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$
in $D(S'_\etale)$.
\item[(4)] For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have
$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$
in $D(\textit{Mod}(S'_\etale, \mathcal{O}_{S'}))$.
\item[(5)] For $K$ in $D((\Sch/T)_\etale)$ we have
$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^{-1}K)$
in $D((\Sch/S')_\etale)$.
\item[(6)] For $K$ in $D((\Sch/T)_\etale, \mathcal{O})$ we have
$g_{big}^*(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^*K)$
in $D(\textit{Mod}(S'_\etale, \mathcal{O}_{S'}))$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from
Lemma \ref{lemma-compare-injectives}
and (\ref{equation-compare-big-small})
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (3) follows from Lemma \ref{lemma-compare-injectives}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (5) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.

\medskip\noindent
Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations
and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$
are the restriction functors. Hence (6) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of modules representing $K$.

\medskip\noindent
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
\ref{sites-cohomology-lemma-derived-pushforward-composition}.
Since the restriction functors $\pi_{S, *}$ and $\pi_{T, *}$
are exact, we conclude.

\medskip\noindent
Part (4) follows from part (6) and part (2) applied to $f' : T' \to S'$.
\end{proof}

Comment #3230 by David Hansen on March 8, 2018 a 5:10 pm UTC

In parts (5) and (6), the functors $Rf'_{small,\ast}$ on the right-hand sides should be replaced by $Rf'_{big,\ast}$.

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