The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 54.82.2. 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. 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. 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. 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 $D((\mathit{Sch}/S')_{\acute{e}tale})$.

  4. 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 54.82.1 and (54.82.1.1) on choosing a K-injective complex of abelian sheaves representing $K$.

Part (3) follows from Lemma 54.82.1 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 $f' : T' \to S'$. $\square$


Comments (2)

Comment #3219 by David Hansen on

In parts (5) and (6), the functors on the right-hand sides should be replaced by .


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