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 54.89.8. Let $K$ be a field. Let $n \geq 1$ be invertible in $K$. Consider a commutative diagram

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ p \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e \\ \mathop{\mathrm{Spec}}(K) & S' \ar[l] & T \ar[l]_ g } \]

of schemes of finite type over $K$ with $X' = X \times _{\mathop{\mathrm{Spec}}(K)} S'$ and $Y = X' \times _{S'} T$. The canonical map

\[ p^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} (f')^{-1}Rg_*F \longrightarrow Rh_*(h^{-1}p^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} e^{-1}F) \]

is an isomorphism for $E$ in $D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$ and $F$ in $D(T_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$.

Proof. We will reduce this to Lemma 54.89.7 using that our functors commute with direct sums. We suggest the reader skip the proof. Recall that derived tensor product commutes with direct sums. Recall that (derived) pullback commutes with direct sums. Recall that $Rh_*$ and $Rg_*$ commute with direct sums, see Lemmas 54.88.2 and 54.88.3 (this is where we use our schemes are of finite type over $K$).

To finish the proof we can argue as follows. First we write $E = \text{hocolim} \tau _{\leq N} E$. Since our functors commute with direct sums, they commute with homotopy colimits. Hence it suffices to prove the lemma for $E$ bounded above. Similarly for $F$ we may assume $F$ is bounded above. Then we can represent $E$ by a bounded above complex $\mathcal{E}^\bullet $ of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Then

\[ \mathcal{E}^\bullet = \mathop{\mathrm{colim}}\nolimits \sigma _{\geq -N}\mathcal{E}^\bullet \]

(stupid truncations). Thus we may assume $\mathcal{E}^\bullet $ is a bounded complex of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. For $F$ we choose a bounded above complex of flat(!) sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Then we reduce to the case where $F$ is represented by a bounded complex of flat sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. At this point Lemma 54.89.7 kicks in and we conclude. $\square$


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