# stacks/stacks-project

Add a case to technical lemma comparing big/small

Thanks to David Hansen
 @@ -18899,7 +18899,9 @@ \section{Comparing big and small topoi} \item $\pi_S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_T^{-1}$ as functors $\Sh(T_\etale) \to \Sh((\Sch/S)_\etale)$, \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for $K$ in $D^+(T_\etale)$ whose cohomology sheaves are torsion, and for $K$ in $D^+(T_\etale)$ whose cohomology sheaves are torsion, \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for $K$ in $D(T_\etale, \mathbf{Z}/n\mathbf{Z})$, and \item $\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\pi_T^{-1}K$ for all $K$ in $D(T_\etale)$ if $f$ is finite. \end{enumerate} @@ -18967,7 +18969,12 @@ \section{Comparing big and small topoi} below and the cohomology sheaves of $K$ are torsion. \medskip\noindent Proof of (3). If $f$ is finite, then the functors The proof of part (3) is the same as the proof of part (2), except we use Lemma \ref{lemma-proper-base-change-mod-n} instead of Lemma \ref{lemma-proper-base-change}. \medskip\noindent Proof of (4). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Proposition \ref{proposition-finite-higher-direct-image-zero} for $f_{small}$. Since any base change $f'$ of $f$ is finite too, @@ -19278,7 +19285,7 @@ \section{Comparing fppf and \'etale topologies} \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-A-and-P}) holds by Lemma \ref{lemma-compare-higher-direct-image-proper} part (3). Lemma \ref{lemma-compare-higher-direct-image-proper} part (4). \medskip\noindent Cohomology on Sites, Property (\ref{sites-cohomology-item-refine-tau-by-P}) @@ -19356,7 +19363,7 @@ \section{Comparing fppf and \'etale topologies} Lemma \ref{lemma-push-pull-fppf-etale} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Then second equality is (3). The third is The second equality is (3). The third is Lemma \ref{lemma-compare-higher-direct-image-proper} part (2). The fourth is (3) again. Thus the base change map $a_Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_X^{-1}K)$