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Add a case to technical lemma comparing big/small

Thanks to David Hansen
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aisejohan committed Oct 7, 2019
1 parent a30d3b6 commit 5806a044afc7b696fe43dbc9be2ef407ab9acdcb
Showing with 11 additions and 4 deletions.
  1. +11 −4 etale-cohomology.tex
@@ -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
$a_Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_X^{-1}K)$

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