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Fix use of wrong lemma + attempt to help XyJax
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aisejohan committed Oct 22, 2018
1 parent 5063650 commit 878b8ac
Showing 1 changed file with 3 additions and 3 deletions.
6 changes: 3 additions & 3 deletions etale-cohomology.tex
Expand Up @@ -15177,13 +15177,13 @@ \section{Smooth base change}
f^{-1}R^qg_*\mathcal{F} \ar[d] \ar[r] &
f^{-1}R^qa_*\mathcal{F}|_W \ar[d] \oplus
f^{-1}R^qb_*\mathcal{F}|_{W'} \ar[d]_{\cong} \ar[r] &
\ldots \\
... \\
\ldots \ar[r] &
R^qk_*e^{-1}\mathcal{F}|_{e^{-1}(W \cap W')} \ar[r] &
R^qh_*e^{-1}\mathcal{F} \ar[r] &
R^qi_*e^{-1}\mathcal{F}|_{e^{-1}(W)} \oplus
R^qj_*e^{-1}\mathcal{F}|_{e^{-1}(W')} \ar[r] &
\ldots
...
}
$$
whose rows are exact sequences by Lemma \ref{lemma-relative-mayer-vietoris}.
Expand Down Expand Up @@ -15243,7 +15243,7 @@ \section{Smooth base change}
We have $\mathcal{I} = \colim (T \to T_i)^{-1}\mathcal{I}_i$
by Lemma \ref{lemma-linus-hamann}. Pulling back by $e$ we get
$e^{-1}\mathcal{I} = \colim (Y \to Y_i)^{-1}e_i^{-1}\mathcal{I}_i$.
By Lemma \ref{lemma-relative-colimit} applied to the system
By Lemma \ref{lemma-relative-colimit-general} applied to the system
of morphisms $Y_i \to X_i$ with limit $Y \to X$ we have
$$
R^qh_*e^{-1}\mathcal{I} =
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