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Typos in sites-cohomology
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aisejohan committed Nov 15, 2020
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Expand Up @@ -222,6 +222,7 @@ Max Lieblich
Bronson Lim
David Benjamin Lim
Joseph Lipman
Zongzhu Lin
Daniel Litt
Huaxin Liu
Hsing Liu
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6 changes: 3 additions & 3 deletions sites-cohomology.tex
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Expand Up @@ -8514,7 +8514,7 @@ \section{Derived lower shriek}

\noindent
In this section we study morphisms $g$ of ringed topoi where besides
$Lg^*$ and $Rg_*$ there also a derived functor $Lg_!$.
$Lg^*$ and $Rg_*$ there also exists a derived functor $Lg_!$.

\begin{lemma}
\label{lemma-pullback-injective-pre-limp}
Expand Down Expand Up @@ -8544,7 +8544,7 @@ \section{Derived lower shriek}
$$
because $g^{-1} = u^p$ by Sites, Lemma \ref{sites-lemma-when-shriek}.
Since $\mathcal{I}$ is an injective
$\mathcal{O}$-module these {\v C}ech cohomology groups vanish, see
$\mathcal{O}_\mathcal{D}$-module these {\v C}ech cohomology groups vanish, see
Lemma \ref{lemma-injective-module-trivial-cech}.
\end{proof}

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H^0(g_!^{Sh}K' \times_{g_!^{Sh}K} \ldots \times_{g_!^{Sh}K} g_!^{Sh}K',
\mathcal{I})
\end{align*}
by our assumption on $g_!^{Sh}$. Since $\mathcal{I}$ is injective module
by our assumption on $g_!^{Sh}$. Since $\mathcal{I}$ is an injective module
it is totally acyclic by Lemma \ref{lemma-direct-image-injective-sheaf}
(applied to the identity). Hence we can use the converse of
Lemma \ref{lemma-characterize-limp} to see that the complex
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